id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-22500 | Volume of Circular Cylinder/Slant Height | Let $\CC$ be a circular cylinder such that:
:the bases of $\CC$ are circles of radius $r$
:the slant height of $\CC$ is $l$
:the inclination of the generatrices of $\CC$ to the base of $\CC$ is $\theta$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = \pi r^2 l \sin \theta$ | Let $h$ denote the height of $\CC$.
From Relation between Slant Height and Height of Cylinder:
:$h = l \sin \theta$
From Volume of Circular Cylinder in terms of Height:
:$\VV = \pi r^2 h$
The result follows.
{{qed}} | Let $\CC$ be a [[Definition:Circular Cylinder|circular cylinder]] such that:
:the [[Definition:Base of Cylinder|bases]] of $\CC$ are [[Definition:Circle|circles]] of [[Definition:Radius of Circle|radius]] $r$
:the [[Definition:Slant Height of Cylinder|slant height]] of $\CC$ is $l$
:the [[Definition:Inclination of Stra... | Let $h$ denote the [[Definition:Height of Cylinder|height]] of $\CC$.
From [[Relation between Slant Height and Height of Cylinder]]:
:$h = l \sin \theta$
From [[Volume of Circular Cylinder in terms of Height]]:
:$\VV = \pi r^2 h$
The result follows.
{{qed}} | Volume of Circular Cylinder/Slant Height | https://proofwiki.org/wiki/Volume_of_Circular_Cylinder/Slant_Height | https://proofwiki.org/wiki/Volume_of_Circular_Cylinder/Slant_Height | [
"Volume of Circular Cylinder"
] | [
"Definition:Circular Solid Figure/Cylinder",
"Definition:Cylinder/Base",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Cylinder/Slant Height",
"Definition:Inclination/Straight Line to Plane",
"Definition:Cylindrical Surface/Generatrix",
"Definition:Cylinder/Base",
"Definition:Volume"
... | [
"Definition:Cylinder/Height",
"Relation between Slant Height and Height of Cylinder",
"Volume of Circular Cylinder/Height"
] |
proofwiki-22501 | Volume of Frustum of Right Circular Cone | Let $F$ be a frustum of a right circular cone.
The volume $\VV$ of $F$ is given as:
:$\VV = \dfrac {\pi h \paren {a^2 + a b + b^2} } 3$
where:
:$a$ and $b$ are the radii of the bases of $F$
:$h$ is the altitude of $F$. | From Volume of Frustum of Cone or Pyramid:
:$\VV = \dfrac {h \paren {A_1 + A_2 + \sqrt {A_1 A_2} } } 3$
where:
:$A_1$ and $A_2$ are the areas of the bases of $F$
:$h$ is the altitude of $F$.
Here we have that $F$ be a frustum of a right circular cone.
Hence the bases of $F$ are circles.
From Area of Circle, the areas o... | Let $F$ be a [[Definition:Frustum|frustum]] of a [[Definition:Right Circular Cone|right circular cone]].
The [[Definition:Volume|volume]] $\VV$ of $F$ is given as:
:$\VV = \dfrac {\pi h \paren {a^2 + a b + b^2} } 3$
where:
:$a$ and $b$ are the [[Definition:Radius of Circle|radii]] of the [[Definition:Base of Frustum|b... | From [[Volume of Frustum of Cone or Pyramid]]:
:$\VV = \dfrac {h \paren {A_1 + A_2 + \sqrt {A_1 A_2} } } 3$
where:
:$A_1$ and $A_2$ are the [[Definition:Area|areas]] of the [[Definition:Base of Frustum|bases]] of $F$
:$h$ is the [[Definition:Altitude of Frustum|altitude]] of $F$.
Here we have that $F$ be a [[Definiti... | Volume of Frustum of Right Circular Cone | https://proofwiki.org/wiki/Volume_of_Frustum_of_Right_Circular_Cone | https://proofwiki.org/wiki/Volume_of_Frustum_of_Right_Circular_Cone | [
"Frusta",
"Volume Formulas"
] | [
"Definition:Frustum",
"Definition:Right Circular Cone",
"Definition:Volume",
"Definition:Circle/Radius",
"Definition:Frustum/Base",
"Definition:Frustum/Altitude"
] | [
"Volume of Frustum of Cone or Pyramid",
"Definition:Area",
"Definition:Frustum/Base",
"Definition:Frustum/Altitude",
"Definition:Frustum",
"Definition:Right Circular Cone",
"Definition:Frustum/Base",
"Definition:Circle",
"Area of Circle",
"Definition:Area",
"Definition:Frustum/Base",
"Volume o... |
proofwiki-22502 | Power Reduction Formulas/Cosine to 6th | :$\cos^6 x = \dfrac {10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x} {32}$ | {{begin-eqn}}
{{eqn | l = \cos 6 x
| r = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1
| c = Sextuple Angle Formula for Cosine
}}
{{eqn | ll= \leadsto
| l = 32 \cos^6 x
| r = \cos 6 x + 48 \cos^4 x - 18 \cos^2 x + 1
| c = rearranging
}}
{{eqn | r = \cos 6 x + 48 \paren {\dfrac {3 + 4 \cos 2 ... | :$\cos^6 x = \dfrac {10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x} {32}$ | {{begin-eqn}}
{{eqn | l = \cos 6 x
| r = 32 \cos^6 x - 48 \cos^4 x + 18 \cos^2 x - 1
| c = [[Sextuple Angle Formula for Cosine]]
}}
{{eqn | ll= \leadsto
| l = 32 \cos^6 x
| r = \cos 6 x + 48 \cos^4 x - 18 \cos^2 x + 1
| c = rearranging
}}
{{eqn | r = \cos 6 x + 48 \paren {\dfrac {3 + 4 \co... | Power Reduction Formulas/Cosine to 6th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_6th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_6th | [
"Cosine Function"
] | [] | [
"Sextuple Angle Formulas/Cosine",
"Power Reduction Formulas/Cosine to 4th",
"Power Reduction Formulas/Cosine Squared",
"Category:Cosine Function"
] |
proofwiki-22503 | Septuple Angle Formulas/Cosine | :$\cos 7 \theta = 64 \cos^7 \theta - 112 \cos^5 \theta + 56 \cos^3 \theta - 7 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 7 \theta + i \sin 7 \theta
| r = \paren {\cos \theta + i \sin \theta}^7
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cos \theta}^7 + \binom 7 1 \paren {\cos \theta}^6 \paren {i \sin \theta} + \binom 7 2 \paren {\cos \theta}^5 \paren {i \sin \theta}^2 + \binom 7 3 \paren {... | :$\cos 7 \theta = 64 \cos^7 \theta - 112 \cos^5 \theta + 56 \cos^3 \theta - 7 \cos \theta$ | {{begin-eqn}}
{{eqn | l = \cos 7 \theta + i \sin 7 \theta
| r = \paren {\cos \theta + i \sin \theta}^7
| c = [[De Moivre's Formula]]
}}
{{eqn | r = \paren {\cos \theta}^7 + \binom 7 1 \paren {\cos \theta}^6 \paren {i \sin \theta} + \binom 7 2 \paren {\cos \theta}^5 \paren {i \sin \theta}^2 + \binom 7 3 \par... | Septuple Angle Formulas/Cosine | https://proofwiki.org/wiki/Septuple_Angle_Formulas/Cosine | https://proofwiki.org/wiki/Septuple_Angle_Formulas/Cosine | [
"Cosine Function",
"Septuple Angle Formulas",
"Septuple Angle Formula for Cosine"
] | [] | [
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Binomial Coefficient",
"Definition:Complex Number/Real Part",
"Sum of Squares of Sine and Cosine",
"Category:Cosine Function",
"Category:Septuple Angle Formulas",
"Category:Septuple Angle Formula for Cosine"
] |
proofwiki-22504 | Power Reduction Formulas/Cosine to 7th | :$\cos^7 x = \dfrac {35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x} {64}$ | {{begin-eqn}}
{{eqn | l = \cos 7 x
| r = 64 \cos^7 x - 112 \cos^5 x + 56 \cos^3 x - 7 \cos x
| c = Septuple Angle Formula for Cosine
}}
{{eqn | ll= \leadsto
| l = 64 \cos^7 x
| r = \cos 7 x + 112 \cos^5 x - 56 \cos^3 x + 7 \cos x
| c = rearranging
}}
{{eqn | r = \cos 7 x + 112 \paren {\dfr... | :$\cos^7 x = \dfrac {35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x} {64}$ | {{begin-eqn}}
{{eqn | l = \cos 7 x
| r = 64 \cos^7 x - 112 \cos^5 x + 56 \cos^3 x - 7 \cos x
| c = [[Septuple Angle Formula for Cosine]]
}}
{{eqn | ll= \leadsto
| l = 64 \cos^7 x
| r = \cos 7 x + 112 \cos^5 x - 56 \cos^3 x + 7 \cos x
| c = rearranging
}}
{{eqn | r = \cos 7 x + 112 \paren {... | Power Reduction Formulas/Cosine to 7th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_7th | https://proofwiki.org/wiki/Power_Reduction_Formulas/Cosine_to_7th | [
"Cosine Function"
] | [] | [
"Septuple Angle Formulas/Cosine",
"Power Reduction Formulas/Cosine to 5th",
"Power Reduction Formulas/Cosine Cubed",
"Category:Cosine Function"
] |
proofwiki-22505 | Extension of Continuous Mapping is Continuous | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T_H = \struct{H, \tau_H}$ be a topological subspace of $T_2$ where $H \subseteq S_2$.
Let $f: S_1 \to H$ be a $\tuple { \tau_1 , \tau_H}$-continuous mapping.
Define a mapping $g: S_1 \to S_2$ by:
:$\forall x \in S_1 : \map g... | Let $i_H$ be the inclusion mapping of $H$ in $S_2$.
Then:
:$g = i_H \circ f$
From Continuity of Composite with Inclusion: Inclusion on Mapping, it follows that $f$ is $\tuple { \tau_1 , \tau_H}$-continuous mapping {{iff}} $i_H \circ f = g$ is $\tuple { \tau_1 , \tau_2}$-continuous.
Since $f$ is continuous by assumption... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T_H = \struct{H, \tau_H}$ be a [[Definition:Subspace Topology|topological subspace]] of $T_2$ where $H \subseteq S_2$.
Let $f: S_1 \to H$ be a [[Definition:Continuous Mapping (Topology)|$\t... | Let $i_H$ be the [[Definition:Inclusion Mapping|inclusion mapping]] of $H$ in $S_2$.
Then:
:$g = i_H \circ f$
From [[Continuity of Composite with Inclusion/Inclusion on Mapping|Continuity of Composite with Inclusion: Inclusion on Mapping]], it follows that $f$ is [[Definition:Continuous Mapping (Topology)|$\tuple { \... | Extension of Continuous Mapping is Continuous | https://proofwiki.org/wiki/Extension_of_Continuous_Mapping_is_Continuous | https://proofwiki.org/wiki/Extension_of_Continuous_Mapping_is_Continuous | [] | [
"Definition:Topological Space",
"Definition:Topological Subspace",
"Definition:Continuous Mapping (Topology)",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)"
] | [
"Definition:Inclusion Mapping",
"Continuity of Composite with Inclusion/Inclusion on Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)"
] |
proofwiki-22506 | Open Continuous Injection is Embedding | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be an open and continuous injection.
Then $f$ is an embedding of $T_1$ into $T_2$. | Let $g: S_1 \to f \sqbrk {S_1}$ be the restriction of $f$ to $S_1 \times f \sqbrk {S_1}$.
It must be shown that $g$ is a homeomorphism. | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be an [[Definition:Open Mapping|open]] and [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Injection|injection]].
Then $f$ is an [[Definition:Embedding (... | Let $g: S_1 \to f \sqbrk {S_1}$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $S_1 \times f \sqbrk {S_1}$.
It must be shown that $g$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]]. | Open Continuous Injection is Embedding | https://proofwiki.org/wiki/Open_Continuous_Injection_is_Embedding | https://proofwiki.org/wiki/Open_Continuous_Injection_is_Embedding | [
"Embeddings (Topology)"
] | [
"Definition:Topological Space",
"Definition:Open Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Injection",
"Definition:Embedding (Topology)"
] | [
"Definition:Restriction/Mapping",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-22507 | Closed Continuous Injection is Embedding | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a closed and continuous injection.
Then $f$ is an embedding of $T_1$ into $T_2$. | Let $g: S_1 \to f \sqbrk {S_1}$ be the restriction of $f$ to $S_1 \times f \sqbrk {S_1}$.
It must be shown that $g$ is a homeomorphism. | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Closed Mapping|closed]] and [[Definition:Continuous Mapping (Topology)|continuous]] [[Definition:Injection|injection]].
Then $f$ is an [[Definition:Embeddin... | Let $g: S_1 \to f \sqbrk {S_1}$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to $S_1 \times f \sqbrk {S_1}$.
It must be shown that $g$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]]. | Closed Continuous Injection is Embedding | https://proofwiki.org/wiki/Closed_Continuous_Injection_is_Embedding | https://proofwiki.org/wiki/Closed_Continuous_Injection_is_Embedding | [
"Embeddings (Topology)"
] | [
"Definition:Topological Space",
"Definition:Closed Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Injection",
"Definition:Embedding (Topology)"
] | [
"Definition:Restriction/Mapping",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-22508 | Surjective Embedding is Homeomorphism | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be topological spaces.
Let $f: S_1 \to S_2$ be a surjective embedding.
Then $f$ is a homeomorphism. | Let $f {\restriction_{S_1 \times f\sqbrk {S_1} }}$ be the restriction of $f$ to its image.
By definition of surjection, $f \sqbrk {S_1} = S_2$.
Therefore, $f {\restriction_{S_1 \times f\sqbrk {S_1} }} = f$
By definition of embedding, $f {\restriction_{S_1 \times f\sqbrk {S_1} }}$ of $f$ to its image is a homeomorphism.... | Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: S_1 \to S_2$ be a [[Definition:Surjection|surjective]] [[Definition:Embedding (Topology)|embedding]].
Then $f$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]]. | Let $f {\restriction_{S_1 \times f\sqbrk {S_1} }}$ be the [[Definition:Restriction of Mapping|restriction]] of $f$ to its [[Definition:Image of Mapping|image]].
By definition of [[Definition:Surjection|surjection]], $f \sqbrk {S_1} = S_2$.
Therefore, $f {\restriction_{S_1 \times f\sqbrk {S_1} }} = f$
By definition o... | Surjective Embedding is Homeomorphism | https://proofwiki.org/wiki/Surjective_Embedding_is_Homeomorphism | https://proofwiki.org/wiki/Surjective_Embedding_is_Homeomorphism | [
"Embeddings (Topology)",
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Topological Space",
"Definition:Surjection",
"Definition:Embedding (Topology)",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Restriction/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Surjection",
"Definition:Embedding (Topology)",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-22509 | Locally Euclidean Subspace of Euclidean Space is Manifold | Let $\R^n$ be an Euclidean space for $n \in \N$.
Let $M = \struct{H, \tau_H}$ be a subspace of $\R^n$, where $H \subseteq \R^n$.
Let $M$ be locally Euclidean of dimension $d$.
Then $M$ is a $d$-manifold. | From Metric Space is Hausdorff, $\R^n$ is a Hausdorff space.
From Subspace of Hausdorff Space is Hausdorff, $M$ is a Hausdorff space.
From Euclidean Space is Second-Countable, $\R^n$ is second-countable.
From Second-Countability is Hereditary, $M$ is second-countable.
Therefore, $M$ is a $d$-manifold.
{{qed}} | Let $\R^n$ be an [[Definition:Euclidean Space|Euclidean space]] for $n \in \N$.
Let $M = \struct{H, \tau_H}$ be a [[Definition:Topological Subspace|subspace]] of $\R^n$, where $H \subseteq \R^n$.
Let $M$ be [[Definition:Locally Euclidean Space|locally Euclidean]] of [[Definition:Dimension of Locally Euclidean Space|d... | From [[Metric Space is Hausdorff]], $\R^n$ is a [[Definition:Hausdorff Space|Hausdorff space]].
From [[Subspace of Hausdorff Space is Hausdorff]], $M$ is a [[Definition:Hausdorff Space|Hausdorff space]].
From [[Euclidean Space is Second-Countable]], $\R^n$ is [[Definition:Second-Countable Space|second-countable]].
F... | Locally Euclidean Subspace of Euclidean Space is Manifold | https://proofwiki.org/wiki/Locally_Euclidean_Subspace_of_Euclidean_Space_is_Manifold | https://proofwiki.org/wiki/Locally_Euclidean_Subspace_of_Euclidean_Space_is_Manifold | [
"Topological Manifolds",
"Locally Euclidean Spaces"
] | [
"Definition:Euclidean Space",
"Definition:Topological Subspace",
"Definition:Locally Euclidean Space",
"Definition:Dimension (Topology)/Locally Euclidean Space",
"Definition:Topological Manifold"
] | [
"Metric Space is T2",
"Definition:T2 Space",
"T2 Property is Hereditary",
"Definition:T2 Space",
"Euclidean Space is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countability is Hereditary",
"Definition:Second-Countable Space",
"Definition:Topological Manifold"
] |
proofwiki-22510 | Graph of Continuous Real Function is Manifold | Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.
Let $f : U \to \R^k$ be a continuous mapping.
Let $\map \Gamma f$ be the graph of $f$ equipped with the subspace topology.
Then $\map \Gamma f$ is a $n$-manifold. | Let $\gamma_f : U \to \R^{n + k}$ be the graph parametrization of $\map \Gamma f$.
From Graph Parametrization of Continuous Mapping is Embedding, $\gamma_f$ is an embedding.
The image of $\gamma_f$ is $\map \Gamma f$.
Therefore, $U$ and $\map \Gamma f$ are homeomorphic.
In other words, for each point in $\map \Gamma f$... | Let $U \subseteq \R^n$ be an [[Definition:Open Subset of Real Euclidean Space|open subset]] of $n$-[[Definition:Dimension of Vector Space|dimensional]] [[Definition:Open Subset of Real Euclidean Space|Euclidean space]].
Let $f : U \to \R^k$ be a [[Definition:Continuous Mapping (Topology)|continuous mapping]].
Let $\... | Let $\gamma_f : U \to \R^{n + k}$ be the [[Definition:Graph Parametrization|graph parametrization]] of $\map \Gamma f$.
From [[Graph Parametrization of Continuous Mapping is Embedding]], $\gamma_f$ is an [[Definition:Topological Embedding|embedding]].
The [[Definition:Image of Mapping|image]] of $\gamma_f$ is $\map \... | Graph of Continuous Real Function is Manifold | https://proofwiki.org/wiki/Graph_of_Continuous_Real_Function_is_Manifold | https://proofwiki.org/wiki/Graph_of_Continuous_Real_Function_is_Manifold | [] | [
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Dimension of Vector Space",
"Definition:Open Set/Real Analysis/Real Euclidean Space",
"Definition:Continuous Mapping (Topology)",
"Definition:Graph of Real Function",
"Definition:Topological Subspace",
"Definition:Topological Manifold... | [
"Definition:Graph Parametrization",
"Graph Parametrization of Continuous Mapping is Embedding",
"Definition:Embedding (Topology)",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Open Neighborhood/Point",
"Definition:Homeomorphism/Topological S... |
proofwiki-22511 | Particular Values of Legendre Polynomials/1 | :$\map {P_n} 1 = 1$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {P_n} 1 = 1$ | :$\map {P_n} 1 = 1$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {P_n} 1 = 1$ | Particular Values of Legendre Polynomials/1 | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/1 | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/1 | [
"Particular Values of Legendre Polynomials"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22512 | Particular Values of Legendre Polynomials/-1 | :$\map {P_n} {-1} = \paren {-1}^n$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {P_n} {-1} = \paren {-1}^n$ | :$\map {P_n} {-1} = \paren {-1}^n$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {P_n} {-1} = \paren {-1}^n$ | Particular Values of Legendre Polynomials/-1 | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/-1 | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/-1 | [
"Particular Values of Legendre Polynomials",
"Proofs by Induction"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22513 | Particular Values of Legendre Polynomials/-x | :$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {P_n} x
| r = \paren {-1}^n \map {P_n} {-x}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\p... | :$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {P_n} x
| r = \paren {-1}^n \map {P_n} {-x}
... | Particular Values of Legendre Polynomials/-x | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/-x | https://proofwiki.org/wiki/Particular_Values_of_Legendre_Polynomials/-x | [
"Particular Values of Legendre Polynomials",
"Proofs by Induction"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Integer Division",
"Definition:Integer Division"
] |
proofwiki-22514 | Legendre Polynomial of Even Index is Even Function | Let $n \in \N$ be an even natural number: $n \in \set {0, 2, 4, \ldots}$
Let $\map {P_n} x$ denote the Legendre polynomial of order $n$.
Then $\map {P_n} x$ is an even function. | From Legendre Polynomial of $-x$:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
When $n$ is even we have:
:$\paren {-1}^n = 1$
and so:
:$\map {P_n} x = \map {P_n} {-x}$
Hence the result by definition of even function.
{{qed}} | Let $n \in \N$ be an [[Definition:Even Integer|even]] [[Definition:Natural Number|natural number]]: $n \in \set {0, 2, 4, \ldots}$
Let $\map {P_n} x$ denote the [[Definition:Legendre Polynomial|Legendre polynomial of order $n$]].
Then $\map {P_n} x$ is an [[Definition:Even Function|even function]]. | From [[Particular Values of Legendre Polynomials/-x|Legendre Polynomial of $-x$]]:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
When $n$ is [[Definition:Even Integer|even]] we have:
:$\paren {-1}^n = 1$
and so:
:$\map {P_n} x = \map {P_n} {-x}$
Hence the result by definition of [[Definition:Even Function|even fu... | Legendre Polynomial of Even Index is Even Function | https://proofwiki.org/wiki/Legendre_Polynomial_of_Even_Index_is_Even_Function | https://proofwiki.org/wiki/Legendre_Polynomial_of_Even_Index_is_Even_Function | [
"Legendre Polynomials",
"Examples of Even Functions"
] | [
"Definition:Even Integer",
"Definition:Natural Numbers",
"Definition:Legendre Polynomial",
"Definition:Even Function"
] | [
"Particular Values of Legendre Polynomials/-x",
"Definition:Even Integer",
"Definition:Even Function"
] |
proofwiki-22515 | Legendre Polynomial of Odd Index is Odd Function | Let $n \in \N$ be an odd natural number: $n \in \set {1, 3, 5, \ldots}$
Let $\map {P_n} x$ denote the Legendre polynomial of order $n$.
Then $\map {P_n} x$ is an odd function. | From Legendre Polynomial of $-x$:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
When $n$ is odd we have:
:$\paren {-1}^n = -1$
and so:
:$\map {P_n} x = -\map {P_n} {-x}$
Hence the result by definition of odd function.
{{qed}} | Let $n \in \N$ be an [[Definition:Odd Integer|odd]] [[Definition:Natural Number|natural number]]: $n \in \set {1, 3, 5, \ldots}$
Let $\map {P_n} x$ denote the [[Definition:Legendre Polynomial|Legendre polynomial of order $n$]].
Then $\map {P_n} x$ is an [[Definition:Odd Function|odd function]]. | From [[Particular Values of Legendre Polynomials/-x|Legendre Polynomial of $-x$]]:
:$\map {P_n} x = \paren {-1}^n \map {P_n} {-x}$
When $n$ is [[Definition:Odd Integer|odd]] we have:
:$\paren {-1}^n = -1$
and so:
:$\map {P_n} x = -\map {P_n} {-x}$
Hence the result by definition of [[Definition:Odd Function|odd func... | Legendre Polynomial of Odd Index is Odd Function | https://proofwiki.org/wiki/Legendre_Polynomial_of_Odd_Index_is_Odd_Function | https://proofwiki.org/wiki/Legendre_Polynomial_of_Odd_Index_is_Odd_Function | [
"Legendre Polynomials",
"Examples of Odd Functions"
] | [
"Definition:Odd Integer",
"Definition:Natural Numbers",
"Definition:Legendre Polynomial",
"Definition:Odd Function"
] | [
"Particular Values of Legendre Polynomials/-x",
"Definition:Odd Integer",
"Definition:Odd Function"
] |
proofwiki-22516 | Generating Function for Associated Legendre Function of the First Kind | Let $\map { {P_n}^m} x$ denote an '''associated Legendre function of the first kind'''.
Then the generating function for ${P_n}^m$ is:
:$\ds \frac {\paren {2 m}! \paren {1 - x^2}^{m / 2} t^m} {2^m m! \paren {1 - 2 t x + t^2}^{m + 1/2} } = \sum_{n \mathop = m}^\infty \map { {P_n}^m} x t^n$ | {{ProofWanted|Proof at https://www.phys.ksu.edu/personal/wysin/notes/legendre.pdf}} | Let $\map { {P_n}^m} x$ denote an '''[[Definition:Associated Legendre Function of the First Kind|associated Legendre function of the first kind]]'''.
Then the [[Definition:Generating Function|generating function]] for ${P_n}^m$ is:
:$\ds \frac {\paren {2 m}! \paren {1 - x^2}^{m / 2} t^m} {2^m m! \paren {1 - 2 t x + t^... | {{ProofWanted|Proof at https://www.phys.ksu.edu/personal/wysin/notes/legendre.pdf}} | Generating Function for Associated Legendre Function of the First Kind | https://proofwiki.org/wiki/Generating_Function_for_Associated_Legendre_Function_of_the_First_Kind | https://proofwiki.org/wiki/Generating_Function_for_Associated_Legendre_Function_of_the_First_Kind | [
"Associated Legendre Functions",
"Examples of Generating Functions"
] | [
"Definition:Associated Legendre Function of the First Kind",
"Definition:Generating Function"
] | [] |
proofwiki-22517 | Stirling's Formula/Refinement | A refinement of Stirling's Formula is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | Let:
:$\ds \map f n := \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } - \paren {1 + \frac 1 {12 n} }$
We need to show:
:$\ds \map f n = \map \OO {\dfrac 1 {n^2} }$
Recall Limit of Error in Stirling's Formula:
:$e^{1 / \paren {12 n + 1} } \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$
Furthermore, observe:
{... | A refinement of [[Stirling's Formula]] is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | Let:
:$\ds \map f n := \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } - \paren {1 + \frac 1 {12 n} }$
We need to show:
:$\ds \map f n = \map \OO {\dfrac 1 {n^2} }$
Recall [[Limit of Error in Stirling's Formula]]:
:$e^{1 / \paren {12 n + 1} } \le \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } \le e^{1 / 12 n}$
Furthermore, ob... | Stirling's Formula/Refinement/Proof 1 | https://proofwiki.org/wiki/Stirling's_Formula/Refinement | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_1 | [
"Stirling's Formula"
] | [
"Stirling's Formula"
] | [
"Limit of Error in Stirling's Formula",
"Exponential of x not less than 1+x",
"Taylor's Theorem",
"Exponential is Strictly Increasing",
"Definition:Euler's Number/Decimal Expansion"
] |
proofwiki-22518 | Stirling's Formula/Refinement | A refinement of Stirling's Formula is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | Let $z\in \R_{>0}$ and $n \in \N_{\ge 0}$
Let $\ds c_n = \ln \map \Gamma {z + n}$
We begin by observing:
{{begin-eqn}}
{{eqn | l = \map \Gamma {z + n}
| r = \map \Gamma {z + 1} \times \paren {z + 1} \times \paren {z + 2} \times \cdots \times \paren {z + n - 1}
| c = Gamma Difference Equation
}}
{{eqn | ll =... | A refinement of [[Stirling's Formula]] is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | Let $z\in \R_{>0}$ and $n \in \N_{\ge 0}$
Let $\ds c_n = \ln \map \Gamma {z + n}$
We begin by observing:
{{begin-eqn}}
{{eqn | l = \map \Gamma {z + n}
| r = \map \Gamma {z + 1} \times \paren {z + 1} \times \paren {z + 2} \times \cdots \times \paren {z + n - 1}
| c = [[Gamma Difference Equation]]
}}
{{eqn ... | Stirling's Formula/Refinement/Proof 2 | https://proofwiki.org/wiki/Stirling's_Formula/Refinement | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_2 | [
"Stirling's Formula"
] | [
"Stirling's Formula"
] | [
"Gamma Difference Equation",
"Sum of Logarithms",
"Definition:Derivative",
"Definition:Finite Difference Operator",
"Primitive of Logarithm of x",
"Definition:Derivative",
"Definition:Finite Difference Operator",
"Primitive of Logarithm of x",
"Sum of Logarithms",
"Power Series Expansion for Logar... |
proofwiki-22519 | Stirling's Formula/Refinement | A refinement of Stirling's Formula is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | From Limit of Error in Stirling's Formula, we have:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$
We also have:
{{begin-eqn}}
{{eqn | l = e^{1 / 12 n}
| r = 1 + \frac 1 {12 n} + \frac 1 {2!} \paren {\frac 1 {12 n} }^2 + \frac 1 {3!} \paren {\frac 1 {12 n} }^3 + \cdots
... | A refinement of [[Stirling's Formula]] is:
:$n! = \sqrt {2 \pi n} \paren {\dfrac n e}^n \paren {1 + \dfrac 1 {12 n} + \map \OO {\dfrac 1 {n^2} } }$ | From [[Limit of Error in Stirling's Formula]], we have:
:$e^{1 / \paren {12 n + 1} } < \dfrac {n!} {\sqrt {2 \pi n} n^n e^{-n} } < e^{1 / 12 n}$
We also have:
{{begin-eqn}}
{{eqn | l = e^{1 / 12 n}
| r = 1 + \frac 1 {12 n} + \frac 1 {2!} \paren {\frac 1 {12 n} }^2 + \frac 1 {3!} \paren {\frac 1 {12 n} }^3 + \cd... | Stirling's Formula/Refinement/Proof 3 | https://proofwiki.org/wiki/Stirling's_Formula/Refinement | https://proofwiki.org/wiki/Stirling's_Formula/Refinement/Proof_3 | [
"Stirling's Formula"
] | [
"Stirling's Formula"
] | [
"Limit of Error in Stirling's Formula",
"Power Series Expansion for Exponential Function",
"Power Series Expansion for Exponential Function",
"Definition:Sufficiently Large",
"Limit of Error in Stirling's Formula"
] |
proofwiki-22520 | Rodrigues' Formula for Hermite Polynomials | :$\map {H_n} x = \paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} }$
where:
:$n \in \N$ is a natural number
:$H_n$ is the $n$th Hermite polynomial. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | :$\map {H_n} x = \paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} }$
where:
:$n \in \N$ is a [[Definition:Natural Number|natural number]]
:$H_n$ is the $n$th [[Definition:Hermite Polynomial|Hermite polynomial]]. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | Rodrigues' Formula for Hermite Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Hermite_Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Hermite_Polynomials | [
"Hermite Polynomials",
"Rodrigues' Formula"
] | [
"Definition:Natural Numbers",
"Definition:Hermite Polynomial"
] | [] |
proofwiki-22521 | Generating Function for Hermite Polynomials | Let $\map {H_n} x$ denote the $n$th Hermite polynomial.
Then the generating function for $H_n$ is:
:$\ds e^{2 t x - t^2} = \sum_{n \mathop = 0}^\infty \dfrac {\map {H_n} x t^n} {n!}$ | {{begin-eqn}}
{{eqn | l = \map {H_n} x
| r = \paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} }
| c = Rodrigues' Formula for Hermite Polynomials
}}
{{eqn | r = \paren {-1}^n \dfrac {n!} {2 i n} \oint_C \dfrac {\map \exp {x^2 - \xi^2} } { {\xi - x}^{n + 1} } \rd \xi
| c = ... | Let $\map {H_n} x$ denote the $n$th [[Definition:Hermite Polynomial|Hermite polynomial]].
Then the [[Definition:Generating Function|generating function]] for $H_n$ is:
:$\ds e^{2 t x - t^2} = \sum_{n \mathop = 0}^\infty \dfrac {\map {H_n} x t^n} {n!}$ | {{begin-eqn}}
{{eqn | l = \map {H_n} x
| r = \paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} }
| c = [[Rodrigues' Formula for Hermite Polynomials]]
}}
{{eqn | r = \paren {-1}^n \dfrac {n!} {2 i n} \oint_C \dfrac {\map \exp {x^2 - \xi^2} } { {\xi - x}^{n + 1} } \rd \xi
| ... | Generating Function for Hermite Polynomials | https://proofwiki.org/wiki/Generating_Function_for_Hermite_Polynomials | https://proofwiki.org/wiki/Generating_Function_for_Hermite_Polynomials | [
"Hermite Polynomials",
"Examples of Generating Functions"
] | [
"Definition:Hermite Polynomial",
"Definition:Generating Function"
] | [
"Rodrigues' Formula for Hermite Polynomials",
"Cauchy's Integral Formula",
"Definition:Contour/Complex Plane",
"Definition:Anticlockwise",
"Integration by Substitution"
] |
proofwiki-22522 | Recurrence Formula for Hermite Polynomials | Let $\map {H_n} x$ denote the Hermite polynomial of order $n$.
Then:
:$\map {H_{n + 1} } x = 2 x \map {H_n} x - 2 n \map {H_{n - 1} } x$ | From Generating Function for Hermite Polynomials, the generating function for $H_n$ is:
:$(1): \quad \ds e^{2 t x - t^2} = \sum_{n \mathop = 0}^\infty \dfrac {\map {H_n} x t^n} {n!}$
Differentiating both sides of $(1)$ {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d t} } {e^{2 t x - t^2} }
... | Let $\map {H_n} x$ denote the [[Definition:Hermite Polynomial|Hermite polynomial of order $n$]].
Then:
:$\map {H_{n + 1} } x = 2 x \map {H_n} x - 2 n \map {H_{n - 1} } x$ | From [[Generating Function for Hermite Polynomials]], the [[Definition:Generating Function|generating function]] for $H_n$ is:
:$(1): \quad \ds e^{2 t x - t^2} = \sum_{n \mathop = 0}^\infty \dfrac {\map {H_n} x t^n} {n!}$
[[Definition:Differentiation|Differentiating]] both sides of $(1)$ {{WRT|Differentiation}} $t$:
... | Recurrence Formula for Hermite Polynomials | https://proofwiki.org/wiki/Recurrence_Formula_for_Hermite_Polynomials | https://proofwiki.org/wiki/Recurrence_Formula_for_Hermite_Polynomials | [
"Hermite Polynomials"
] | [
"Definition:Hermite Polynomial"
] | [
"Generating Function for Hermite Polynomials",
"Definition:Generating Function",
"Definition:Differentiation",
"Derivative of Exponential Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Translation of Index Variable of Summation",
"Definiti... |
proofwiki-22523 | Recurrence Formula for Hermite Polynomials using Derivative | Let $\map {H_n} x$ denote the Hermite polynomial of order $n$.
Then:
:$\dfrac \d {\d x} \map {H_n} x = 2 n \map {H_{n - 1} } x$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \map {H_n} x
| r = \map {\dfrac \d {\d x} } {\paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} } }
| c = Rodrigues' Formula for Hermite Polynomials
}}
{{eqn | r = \paren {-1}^n \paren {2 x \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } ... | Let $\map {H_n} x$ denote the [[Definition:Hermite Polynomial|Hermite polynomial of order $n$]].
Then:
:$\dfrac \d {\d x} \map {H_n} x = 2 n \map {H_{n - 1} } x$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \map {H_n} x
| r = \map {\dfrac \d {\d x} } {\paren {-1}^n \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n} } {\map \exp {-x^2} } }
| c = [[Rodrigues' Formula for Hermite Polynomials]]
}}
{{eqn | r = \paren {-1}^n \paren {2 x \map \exp {x^2} \map {\dfrac {\d^n} {\d x^n... | Recurrence Formula for Hermite Polynomials using Derivative | https://proofwiki.org/wiki/Recurrence_Formula_for_Hermite_Polynomials_using_Derivative | https://proofwiki.org/wiki/Recurrence_Formula_for_Hermite_Polynomials_using_Derivative | [
"Hermite Polynomials"
] | [
"Definition:Hermite Polynomial"
] | [
"Rodrigues' Formula for Hermite Polynomials",
"Derivative of Exponential Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Rodrigues' Formula for Hermite Polynomials",
"Recurrence Formula for Hermite Polynomials"
] |
proofwiki-22524 | Closed Form for Hermite Polynomials | {{begin-eqn}}
{{eqn | l = \map {H_n} x
| r = \paren {2 x}^n - \dfrac {n \paren {n - 1} } {1!} \paren {2 x}^{n - 2} + \dfrac {n \paren {n - 1} \paren {n - 2} \paren {n - 3} } {2!} \paren {2 x}^{n - 4} \cdots
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^{\floor {n / 2} } \paren {-1}^k \dfrac {n^\underline {2 k}... | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {H_n} x = \ds \sum_{k \mathop = 0}^{\floor {n / 2} } \paren {-1}^k \dfrac {n^\underline {2 k} } {k!} \paren {2 x}^{n - 2 k}$ | {{begin-eqn}}
{{eqn | l = \map {H_n} x
| r = \paren {2 x}^n - \dfrac {n \paren {n - 1} } {1!} \paren {2 x}^{n - 2} + \dfrac {n \paren {n - 1} \paren {n - 2} \paren {n - 3} } {2!} \paren {2 x}^{n - 4} \cdots
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^{\floor {n / 2} } \paren {-1}^k \dfrac {n^\underline {2 k}... | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {H_n} x = \ds \sum_{k \mathop = 0}^{\floor {n / 2} } \paren {-1}^k \dfrac {n^\underline {2 k} } {k!} \paren {2 x}^{n - 2 k}$ | Closed Form for Hermite Polynomials | https://proofwiki.org/wiki/Closed_Form_for_Hermite_Polynomials | https://proofwiki.org/wiki/Closed_Form_for_Hermite_Polynomials | [
"Hermite Polynomials"
] | [
"Definition:Falling Factorial"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22525 | Rodrigues' Formula for Laguerre Polynomials | :$\map {L_n} x = e^x \map {\dfrac {\d^n} {\d x^n} } {x^n e^{-x} }$
where:
:$n \in \N$ is a natural number
:$L_n$ is the $n$th Laguerre polynomial. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | :$\map {L_n} x = e^x \map {\dfrac {\d^n} {\d x^n} } {x^n e^{-x} }$
where:
:$n \in \N$ is a [[Definition:Natural Number|natural number]]
:$L_n$ is the $n$th [[Definition:Laguerre Polynomial|Laguerre polynomial]]. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | Rodrigues' Formula for Laguerre Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Laguerre_Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Laguerre_Polynomials | [
"Laguerre Polynomials",
"Rodrigues' Formula"
] | [
"Definition:Natural Numbers",
"Definition:Laguerre's Differential Equation/Laguerre Polynomial"
] | [] |
proofwiki-22526 | Power Series Expansion for Chebyshev Polynomial of the First Kind | The $n$th '''Chebyshev polynomial of the first kind''' can be expressed as a power series expansion in the form:
{{begin-eqn}}
{{eqn | l = \map {T_n} x
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \dbinom n {2 k} x^{n - 2 k} \paren {1 - x^2}^k
}}
{{eqn | r = x^n - \dbinom n 2 x^{n - 2} \paren {1 - x^2} + \dbin... | From the Definition of Chebyshev Polynomial of the First Kind, we have:
:$\map {T_n} {\cos \theta} = \map \cos {n \theta}$
From De Moivre's Formula, we have:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \in \Z_{>0}$, we use the Binomial Theorem on the {{RHS}}, resulting in:
:$\ds \c... | The $n$th '''[[Definition:Chebyshev Polynomial of the First Kind|Chebyshev polynomial of the first kind]]''' can be expressed as a [[Definition:Power Series|power series expansion]] in the form:
{{begin-eqn}}
{{eqn | l = \map {T_n} x
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \dbinom n {2 k} x^{n - 2 k} \pa... | From the [[Definition:Chebyshev Polynomial of the First Kind/Also presented as|Definition of Chebyshev Polynomial of the First Kind]], we have:
:$\map {T_n} {\cos \theta} = \map \cos {n \theta}$
From [[De Moivre's Formula]], we have:
:$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$
As $n \i... | Power Series Expansion for Chebyshev Polynomial of the First Kind | https://proofwiki.org/wiki/Power_Series_Expansion_for_Chebyshev_Polynomial_of_the_First_Kind | https://proofwiki.org/wiki/Power_Series_Expansion_for_Chebyshev_Polynomial_of_the_First_Kind | [
"Chebyshev Polynomials of the First Kind",
"Examples of Power Series"
] | [
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Power Series"
] | [
"Definition:Chebyshev Polynomial of the First Kind/Also presented as",
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Even Integer",
"Definition:Real Number",
"Definition:Complex Number/Real Part",
"Definition:Even Integer",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-22527 | Particular Values of Chebyshev Polynomials of the First Kind/-x | :$\map {T_n} {-x} = \paren {-1}^n \map {T_n} x$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {T_n} {-x} = \paren {-1}^n \map {T_n} x$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\p... | :$\map {T_n} {-x} = \paren {-1}^n \map {T_n} x$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {T_n} {-x} = \paren {-1}^n \map {T_n} x$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
... | Particular Values of Chebyshev Polynomials of the First Kind/-x | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/-x | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/-x | [
"Particular Values of Chebyshev Polynomials of the First Kind",
"Proofs by Induction"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Integer Division"
] |
proofwiki-22528 | Particular Values of Chebyshev Polynomials of the First Kind/1 | :$\map {T_n} 1 = 1$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {T_n} 1 = 1$ | :$\map {T_n} 1 = 1$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {T_n} 1 = 1$ | Particular Values of Chebyshev Polynomials of the First Kind/1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/1 | [
"Particular Values of Chebyshev Polynomials of the First Kind"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22529 | Particular Values of Chebyshev Polynomials of the First Kind/-1 | :$\map {T_n} {-1} = \paren {-1}^n$ | {{begin-eqn}}
{{eqn | l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
| c = Particular Values of Chebyshev Polynomials of the First Kind: $-x$
}}
{{eqn | ll= \leadsto
| l = \map {T_n} {-1}
| r = \paren {-1}^n \map {T_n} 1
| c = setting $x = 1$
}}
{{eqn | r = \paren {-1}^n
| c = ... | :$\map {T_n} {-1} = \paren {-1}^n$ | {{begin-eqn}}
{{eqn | l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
| c = [[Particular Values of Chebyshev Polynomials of the First Kind/-x|Particular Values of Chebyshev Polynomials of the First Kind: $-x$]]
}}
{{eqn | ll= \leadsto
| l = \map {T_n} {-1}
| r = \paren {-1}^n \map {T_n} 1
... | Particular Values of Chebyshev Polynomials of the First Kind/-1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/-1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/-1 | [
"Particular Values of Chebyshev Polynomials of the First Kind"
] | [] | [
"Particular Values of Chebyshev Polynomials of the First Kind/-x",
"Particular Values of Chebyshev Polynomials of the First Kind/1"
] |
proofwiki-22530 | Particular Values of Chebyshev Polynomials of the First Kind/0 | :$\map {T_n} 0 = \begin {cases} \paren {-1}^{n / 2} & : \text {$n$ even} \\ 0 & : \text {$n$ odd} \end {cases}$ | === Even Order ===
{{:Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order}} | :$\map {T_n} 0 = \begin {cases} \paren {-1}^{n / 2} & : \text {$n$ even} \\ 0 & : \text {$n$ odd} \end {cases}$ | === [[Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order|Even Order]] ===
{{:Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order}} | Particular Values of Chebyshev Polynomials of the First Kind/0 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0 | [
"Particular Values of Chebyshev Polynomials of the First Kind"
] | [] | [
"Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order"
] |
proofwiki-22531 | Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order | Let $\map {T_n} x$ denote the Chebyshev polynomial of the first kind of order $n$.
Let $n = 2 m$ for some $m \in \N$.
Then:
:$\map {T_n} 0 = \paren {-1}^m$ | Let $n = 2 m$ for some $m \in \N$.
The proof proceeds by induction.
For all $m \in \Z_{\ge 0}$, let $\map P m$ be the proposition:
:$\map {T_{2 m} } 0 = \paren {-1}^m$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \map {T_0} 0
| r = \paren {-1}
| c = Chebyshev Polynomial ... | Let $\map {T_n} x$ denote the [[Definition:Chebyshev Polynomial of the First Kind|Chebyshev polynomial of the first kind of order $n$]].
Let $n = 2 m$ for some $m \in \N$.
Then:
:$\map {T_n} 0 = \paren {-1}^m$ | Let $n = 2 m$ for some $m \in \N$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $m \in \Z_{\ge 0}$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\map {T_{2 m} } 0 = \paren {-1}^m$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l... | Particular Values of Chebyshev Polynomials of the First Kind/0/Even Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0/Even_Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0/Even_Order | [
"Particular Values of Chebyshev Polynomials of the First Kind"
] | [
"Definition:Chebyshev Polynomials/First Kind"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Chebyshev Polynomial of the First Kind/Examples/T0",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Recurrence Formula for Chebyshev Polynomials of the First Kind",
"Particular Val... |
proofwiki-22532 | Particular Values of Chebyshev Polynomials of the First Kind/0/Odd Order | Let $\map {T_n} x$ denote the Chebyshev polynomial of the first kind of order $n$.
Let $n = 2 k + 1$ for some $k \in \N$.
Then:
:$\map {T_n} 0 = 0$ | Let $n = 2 k + 1$ for some $k \in \N$.
{{begin-eqn}}
{{eqn | q = \forall x \in \Dom {T_n}
| l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
| c = Particular Values of Chebyshev Polynomials of the First Kind: $-x$
}}
{{eqn | ll= \leadsto
| l = \map {T_n} 0
| r = \paren {-1}^n \map {T_n... | Let $\map {T_n} x$ denote the [[Definition:Chebyshev Polynomial of the First Kind|Chebyshev polynomial of the first kind of order $n$]].
Let $n = 2 k + 1$ for some $k \in \N$.
Then:
:$\map {T_n} 0 = 0$ | Let $n = 2 k + 1$ for some $k \in \N$.
{{begin-eqn}}
{{eqn | q = \forall x \in \Dom {T_n}
| l = \map {T_n} {-x}
| r = \paren {-1}^n \map {T_n} x
| c = [[Particular Values of Chebyshev Polynomials of the First Kind/-x|Particular Values of Chebyshev Polynomials of the First Kind: $-x$]]
}}
{{eqn | ll= ... | Particular Values of Chebyshev Polynomials of the First Kind/0/Odd Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0/Odd_Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_First_Kind/0/Odd_Order | [
"Particular Values of Chebyshev Polynomials of the First Kind"
] | [
"Definition:Chebyshev Polynomials/First Kind"
] | [
"Particular Values of Chebyshev Polynomials of the First Kind/-x",
"Definition:Odd Integer",
"Definition:Number"
] |
proofwiki-22533 | Recurrence Formula for Chebyshev Polynomials of the Second Kind | Let $\map {U_n} x$ denote the Chebyshev polynomials of the second kind of order $n$.
Then:
:$\map {U_n} x = \begin {cases} 1 & : n = 0 \\ 2 x & : n = 1 \\ 2 x \map {U_{n - 1} } x - \map {U_{n - 2} } x & : n > 1 \end {cases}$ | From the {{Defof|Chebyshev Polynomial of the Second Kind}}, we have:
:$\map {U_n} x = \dfrac {\map \sin {\paren {n + 1} \arccos x} } {\map \sin {\arccos x} }$
For $n = 0$, we have:
{{begin-eqn}}
{{eqn | l = \map {U_0} x
| r = \dfrac {\map \sin {\paren {0 + 1} \arccos x} } {\map \sin {\arccos x} }
| c = {{De... | Let $\map {U_n} x$ denote the [[Definition:Chebyshev Polynomial of the Second Kind|Chebyshev polynomials of the second kind of order $n$]].
Then:
:$\map {U_n} x = \begin {cases} 1 & : n = 0 \\ 2 x & : n = 1 \\ 2 x \map {U_{n - 1} } x - \map {U_{n - 2} } x & : n > 1 \end {cases}$ | From the {{Defof|Chebyshev Polynomial of the Second Kind}}, we have:
:$\map {U_n} x = \dfrac {\map \sin {\paren {n + 1} \arccos x} } {\map \sin {\arccos x} }$
For $n = 0$, we have:
{{begin-eqn}}
{{eqn | l = \map {U_0} x
| r = \dfrac {\map \sin {\paren {0 + 1} \arccos x} } {\map \sin {\arccos x} }
| c = ... | Recurrence Formula for Chebyshev Polynomials of the Second Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Chebyshev_Polynomials_of_the_Second_Kind | https://proofwiki.org/wiki/Recurrence_Formula_for_Chebyshev_Polynomials_of_the_Second_Kind | [
"Recurrence Formula for Chebyshev Polynomials of the Second Kind",
"Chebyshev Polynomials of the Second Kind"
] | [
"Definition:Chebyshev Polynomials/Second Kind"
] | [
" Double Angle Formula for Sine"
] |
proofwiki-22534 | Power Series Expansion for Chebyshev Polynomial of the Second Kind | The $n$th '''Chebyshev polynomial of the second kind''' can be expressed as a power series expansion in the form:
{{begin-eqn}}
{{eqn | l = \map {U_n} x
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \dbinom {n + 1} {2 k + 1} x^{n - 2 k} \paren {1 - x^2}^k
}}
{{eqn | r = \dbinom {n + 1} 1 x^n - \dbinom {n + 1} 3... | From the Definition of Chebyshev Polynomial of the Second Kind, we have:
:$\map {U_n} {\cos \theta} \sin \theta = \map \sin {\paren {n + 1} \theta}$
From De Moivre's Formula, we have:
:$\map \cos {\paren {n + 1} \theta} + i \map \sin {\paren {n + 1} \theta} = \paren {\cos \theta + i \sin \theta}^{n + 1}$
As $n \in \Z_{... | The $n$th '''[[Definition:Chebyshev Polynomial of the Second Kind|Chebyshev polynomial of the second kind]]''' can be expressed as a [[Definition:Power Series|power series expansion]] in the form:
{{begin-eqn}}
{{eqn | l = \map {U_n} x
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \dbinom {n + 1} {2 k + 1} x^{... | From the [[Definition:Chebyshev Polynomial of the Second Kind/Also presented as|Definition of Chebyshev Polynomial of the Second Kind]], we have:
:$\map {U_n} {\cos \theta} \sin \theta = \map \sin {\paren {n + 1} \theta}$
From [[De Moivre's Formula]], we have:
:$\map \cos {\paren {n + 1} \theta} + i \map \sin {\paren ... | Power Series Expansion for Chebyshev Polynomial of the Second Kind | https://proofwiki.org/wiki/Power_Series_Expansion_for_Chebyshev_Polynomial_of_the_Second_Kind | https://proofwiki.org/wiki/Power_Series_Expansion_for_Chebyshev_Polynomial_of_the_Second_Kind | [
"Chebyshev Polynomials of the Second Kind",
"Examples of Power Series"
] | [
"Definition:Chebyshev Polynomials/Second Kind",
"Definition:Power Series"
] | [
"Definition:Chebyshev Polynomial of the Second Kind/Also presented as",
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Odd Integer",
"Definition:Imaginary Number",
"Definition:Complex Number/Imaginary Part",
"Definition:Odd Integer",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-22535 | Particular Values of Chebyshev Polynomials of the Second Kind/-x | :$\map {U_n} {-x} = \paren {-1}^n \map {U_n} x$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {U_n} {-x} = \paren {-1}^n \map {U_n} x$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {U_n} {-x}
| r = \paren {-1}^n \map {U_n} x
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\p... | :$\map {U_n} {-x} = \paren {-1}^n \map {U_n} x$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {U_n} {-x} = \paren {-1}^n \map {U_n} x$
First we note that:
{{begin-eqn}}
{{eqn | l = \map {U_n} {-x}
| r = \paren {-1}^n \map {U_n} x
... | Particular Values of Chebyshev Polynomials of the Second Kind/-x | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/-x | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/-x | [
"Particular Values of Chebyshev Polynomials of the Second Kind",
"Proofs by Induction"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Integer Division"
] |
proofwiki-22536 | Particular Values of Chebyshev Polynomials of the Second Kind/1 | :$\map {U_n} 1 = n + 1$ | The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\map {U_n} 1 = n + 1$ | :$\map {U_n} 1 = n + 1$ | The proof proceeds by [[Principle of Strong Induction|strong induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\map {U_n} 1 = n + 1$ | Particular Values of Chebyshev Polynomials of the Second Kind/1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/1 | [
"Particular Values of Chebyshev Polynomials of the Second Kind"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition"
] |
proofwiki-22537 | Particular Values of Chebyshev Polynomials of the Second Kind/-1 | :$\map {U_n} {-1} = \paren {-1}^n \paren {n + 1}$ | {{begin-eqn}}
{{eqn | l = \map {U_n} {-x}
| r = \paren {-1}^n \map {U_n} x
| c = Particular Values of Chebyshev Polynomials of the Second Kind: $-x$
}}
{{eqn | ll= \leadsto
| l = \map {U_n} {-1}
| r = \paren {-1}^n \map {U_n} 1
| c = setting $x = 1$
}}
{{eqn | r = \paren {-1}^n \paren {n +... | :$\map {U_n} {-1} = \paren {-1}^n \paren {n + 1}$ | {{begin-eqn}}
{{eqn | l = \map {U_n} {-x}
| r = \paren {-1}^n \map {U_n} x
| c = [[Particular Values of Chebyshev Polynomials of the Second Kind/-x|Particular Values of Chebyshev Polynomials of the Second Kind: $-x$]]
}}
{{eqn | ll= \leadsto
| l = \map {U_n} {-1}
| r = \paren {-1}^n \map {U_n} 1... | Particular Values of Chebyshev Polynomials of the Second Kind/-1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/-1 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/-1 | [
"Particular Values of Chebyshev Polynomials of the Second Kind"
] | [] | [
"Particular Values of Chebyshev Polynomials of the Second Kind/-x",
"Particular Values of Chebyshev Polynomials of the Second Kind/1"
] |
proofwiki-22538 | Particular Values of Chebyshev Polynomials of the Second Kind/0 | :$\map {U_n} 0 = \begin {cases} \paren {-1}^{n / 2} & : \text {$n$ even} \\ 0 & : \text {$n$ odd} \end {cases}$ | === Even Order ===
{{:Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order}} | :$\map {U_n} 0 = \begin {cases} \paren {-1}^{n / 2} & : \text {$n$ even} \\ 0 & : \text {$n$ odd} \end {cases}$ | === [[Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order|Even Order]] ===
{{:Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order}} | Particular Values of Chebyshev Polynomials of the Second Kind/0 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/0 | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/0 | [
"Particular Values of Chebyshev Polynomials of the Second Kind"
] | [] | [
"Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order"
] |
proofwiki-22539 | Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order | Let $\map {U_n} x$ denote the Chebyshev polynomial of the second kind of order $n$.
Let $n = 2 m$ for some $m \in \N$.
Then:
:$\map {U_n} 0 = \paren {-1}^m$ | Let $n = 2 m$ for some $m \in \N$.
The proof proceeds by induction.
For all $m \in \Z_{\ge 0}$, let $\map P m$ be the proposition:
:$\map {U_{2 m} } 0 = \paren {-1}^m$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \map {U_0} 0
| r = \paren {-1}
| c = Chebyshev Polynomial ... | Let $\map {U_n} x$ denote the [[Definition:Chebyshev Polynomial of the Second Kind|Chebyshev polynomial of the second kind of order $n$]].
Let $n = 2 m$ for some $m \in \N$.
Then:
:$\map {U_n} 0 = \paren {-1}^m$ | Let $n = 2 m$ for some $m \in \N$.
The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $m \in \Z_{\ge 0}$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
:$\map {U_{2 m} } 0 = \paren {-1}^m$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l... | Particular Values of Chebyshev Polynomials of the Second Kind/0/Even Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/0/Even_Order | https://proofwiki.org/wiki/Particular_Values_of_Chebyshev_Polynomials_of_the_Second_Kind/0/Even_Order | [
"Particular Values of Chebyshev Polynomials of the Second Kind"
] | [
"Definition:Chebyshev Polynomials/Second Kind"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Chebyshev Polynomial of the Second Kind/Examples/U0",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Recurrence Formula for Chebyshev Polynomials of the First Kind",
"Particular Va... |
proofwiki-22540 | General Solution to Chebyshev's Differential Equation | Consider '''Chebyshev's differential equation''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The general solution to $(1)$ is given by:
:$y = \begin {cases} A \map {T_n} x + B \sqrt {1 - x^2} \, \map {U_{n - 1} } x + C \map \cos {n \arcsin x} + D \ma... | First a {{Lemma|General Solution to Chebyshev's Differential Equation}}:
{{:General Solution to Chebyshev's Differential Equation/Lemma}}{{qed|lemma}}
Let $n = 0$.
In our Lemma, we assumed that either $x = \sin \theta$ or $x = \cos \theta$.
Assuming $x = \cos \theta$, then:
{{begin-eqn}}
{{eqn | ll = \leadsto
| l... | Consider '''[[Definition:Chebyshev's Differential Equation|Chebyshev's differential equation]]''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ is given by:
:$y =... | First a {{Lemma|General Solution to Chebyshev's Differential Equation}}:
{{:General Solution to Chebyshev's Differential Equation/Lemma}}{{qed|lemma}}
Let $n = 0$.
In our [[General Solution to Chebyshev's Differential Equation/Lemma|Lemma]], we assumed that either $x = \sin \theta$ or $x = \cos \theta$.
Assuming $x... | General Solution to Chebyshev's Differential Equation | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation | [
"General Solution to Chebyshev's Differential Equation",
"Chebyshev's Differential Equation",
"Chebyshev Polynomials",
"Ordinary Differential Equations",
"Differential Equations"
] | [
"Definition:Chebyshev's Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomials/Second Kind"
] | [
"General Solution to Chebyshev's Differential Equation/Lemma",
"Primitive of Constant",
"Primitive of Constant",
"Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE",
"Definition:Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE",
"Definition:Differential Equation/Soluti... |
proofwiki-22541 | General Solution to Chebyshev's Differential Equation | Consider '''Chebyshev's differential equation''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The general solution to $(1)$ is given by:
:$y = \begin {cases} A \map {T_n} x + B \sqrt {1 - x^2} \, \map {U_{n - 1} } x + C \map \cos {n \arcsin x} + D \ma... | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = -\sin \theta \rd \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d \theta} {\d x}
| r = -\frac 1 {\sin \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2 \th... | Consider '''[[Definition:Chebyshev's Differential Equation|Chebyshev's differential equation]]''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ is given by:
:$y =... | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \cos \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = -\sin \theta \rd \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d \theta} {\d x}
| r = -\frac 1 {\sin \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2 \t... | General Solution to Chebyshev's Differential Equation/Lemma/Proof 1 | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation/Lemma/Proof_1 | [
"General Solution to Chebyshev's Differential Equation",
"Chebyshev's Differential Equation",
"Chebyshev Polynomials",
"Ordinary Differential Equations",
"Differential Equations"
] | [
"Definition:Chebyshev's Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomials/Second Kind"
] | [
"Derivative of Cosecant Function",
"Derivative of Composite Function",
"Definition:Chebyshev's Differential Equation",
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Sum of Squares of Sine and Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22542 | General Solution to Chebyshev's Differential Equation | Consider '''Chebyshev's differential equation''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The general solution to $(1)$ is given by:
:$y = \begin {cases} A \map {T_n} x + B \sqrt {1 - x^2} \, \map {U_{n - 1} } x + C \map \cos {n \arcsin x} + D \ma... | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \sin \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \cos \theta \rd \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d \theta} {\d x}
| r = \frac 1 {\cos \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2 \thet... | Consider '''[[Definition:Chebyshev's Differential Equation|Chebyshev's differential equation]]''':
:$(1): \quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$
where $n \in \N$.
The [[Definition:General Solution to Differential Equation|general solution]] to $(1)$ is given by:
:$y =... | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \sin \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \d x
| r = \cos \theta \rd \theta
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d \theta} {\d x}
| r = \frac 1 {\cos \theta}
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2 \the... | General Solution to Chebyshev's Differential Equation/Lemma/Proof 2 | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation | https://proofwiki.org/wiki/General_Solution_to_Chebyshev's_Differential_Equation/Lemma/Proof_2 | [
"General Solution to Chebyshev's Differential Equation",
"Chebyshev's Differential Equation",
"Chebyshev Polynomials",
"Ordinary Differential Equations",
"Differential Equations"
] | [
"Definition:Chebyshev's Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Chebyshev Polynomials/First Kind",
"Definition:Chebyshev Polynomials/Second Kind"
] | [
"Derivative of Secant Function",
"Derivative of Composite Function",
"Definition:Chebyshev's Differential Equation",
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Sum of Squares of Sine and Cosine",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-22543 | Relationship between Chebyshev Polynomial of the First and Second Kind/Formulation 2 | :$\paren {1 - x^2} \map {U_{n - 1} } x = x \map {T_n} x - \map {T_{n + 1} } x$ | Let $x = \cos \theta$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T_n} x
| r = \map \cos {n \arccos x}
| c = {{Defof|Chebyshev Polynomial of the First Kind}}
}}
{{eqn | ll = \leadsto
| l = \map {T_n} {\cos \theta}
| r = \map \cos {n \arccos \cos \theta}
| c = $x \to \cos \theta$
}}
{{eqn | ll =... | :$\paren {1 - x^2} \map {U_{n - 1} } x = x \map {T_n} x - \map {T_{n + 1} } x$ | Let $x = \cos \theta$.
Then:
{{begin-eqn}}
{{eqn | l = \map {T_n} x
| r = \map \cos {n \arccos x}
| c = {{Defof|Chebyshev Polynomial of the First Kind}}
}}
{{eqn | ll = \leadsto
| l = \map {T_n} {\cos \theta}
| r = \map \cos {n \arccos \cos \theta}
| c = $x \to \cos \theta$
}}
{{eqn | ll ... | Relationship between Chebyshev Polynomial of the First and Second Kind/Formulation 2 | https://proofwiki.org/wiki/Relationship_between_Chebyshev_Polynomial_of_the_First_and_Second_Kind/Formulation_2 | https://proofwiki.org/wiki/Relationship_between_Chebyshev_Polynomial_of_the_First_and_Second_Kind/Formulation_2 | [
"Relationship between Chebyshev Polynomial of the First and Second Kind"
] | [] | [
"Cosine of Sum",
"Cosine of Sum",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-22544 | Legendre Polynomial in terms of Gaussian Hypergeometric Function | :$\map {P_n} x = \map F {n + 1, -n; 1; \dfrac {1 - x} 2}$ | From Solution to Hypergeometric Differential Equation, we have:
{{:Solution to Hypergeometric Differential Equation}}
Inputting $\map F {n + 1, -n; 1; \dfrac {1 - x} 2}$ into the hypergeometric differential equation, we obtain:
{{begin-eqn}}
{{eqn | l = 0
| r = x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + \paren {... | :$\map {P_n} x = \map F {n + 1, -n; 1; \dfrac {1 - x} 2}$ | From [[Solution to Hypergeometric Differential Equation]], we have:
{{:Solution to Hypergeometric Differential Equation}}
Inputting $\map F {n + 1, -n; 1; \dfrac {1 - x} 2}$ into the [[Definition:Hypergeometric Differential Equation|hypergeometric differential equation]], we obtain:
{{begin-eqn}}
{{eqn | l = 0
|... | Legendre Polynomial in terms of Gaussian Hypergeometric Function | https://proofwiki.org/wiki/Legendre_Polynomial_in_terms_of_Gaussian_Hypergeometric_Function | https://proofwiki.org/wiki/Legendre_Polynomial_in_terms_of_Gaussian_Hypergeometric_Function | [
"Legendre Polynomials",
"Gaussian Hypergeometric Function",
"Hypergeometric Functions"
] | [] | [
"Solution to Hypergeometric Differential Equation",
"Definition:Hypergeometric Differential Equation",
"Solution to Hypergeometric Differential Equation",
"Definition:Legendre's Differential Equation",
"Definition:Hypergeometric Differential Equation",
"Definition:Legendre's Differential Equation",
"Def... |
proofwiki-22545 | Laplace Transform of Exponential times Hyperbolic Sine | :$\map {\laptrans {e^{b t} \sinh a t} } s = \dfrac a {\paren {s - b}^2 - a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \sinh a t} } s
| r = \map {\laptrans {\sinh a t} } {s - b}
| c = First Translation Property of Laplace Transforms
}}
{{eqn | r = \frac a {\paren {s - b}^2 - a^2}
| c = Laplace Transform of Hyperbolic Sine
}}
{{end-eqn}}
{{qed}} | :$\map {\laptrans {e^{b t} \sinh a t} } s = \dfrac a {\paren {s - b}^2 - a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \sinh a t} } s
| r = \map {\laptrans {\sinh a t} } {s - b}
| c = [[First Translation Property of Laplace Transforms]]
}}
{{eqn | r = \frac a {\paren {s - b}^2 - a^2}
| c = [[Laplace Transform of Hyperbolic Sine]]
}}
{{end-eqn}}
{{qed}} | Laplace Transform of Exponential times Hyperbolic Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Hyperbolic_Sine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Hyperbolic_Sine | [
"Laplace Transform of Exponential times Hyperbolic Sine",
"Laplace Transforms involving Exponential Function",
"Laplace Transforms involving Hyperbolic Sine Function",
"Examples of Laplace Transforms"
] | [] | [
"First Translation Property of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine"
] |
proofwiki-22546 | Laplace Transform of Exponential times Hyperbolic Cosine | :$\map {\laptrans {e^{b t} \cosh a t} } s = \dfrac {s - b} {\paren {s - b}^2 - a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \cosh a t} } s
| r = \map {\laptrans {\cosh a t} } {s - b}
| c = First Translation Property of Laplace Transforms
}}
{{eqn | r = \frac {s - b} {\paren {s - b}^2 - a^2}
| c = Laplace Transform of Hyperbolic Cosine
}}
{{end-eqn}}
{{qed}} | :$\map {\laptrans {e^{b t} \cosh a t} } s = \dfrac {s - b} {\paren {s - b}^2 - a^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {e^{b t} \cosh a t} } s
| r = \map {\laptrans {\cosh a t} } {s - b}
| c = [[First Translation Property of Laplace Transforms]]
}}
{{eqn | r = \frac {s - b} {\paren {s - b}^2 - a^2}
| c = [[Laplace Transform of Hyperbolic Cosine]]
}}
{{end-eqn}}
{{qed}} | Laplace Transform of Exponential times Hyperbolic Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Hyperbolic_Cosine | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_times_Hyperbolic_Cosine | [
"Laplace Transforms involving Exponential Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Examples of Laplace Transforms"
] | [] | [
"First Translation Property of Laplace Transforms",
"Laplace Transform of Hyperbolic Cosine"
] |
proofwiki-22547 | Laplace Transform of Difference between Exponentials | :$\map {\laptrans {\dfrac {e^{b t} - e^{a t} } {b - a} } } s = \dfrac 1 {\paren {s - a} \paren {s - b} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{b t} - e^{a t} } {b - a} } } s
| r = \dfrac 1 {b - a} \paren {\map {\laptrans {e^{b t} } } s - \map {\laptrans {e^{a t} } } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {b - a} \paren {\dfrac 1 {s - b} - \dfrac 1 {s - a} }
... | :$\map {\laptrans {\dfrac {e^{b t} - e^{a t} } {b - a} } } s = \dfrac 1 {\paren {s - a} \paren {s - b} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{b t} - e^{a t} } {b - a} } } s
| r = \dfrac 1 {b - a} \paren {\map {\laptrans {e^{b t} } } s - \map {\laptrans {e^{a t} } } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {b - a} \paren {\dfrac 1 {s - b} - \dfrac 1 {s - a} ... | Laplace Transform of Difference between Exponentials | https://proofwiki.org/wiki/Laplace_Transform_of_Difference_between_Exponentials | https://proofwiki.org/wiki/Laplace_Transform_of_Difference_between_Exponentials | [
"Laplace Transforms involving Exponential Function",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential",
"Definition:Common Denominator"
] |
proofwiki-22548 | Laplace Transform of b e^bt - a e^at over b - a | :$\map {\laptrans {\dfrac {b e^{b t} - a e^{a t} } {b - a} } } s = \dfrac s {\paren {s - a} \paren {s - b} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {b e^{b t} - a e^{a t} } {b - a} } } s
| r = \dfrac 1 {b - a} \paren {b \map {\laptrans {e^{b t} } } s - a \map {\laptrans {e^{a t} } } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {b - a} \paren {\dfrac b {s - b} - \dfrac a {s -... | :$\map {\laptrans {\dfrac {b e^{b t} - a e^{a t} } {b - a} } } s = \dfrac s {\paren {s - a} \paren {s - b} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {b e^{b t} - a e^{a t} } {b - a} } } s
| r = \dfrac 1 {b - a} \paren {b \map {\laptrans {e^{b t} } } s - a \map {\laptrans {e^{a t} } } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {b - a} \paren {\dfrac b {s - b} - \dfrac a ... | Laplace Transform of b e^bt - a e^at over b - a | https://proofwiki.org/wiki/Laplace_Transform_of_b_e^bt_-_a_e^at_over_b_-_a | https://proofwiki.org/wiki/Laplace_Transform_of_b_e^bt_-_a_e^at_over_b_-_a | [
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential",
"Definition:Common Denominator"
] |
proofwiki-22549 | Laplace Transform of sine a t - a t cosine a t over 2 a^3 | :$\map {\laptrans {\dfrac {\sin a t - a t \cos a t} {2 a^3} } } s = \dfrac 1 {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sin a t - a t \cos a t} {2 a^3} } } s
| r = \dfrac 1 {2 a^3} \paren {\map {\laptrans {\sin a t} } s - a \map {\laptrans {t \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {\dfrac a {s^2 + a^2} - a \map {... | :$\map {\laptrans {\dfrac {\sin a t - a t \cos a t} {2 a^3} } } s = \dfrac 1 {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sin a t - a t \cos a t} {2 a^3} } } s
| r = \dfrac 1 {2 a^3} \paren {\map {\laptrans {\sin a t} } s - a \map {\laptrans {t \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {\dfrac a {s^2 + a^2} - a \m... | Laplace Transform of sine a t - a t cosine a t over 2 a^3 | https://proofwiki.org/wiki/Laplace_Transform_of_sine_a_t_-_a_t_cosine_a_t_over_2_a^3 | https://proofwiki.org/wiki/Laplace_Transform_of_sine_a_t_-_a_t_cosine_a_t_over_2_a^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Sine",
"Laplace Transform of t cosine a t",
"Definition:Common Denominator"
] |
proofwiki-22550 | Laplace Transform of t sine a t over 2 a | :$\map {\laptrans {\dfrac {t \sin a t} {2 a} } } s = \dfrac s {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sin a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t \sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac {2 a s} {\paren {s^2 + a^2}^2} }
| c = Laplace Transform of $t \sin a t$... | :$\map {\laptrans {\dfrac {t \sin a t} {2 a} } } s = \dfrac s {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sin a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t \sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac {2 a s} {\paren {s^2 + a^2}^2} }
| c = [[Laplace Transform of t sine... | Laplace Transform of t sine a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t_over_2_a | [
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sine a t"
] |
proofwiki-22551 | Laplace Transform of sine a t + a t cosine a t over 2 a | :$\map {\laptrans {\dfrac {\sin a t + a t \cos a t} {2 a} } } s = \dfrac {s^2} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sin a t + a t \cos a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sin a t} } s + a \map {\laptrans {t \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 + a^2} + a \map {\laptr... | :$\map {\laptrans {\dfrac {\sin a t + a t \cos a t} {2 a} } } s = \dfrac {s^2} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sin a t + a t \cos a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sin a t} } s + a \map {\laptrans {t \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 + a^2} + a \map {\l... | Laplace Transform of sine a t + a t cosine a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_sine_a_t_+_a_t_cosine_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_sine_a_t_+_a_t_cosine_a_t_over_2_a | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Sine",
"Laplace Transform of t cosine a t",
"Definition:Common Denominator"
] |
proofwiki-22552 | Laplace Transform of cosine a t - half a t sine a t | :$\map {\laptrans {\cos a t - \dfrac 1 2 a t \sin a t} } s = \dfrac {s^3} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cos a t - \dfrac 1 2 a t \sin a t} } s
| r = \map {\laptrans {\cos a t} } s - \dfrac a 2 \map {\laptrans {t \sin a t} } s
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac s {s^2 + a^2} - \dfrac a 2 \map {\laptrans {t \sin a t} } s
| c = L... | :$\map {\laptrans {\cos a t - \dfrac 1 2 a t \sin a t} } s = \dfrac {s^3} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cos a t - \dfrac 1 2 a t \sin a t} } s
| r = \map {\laptrans {\cos a t} } s - \dfrac a 2 \map {\laptrans {t \sin a t} } s
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac s {s^2 + a^2} - \dfrac a 2 \map {\laptrans {t \sin a t} } s
| c... | Laplace Transform of cosine a t - half a t sine a t | https://proofwiki.org/wiki/Laplace_Transform_of_cosine_a_t_-_half_a_t_sine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_cosine_a_t_-_half_a_t_sine_a_t | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Cosine",
"Laplace Transform of t sine a t",
"Definition:Common Denominator"
] |
proofwiki-22553 | Laplace Transform of t cosine a t | :$\map {\laptrans {t \cos a t} } s = \dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cos a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cos a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 + a^2} }
| c = Laplace Transform of Cosine
}}
{{eqn | r = -\dfrac {\paren {s^2 + a^2... | :$\map {\laptrans {t \cos a t} } s = \dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cos a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cos a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 + a^2} }
| c = [[Laplace Transform of Cosine]]
}}
{{eqn | r = -\dfrac {\paren {s... | Laplace Transform of t cosine a t | https://proofwiki.org/wiki/Laplace_Transform_of_t_cosine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t_cosine_a_t | [
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of Cosine",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives"
] |
proofwiki-22554 | Laplace Transform of t sine a t | :$\map {\laptrans {t \sin a t} } s = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sin a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sin a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 + a^2} }
| c = Laplace Transform of Sine
}}
{{eqn | r = -\dfrac {\paren {s^2 + a^2} ... | :$\map {\laptrans {t \sin a t} } s = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sin a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sin a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 + a^2} }
| c = [[Laplace Transform of Sine]]
}}
{{eqn | r = -\dfrac {\paren {s^2... | Laplace Transform of t sine a t/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t/Proof_1 | [
"Laplace Transform of t sine a t",
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of Sine",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives"
] |
proofwiki-22555 | Laplace Transform of t sine a t | :$\map {\laptrans {t \sin a t} } s = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$ | We have:
{{begin-eqn}}
{{eqn | l = \laptrans {\cos a t}
| r = \int_0^\infty e^{-s t} \cos a t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | n = 1
| r = \dfrac s {s^2 + a^2}
| c = Laplace Transform of Cosine
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d a} \int_0^\infty e^{-s... | :$\map {\laptrans {t \sin a t} } s = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$ | We have:
{{begin-eqn}}
{{eqn | l = \laptrans {\cos a t}
| r = \int_0^\infty e^{-s t} \cos a t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | n = 1
| r = \dfrac s {s^2 + a^2}
| c = [[Laplace Transform of Cosine]]
}}
{{end-eqn}}
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d a} \int_0^\inf... | Laplace Transform of t sine a t/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t_sine_a_t/Proof_2 | [
"Laplace Transform of t sine a t",
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Laplace Transform of Cosine",
"Derivative of Integral",
"Derivative of Cosine Function",
"Quotient Rule for Derivatives"
] |
proofwiki-22556 | Laplace Transform of a t cosh a t - sinh a t over 2 a^3 | :$\map {\laptrans {\dfrac {a t \cosh a t - \sinh a t} {2 a^3} } } s = \dfrac 1 {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {a t \cosh a t - \sinh a t} {2 a^3} } } s
| r = \dfrac 1 {2 a^3} \paren {a \map {\laptrans {t \cosh a t} } s - \map {\laptrans {\sinh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {a \map {\laptrans {t \cosh ... | :$\map {\laptrans {\dfrac {a t \cosh a t - \sinh a t} {2 a^3} } } s = \dfrac 1 {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {a t \cosh a t - \sinh a t} {2 a^3} } } s
| r = \dfrac 1 {2 a^3} \paren {a \map {\laptrans {t \cosh a t} } s - \map {\laptrans {\sinh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a^3} \paren {a \map {\laptrans {t \c... | Laplace Transform of a t cosh a t - sinh a t over 2 a^3 | https://proofwiki.org/wiki/Laplace_Transform_of_a_t_cosh_a_t_-_sinh_a_t_over_2_a^3 | https://proofwiki.org/wiki/Laplace_Transform_of_a_t_cosh_a_t_-_sinh_a_t_over_2_a^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of t cosh a t",
"Definition:Common Denominator"
] |
proofwiki-22557 | Laplace Transform of t sinh a t over 2 a | :$\map {\laptrans {\dfrac {t \sinh a t} {2 a} } } s = \dfrac s {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sinh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t \sinh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac {2 a s} {\paren {s^2 - a^2}^2} }
| c = Laplace Transform of $t \sinh a... | :$\map {\laptrans {\dfrac {t \sinh a t} {2 a} } } s = \dfrac s {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sinh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t \sinh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac {2 a s} {\paren {s^2 - a^2}^2} }
| c = [[Laplace Transform of t si... | Laplace Transform of t sinh a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_t_sinh_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_t_sinh_a_t_over_2_a | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sinh a t"
] |
proofwiki-22558 | Laplace Transform of t sinh a t | :$\map {\laptrans {t \sinh a t} } s = \dfrac {2 a s} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sinh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 - a^2} }
| c = Laplace Transform of Hyperbolic Sine
}}
{{eqn | r = -\dfrac {\paren... | :$\map {\laptrans {t \sinh a t} } s = \dfrac {2 a s} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\sinh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac a {s^2 - a^2} }
| c = [[Laplace Transform of Hyperbolic Sine]]
}}
{{eqn | r = -\dfrac... | Laplace Transform of t sinh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t_sinh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t_sinh_a_t | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of Hyperbolic Sine",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Category:Laplace Transforms involving Hyperbolic Sine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Category:Examples of Laplace Transform... |
proofwiki-22559 | Laplace Transform of sinh a t + a t cosh a t over 2 a | :$\map {\laptrans {\dfrac {\sinh a t + a t \cosh a t} {2 a} } } s = \dfrac {s^2} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sinh a t + a t \cosh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sinh a t} } s + a \map {\laptrans {t \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 - a^2} + a \map {\l... | :$\map {\laptrans {\dfrac {\sinh a t + a t \cosh a t} {2 a} } } s = \dfrac {s^2} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\sinh a t + a t \cosh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {\sinh a t} } s + a \map {\laptrans {t \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {\dfrac a {s^2 - a^2} + a \map... | Laplace Transform of sinh a t + a t cosh a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_sinh_a_t_+_a_t_cosh_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_sinh_a_t_+_a_t_cosh_a_t_over_2_a | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of t cosh a t",
"Definition:Common Denominator"
] |
proofwiki-22560 | Submodule is Subgroup | Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {G, +_G}$ be an abelian group.
Let $\struct {G, +_G, \circ_G}_R$ be an $R$-module.
Let $\struct {H, +_H, \circ_H}_R$ be an $R$-submodule of $\struct {G, +_G, \circ_G}_R$.
Then $H$ is a subgroup of $G$. | By definition of an $R$-submodule, $H$ must be a subset of $G$.
By definition of an $R$-submodule, $\struct {H, +_H, \circ_H}_R$ must be a $R$-module.
By definition of an $R$-module, $H$ must be a group.
Hence, by definition of a subgroup, $H$ is a subgroup of $G$.
{{qed}}
<!-- no sources because most authors require a... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $\struct {G, +_G}$ be an [[Definition:Abelian Group|abelian group]].
Let $\struct {G, +_G, \circ_G}_R$ be an [[Definition:Module over Ring|$R$-module]].
Let $\struct {H, +_H, \circ_H}_R$ be an [[Definition:Submodule|$R$-submodule]] of ... | By definition of an [[Definition:Submodule|$R$-submodule]], $H$ must be a [[Definition:Subset|subset]] of $G$.
By definition of an [[Definition:Submodule|$R$-submodule]], $\struct {H, +_H, \circ_H}_R$ must be a [[Definition:Module over Ring|$R$-module]].
By definition of an [[Definition:Module over Ring|$R$-module]],... | Submodule is Subgroup | https://proofwiki.org/wiki/Submodule_is_Subgroup | https://proofwiki.org/wiki/Submodule_is_Subgroup | [
"Submodules",
"Subgroups"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Abelian Group",
"Definition:Module over Ring",
"Definition:Submodule",
"Definition:Subgroup"
] | [
"Definition:Submodule",
"Definition:Subset",
"Definition:Submodule",
"Definition:Module over Ring",
"Definition:Module over Ring",
"Definition:Group",
"Definition:Subgroup",
"Definition:Subgroup",
"Category:Submodules",
"Category:Subgroups"
] |
proofwiki-22561 | Laplace Transform of cosh a t + half a t sinh a t | :$\map {\laptrans {\cosh a t + \dfrac 1 2 a t \sinh a t} } s = \dfrac {s^3} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cosh a t + \dfrac 1 2 a t \sinh a t} } s
| r = \map {\laptrans {\cosh a t} } s + \dfrac a 2 \map {\laptrans {t \sinh a t} } s
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac s {s^2 - a^2} + \dfrac a 2 \map {\laptrans {t \sinh a t} } s
| ... | :$\map {\laptrans {\cosh a t + \dfrac 1 2 a t \sinh a t} } s = \dfrac {s^3} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\cosh a t + \dfrac 1 2 a t \sinh a t} } s
| r = \map {\laptrans {\cosh a t} } s + \dfrac a 2 \map {\laptrans {t \sinh a t} } s
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac s {s^2 - a^2} + \dfrac a 2 \map {\laptrans {t \sinh a t} } s
... | Laplace Transform of cosh a t + half a t sinh a t | https://proofwiki.org/wiki/Laplace_Transform_of_cosh_a_t_+_half_a_t_sinh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_cosh_a_t_+_half_a_t_sinh_a_t | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transform of t sinh a t",
"Definition:Common Denominator"
] |
proofwiki-22562 | Laplace Transform of t cosh a t | :$\map {\laptrans {t \cosh a t} } s = \dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cosh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 - a^2} }
| c = Laplace Transform of Hyperbolic Cosine
}}
{{eqn | r = -\dfrac {\par... | :$\map {\laptrans {t \cosh a t} } s = \dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {\map {\laptrans {-\cosh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = -\map {\dfrac \d {\d s} } {\dfrac s {s^2 - a^2} }
| c = [[Laplace Transform of Hyperbolic Cosine]]
}}
{{eqn | r = -\dfr... | Laplace Transform of t cosh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t_cosh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t_cosh_a_t | [
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of Hyperbolic Cosine",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives"
] |
proofwiki-22563 | Size of Maximal Planar Graph | Let $n$ be the order of a simple planar graph $G$ such that $n \ge 3$.
Let $\size G$ denote the size of $G$.
Then:
:$\size G = 3 n - 6$
{{iff}}:
:$G$ is maximal.
That is, a simple planar graph with $n$ vertices can have no more than $3 n - 6$ edges. | {{ProofWanted}}
Category:Maximal Planar Graphs
3w0m06we1np9zjn4zjr6kac731ylv21 | Let $n$ be the [[Definition:Order of Graph|order]] of a [[Definition:Simple|simple]] [[Definition:Planar Graph|planar graph]] $G$ such that $n \ge 3$.
Let $\size G$ denote the [[Definition:Size of Graph|size]] of $G$.
Then:
:$\size G = 3 n - 6$
{{iff}}:
:$G$ is [[Definition:Maximal Planar Graph|maximal]].
That is,... | {{ProofWanted}}
[[Category:Maximal Planar Graphs]]
3w0m06we1np9zjn4zjr6kac731ylv21 | Size of Maximal Planar Graph | https://proofwiki.org/wiki/Size_of_Maximal_Planar_Graph | https://proofwiki.org/wiki/Size_of_Maximal_Planar_Graph | [
"Maximal Planar Graphs"
] | [
"Definition:Graph (Graph Theory)/Order",
"Definition:Simple",
"Definition:Planar Graph",
"Definition:Graph (Graph Theory)/Size",
"Definition:Maximal Planar Graph",
"Definition:Simple",
"Definition:Planar Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge"
] | [
"Category:Maximal Planar Graphs"
] |
proofwiki-22564 | Laplace Transform of t^2 sine a t | :$\map {\laptrans {t^2 \sin a t} } s = 2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \sin a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \sin a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {-2 a s} {\paren {s^2 + a^2}^2} }
| c = Laplace Transform of $t \sin a t$
}}
{{eqn | r =... | :$\map {\laptrans {t^2 \sin a t} } s = 2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \sin a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \sin a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {-2 a s} {\paren {s^2 + a^2}^2} }
| c = [[Laplace Transform of t sine a t|Laplace Tr... | Laplace Transform of t^2 sine a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sine_a_t | [
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t sine a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Sine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Category:Example... |
proofwiki-22565 | Inverse Laplace Transform of 1 over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t - 3 a t \cos a t} {8 a^5} } } s = \dfrac 1 {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t - 3 a t \cos a t} {8 a^5} } } s
| r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s - 3 a \map {\laptrans {t \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
... | :$\map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t - 3 a t \cos a t} {8 a^5} } } s = \dfrac 1 {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t - 3 a t \cos a t} {8 a^5} } } s
| r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s - 3 a \map {\laptrans {t \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]... | Inverse Laplace Transform of 1 over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Sine",
"Laplace Transform of t cosine a t",
"Laplace Transform of t^2 sine a t",
"Definition:Common Denominator"
] |
proofwiki-22566 | Laplace Transform of t^2 cosine a t | :$\map {\laptrans {t^2 \cos a t} } s = \dfrac {2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cos a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cos a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2} }
| c = Laplace Transform of $t \cos a t$
}}
{{eqn |... | :$\map {\laptrans {t^2 \cos a t} } s = \dfrac {2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cos a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cos a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {s^2 - a^2} {\paren {s^2 + a^2}^2} }
| c = [[Laplace Transform of t cosine a t|Lapl... | Laplace Transform of t^2 cosine a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosine_a_t | [
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t cosine a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function"
] |
proofwiki-22567 | Inverse Laplace Transform of s over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {t \sin a t - a t^2 \cos a t} {8 a^3} } } s = \dfrac s {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sin a t - a t^2 \cos a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {t \sin a t} } s - a \map {\laptrans {t^2 \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s} {\paren ... | :$\map {\laptrans {\dfrac {t \sin a t - a t^2 \cos a t} {8 a^3} } } s = \dfrac s {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t \sin a t - a t^2 \cos a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {t \sin a t} } s - a \map {\laptrans {t^2 \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {\dfrac {2 a s} {\pa... | Inverse Laplace Transform of s over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sine a t",
"Laplace Transform of t^2 cosine a t",
"Definition:Common Denominator"
] |
proofwiki-22568 | Inverse Laplace Transform of s^2 over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {\paren {1 + a^2 t^2} \sin a t - a t \cos a t} {8 a^3} } } s = \dfrac {s^2} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {1 + a^2 t^2} \sin a t - a t \cos a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {\sin a t} } s + a^2 \map {\laptrans {t^2 \sin a t} } s - a \map {\laptrans {t \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn ... | :$\map {\laptrans {\dfrac {\paren {1 + a^2 t^2} \sin a t - a t \cos a t} {8 a^3} } } s = \dfrac {s^2} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {1 + a^2 t^2} \sin a t - a t \cos a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {\map {\laptrans {\sin a t} } s + a^2 \map {\laptrans {t^2 \sin a t} } s - a \map {\laptrans {t \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{... | Inverse Laplace Transform of s^2 over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Sine",
"Laplace Transform of t cosine a t",
"Laplace Transform of t^2 sine a t",
"Definition:Common Denominator"
] |
proofwiki-22569 | Inverse Laplace Transform of s^3 over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {3 t \sin a t + a t^2 \cos a t} {8 a} } } s = \dfrac {s^3} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sin a t + a t^2 \cos a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sin a t} } s + a \map {\laptrans {t^2 \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac {2 a s} {\paren ... | :$\map {\laptrans {\dfrac {3 t \sin a t + a t^2 \cos a t} {8 a} } } s = \dfrac {s^3} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sin a t + a t^2 \cos a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sin a t} } s + a \map {\laptrans {t^2 \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac {2 a s} {\pa... | Inverse Laplace Transform of s^3 over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sine a t",
"Laplace Transform of t^2 cosine a t",
"Definition:Common Denominator"
] |
proofwiki-22570 | Inverse Laplace Transform of s^4 over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t + 5 a t \cos a t} {8 a} } } s = \dfrac {s^4} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t + 5 a t \cos a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s + 5 a \map {\laptrans {t \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eq... | :$\map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t + 5 a t \cos a t} {8 a} } } s = \dfrac {s^4} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 - a^2 t^2} \sin a t + 5 a t \cos a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sin a t} } s - a^2 \map {\laptrans {t^2 \sin a t} } s + 5 a \map {\laptrans {t \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
... | Inverse Laplace Transform of s^4 over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^4_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^4_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Sine",
"Laplace Transform of t cosine a t",
"Laplace Transform of t^2 sine a t",
"Definition:Common Denominator"
] |
proofwiki-22571 | Inverse Laplace Transform of s^5 over (s^2 + a^2)^3 | :$\map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s = \dfrac {s^5} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s
| r = \dfrac 1 8 \paren {8 \map {\laptrans {\cos a t} } s - a^2 \map {\laptrans {t^2 \cos a t} } s - 7 a \map {\laptrans {t \sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \... | :$\map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s = \dfrac {s^5} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 - a^2 t^2} \cos a t - 7 a t \sin a t} 8} } s
| r = \dfrac 1 8 \paren {8 \map {\laptrans {\cos a t} } s - a^2 \map {\laptrans {t^2 \cos a t} } s - 7 a \map {\laptrans {t \sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r... | Inverse Laplace Transform of s^5 over (s^2 + a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^5_over_(s^2_+_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^5_over_(s^2_+_a^2)^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Cosine",
"Laplace Transform of t sine a t",
"Laplace Transform of t^2 cosine a t",
"Definition:Common Denominator"
] |
proofwiki-22572 | Laplace Transform of t^2 sine a t over 2 a | :$\map {\laptrans {\dfrac {t^2 \sin a t} {2 a} } } s = \dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \sin a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t^2 \sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} } }
| c = Laplace T... | :$\map {\laptrans {\dfrac {t^2 \sin a t} {2 a} } } s = \dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \sin a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t^2 \sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} } }
| c = [[Lap... | Laplace Transform of t^2 sine a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sine_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sine_a_t_over_2_a | [
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^2 sine a t"
] |
proofwiki-22573 | Laplace Transform of t^2 cosine a t over 2 | :$\map {\laptrans {\dfrac {t^2 \cos a t} 2} } s = \dfrac {s^3 - 3 a^2 s} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \cos a t} 2} } s
| r = \dfrac 1 2 \paren {\map {\laptrans {t^2 \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac {2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3} }
| c = Laplace Transform of $... | :$\map {\laptrans {\dfrac {t^2 \cos a t} 2} } s = \dfrac {s^3 - 3 a^2 s} {\paren {s^2 + a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \cos a t} 2} } s
| r = \dfrac 1 2 \paren {\map {\laptrans {t^2 \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac {2 s \paren {s^2 - 3 a^2} } {\paren {s^2 + a^2}^3} }
| c = [[Laplace Transfor... | Laplace Transform of t^2 cosine a t over 2 | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosine_a_t_over_2 | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosine_a_t_over_2 | [
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^2 cosine a t"
] |
proofwiki-22574 | Laplace Transform of t^3 cosine a t | :$\map {\laptrans {t^3 \cos a t} } s = \dfrac {6 \paren {s^4 - 6 a^2 s^2 + a^4} } {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cos a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cos a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3} }
| c = Laplace Transform of $t^2 \cos a t$
... | :$\map {\laptrans {t^3 \cos a t} } s = \dfrac {6 \paren {s^4 - 6 a^2 s^2 + a^4} } {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cos a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cos a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3} }
| c = [[Laplace Transform of t^2 cosin... | Laplace Transform of t^3 cosine a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosine_a_t | [
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t^2 cosine a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Cosine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Category:E... |
proofwiki-22575 | Laplace Transform of t^3 cosine a t over 6 | :$\map {\laptrans {\dfrac {t^3 \cos a t} 6} } s = \dfrac {s^4 - 6 a^2 s^2 + a^4} {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cos a t} 6} } s
| r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cos a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 - 6 a^2 s^2 + a^4} } {\paren {s^2 + a^2}^4} }
| c = Laplace Transf... | :$\map {\laptrans {\dfrac {t^3 \cos a t} 6} } s = \dfrac {s^4 - 6 a^2 s^2 + a^4} {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cos a t} 6} } s
| r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cos a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 - 6 a^2 s^2 + a^4} } {\paren {s^2 + a^2}^4} }
| c = [[Laplace ... | Laplace Transform of t^3 cosine a t over 6 | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosine_a_t_over_6 | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosine_a_t_over_6 | [
"Laplace Transforms involving Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^3 cosine a t"
] |
proofwiki-22576 | Laplace Transform of t^3 sine a t over 24 a | :$\map {\laptrans {\dfrac {t^3 \sin a t} {24 a} } } s = \dfrac {s^3 - a^2 s} {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sin a t} {24 a} } } s
| r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sin a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {24 a} \paren {\dfrac {24 a s \paren {s^2 - a^2} } {\paren {s^2 + a^2}^4} }
| c = Lapla... | :$\map {\laptrans {\dfrac {t^3 \sin a t} {24 a} } } s = \dfrac {s^3 - a^2 s} {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sin a t} {24 a} } } s
| r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sin a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {24 a} \paren {\dfrac {24 a s \paren {s^2 - a^2} } {\paren {s^2 + a^2}^4} }
| c = [... | Laplace Transform of t^3 sine a t over 24 a | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sine_a_t_over_24_a | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sine_a_t_over_24_a | [
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^3 sine a t"
] |
proofwiki-22577 | Laplace Transform of t^3 sine a t | :$\map {\laptrans {t^3 \sin a t} } s = \dfrac {24 a s \paren {s^2 - a^2} } {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sin a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sin a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} } }
| c = Laplace Transform of $t^2 ... | :$\map {\laptrans {t^3 \sin a t} } s = \dfrac {24 a s \paren {s^2 - a^2} } {\paren {s^2 + a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sin a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sin a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-2 a \paren {\dfrac {3 s^2 - a^2} {\paren {s^2 + a^2}^3} } }
| c = [[Laplace Transform of... | Laplace Transform of t^3 sine a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sine_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sine_a_t | [
"Laplace Transforms involving Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t^2 sine a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Sine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Category:Examp... |
proofwiki-22578 | Inverse Laplace Transform of 1 over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s = \dfrac 1 {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s
| r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s - 3 a \map {\laptrans {t \cosh a t} } s}
| c = Linear Combination of Laplace Transform... | :$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s = \dfrac 1 {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t - 3 a t \cosh a t} {8 a^5} } } s
| r = \dfrac 1 {8 a^5} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s - 3 a \map {\laptrans {t \cosh a t} } s}
| c = [[Linear Combination of Laplace Transfo... | Inverse Laplace Transform of 1 over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of t cosh a t",
"Laplace Transform of t^2 sinh a t",
"Definition:Common Denominator"
] |
proofwiki-22579 | Inverse Laplace Transform of s over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s = \dfrac s {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {a \map {\laptrans {t^2 \cosh a t} - \map {\laptrans {t \sinh a t} } s} s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a \map {\laptrans {... | :$\map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s = \dfrac s {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {a t^2 \cosh a t - t \sinh a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {a \map {\laptrans {t^2 \cosh a t} - \map {\laptrans {t \sinh a t} } s} s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {8 a^3} \paren {a \map {\laptra... | Inverse Laplace Transform of s over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sinh a t",
"Laplace Transform of t^2 cosh a t",
"Definition:Common Denominator"
] |
proofwiki-22580 | Laplace Transform of t^2 cosh a t | :$\map {\laptrans {t^2 \cosh a t} } s = \dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cosh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2} }
| c = Laplace Transform of $t \cosh a t$
}}
{{eq... | :$\map {\laptrans {t^2 \cosh a t} } s = \dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \cosh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {s^2 + a^2} {\paren {s^2 - a^2}^2} }
| c = [[Laplace Transform of t cosh a t|Lapl... | Laplace Transform of t^2 cosh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosh_a_t | [
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t cosh a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Hyperbolic Cosine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Cat... |
proofwiki-22581 | Inverse Laplace Transform of s^2 over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s = \dfrac {s^2} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {a^2 \map {\laptrans {t^2 \sinh a t} } s - \map {\laptrans {\sinh a t} } s + a \map {\laptrans {t \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{... | :$\map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s = \dfrac {s^2} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {a^2 t^2 - 1} \sinh a t + a t \cosh a t} {8 a^3} } } s
| r = \dfrac 1 {8 a^3} \paren {a^2 \map {\laptrans {t^2 \sinh a t} } s - \map {\laptrans {\sinh a t} } s + a \map {\laptrans {t \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
... | Inverse Laplace Transform of s^2 over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of t cosh a t",
"Laplace Transform of t^2 sinh a t",
"Definition:Common Denominator"
] |
proofwiki-22582 | Poisson Bracket satisfies Jacobi Identity | For $n \in \N$, let:
:$\map A {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$
:$\map B {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$
be real functions, dependent on canonical variables.
Let $\sqbrk {A, B}$ denote the '''Poisson bracket''' of $A$ and $B$.
Then:
:$\sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} }... | The Poisson bracket of two functions $f$ and $g$ on a phase space with coordinates $\tuple {\mathbf q, \mathbf p, t} = \tuple {q_1, \dots, q_n, p_1, \dots, p_n, t}$ is defined as:
:$\ds \sqbrk {f, g} = \sum_i \paren {\frac {\partial f} {\partial q_i} \frac {\partial g} {\partial p_i} - \frac {\partial f} {\partial p_i}... | For $n \in \N$, let:
:$\map A {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$
:$\map B {x, \mathbf y, \mathbf p}: \R^{2 n + 1} \to \R$
be [[Definition:Real Function|real functions]], dependent on [[Definition:Canonical Variable|canonical variables]].
Let $\sqbrk {A, B}$ denote the '''[[Definition:Poisson Bracket|Po... | The [[Definition:Poisson Bracket|Poisson bracket]] of two functions $f$ and $g$ on a [[Definition:Phase Space|phase space]] with [[Definition:Coordinate of Ordered Tuple|coordinates]] $\tuple {\mathbf q, \mathbf p, t} = \tuple {q_1, \dots, q_n, p_1, \dots, p_n, t}$ is defined as:
:$\ds \sqbrk {f, g} = \sum_i \paren {\f... | Poisson Bracket satisfies Jacobi Identity/Proof 1 | https://proofwiki.org/wiki/Poisson_Bracket_satisfies_Jacobi_Identity | https://proofwiki.org/wiki/Poisson_Bracket_satisfies_Jacobi_Identity/Proof_1 | [
"Poisson Bracket satisfies Jacobi Identity",
"Poisson Brackets",
"Jacobi Identity"
] | [
"Definition:Real Function",
"Definition:Canonical Variable",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Jacobi Identity"
] | [
"Definition:Poisson Bracket",
"Definition:Phase Space",
"Definition:Cartesian Product/Coordinate",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Jacobi Identity"
] |
proofwiki-22583 | Inverse Laplace Transform of s^3 over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s = \dfrac {s^3} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sinh a t} } s + a \map {\laptrans {t^2 \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac {2 a s} {\pa... | :$\map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s = \dfrac {s^3} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {3 t \sinh a t + a t^2 \cosh a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {t \sinh a t} } s + a \map {\laptrans {t^2 \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {8 a} \paren {3 \dfrac {2 a s} ... | Inverse Laplace Transform of s^3 over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^3_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t sinh a t",
"Laplace Transform of t^2 cosh a t",
"Definition:Common Denominator"
] |
proofwiki-22584 | Inverse Laplace Transform of s^4 over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s = \dfrac {s^4} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s + 5 a \map {\laptrans {t \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}... | :$\map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s = \dfrac {s^4} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {3 + a^2 t^2} \sinh a t + 5 a t \cosh a t} {8 a} } } s
| r = \dfrac 1 {8 a} \paren {3 \map {\laptrans {\sinh a t} } s + a^2 \map {\laptrans {t^2 \sinh a t} } s + 5 a \map {\laptrans {t \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]... | Inverse Laplace Transform of s^4 over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^4_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^4_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Sine",
"Laplace Transform of t cosh a t",
"Laplace Transform of t^2 sinh a t",
"Definition:Common Denominator"
] |
proofwiki-22585 | Quantum Mechanics Commutator satisfies Jacobi Identity | Let $A, B, C$ be operators acting on the Hilbert Space of some quantum system.
Let $\sqbrk {\, \cdot, \cdot \,}$ denotes the commutator of these operators.
Then:
:$\sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} } + \sqbrk {C, \sqbrk {A, B} } = 0$
That is, the commutator satisfies the Jacobi identity. | By expanding the definition of the commutator:
:$\sqbrk {A, B} = AB - BA$
into the definition of Jacobi identity, we obtain:
{{begin-eqn}}
{{eqn | l = \sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} } + \sqbrk {C, \sqbrk {A, B} }
| r = \sqbrk {A, \paren {B C - C B} } + \sqbrk {B, \paren {C A - A C} } + \sqb... | Let $A, B, C$ be [[Definition:Linear Operator|operators]] acting on the [[Definition:Hilbert Space|Hilbert Space]] of some [[Definition:Quantum Mechanics|quantum system]].
Let $\sqbrk {\, \cdot, \cdot \,}$ denotes the [[Definition:Commutator on Algebra|commutator]] of these [[Definition:Linear Operator|operators]].
... | By expanding the definition of the [[Definition:Commutator on Algebra|commutator]]:
:$\sqbrk {A, B} = AB - BA$
into the definition of [[Definition:Jacobi Identity|Jacobi identity]], we obtain:
{{begin-eqn}}
{{eqn | l = \sqbrk {A, \sqbrk {B, C} } + \sqbrk {B, \sqbrk {C, A} } + \sqbrk {C, \sqbrk {A, B} }
| r = \sqb... | Quantum Mechanics Commutator satisfies Jacobi Identity | https://proofwiki.org/wiki/Quantum_Mechanics_Commutator_satisfies_Jacobi_Identity | https://proofwiki.org/wiki/Quantum_Mechanics_Commutator_satisfies_Jacobi_Identity | [
"Jacobi Identity",
"Quantum Mechanics"
] | [
"Definition:Linear Operator",
"Definition:Hilbert Space",
"Definition:Quantum Mechanics",
"Definition:Commutator/Algebra",
"Definition:Linear Operator",
"Definition:Commutator/Algebra",
"Definition:Jacobi Identity"
] | [
"Definition:Commutator/Algebra",
"Definition:Jacobi Identity",
"Definition:Commutator/Algebra",
"Definition:Jacobi Identity"
] |
proofwiki-22586 | Inverse Laplace Transform of s^5 over (s^2 - a^2)^3 | :$\map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s = \dfrac {s^5} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s
| r = \dfrac 1 8 \paren {8 \map {\laptrans {\cosh a t} } s + a^2 \map {\laptrans {t^2 \cosh a t} } s + 7 a \map {\laptrans {t \sinh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | ... | :$\map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s = \dfrac {s^5} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {\paren {8 + a^2 t^2} \cosh a t + 7 a t \sinh a t} 8} } s
| r = \dfrac 1 8 \paren {8 \map {\laptrans {\cosh a t} } s + a^2 \map {\laptrans {t^2 \cosh a t} } s + 7 a \map {\laptrans {t \sinh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eq... | Inverse Laplace Transform of s^5 over (s^2 - a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^5_over_(s^2_-_a^2)^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^5_over_(s^2_-_a^2)^3 | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Hyperbolic Cosine",
"Laplace Transform of t sinh a t",
"Laplace Transform of t^2 cosh a t",
"Definition:Common Denominator"
] |
proofwiki-22587 | Laplace Transform of t^2 sinh a t over 2 a | :$\map {\laptrans {\dfrac {t^2 \sinh a t} {2 a} } } s = \dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \sinh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t^2 \sinh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {2 a} \paren {2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } }
| c = Laplace... | :$\map {\laptrans {\dfrac {t^2 \sinh a t} {2 a} } } s = \dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \sinh a t} {2 a} } } s
| r = \dfrac 1 {2 a} \paren {\map {\laptrans {t^2 \sinh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {2 a} \paren {2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } }
| c = [[L... | Laplace Transform of t^2 sinh a t over 2 a | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sinh_a_t_over_2_a | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sinh_a_t_over_2_a | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^2 sinh a t"
] |
proofwiki-22588 | Poisson Bracket satisfies Jacobi Identity/Proof 1 | For $n \in \N$, let:
:$\map f {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map g {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map h {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
be real functions, dependent on time and the canonical variables $\tuple {\mathbf q, \mathbf p}$.
Let $\sqbrk {f, g}$ denote the... | The Poisson bracket of two functions $f$ and $g$ on a phase space with coordinates $\tuple {\mathbf q, \mathbf p, t} = \tuple {q_1, \dots, q_n, p_1, \dots, p_n, t}$ is defined as:
:$\ds \sqbrk {f, g} = \sum_i \paren {\frac {\partial f} {\partial q_i} \frac {\partial g} {\partial p_i} - \frac {\partial f} {\partial p_i}... | For $n \in \N$, let:
:$\map f {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map g {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
:$\map h {\mathbf q, \mathbf p, t}: \R^{2 n + 1} \to \R$
be [[Definition:Real Function|real functions]], dependent on time and the [[Definition:Canonical Variable|canonical variables]] ... | The [[Definition:Poisson Bracket|Poisson bracket]] of two functions $f$ and $g$ on a [[Definition:Phase Space|phase space]] with [[Definition:Coordinate of Ordered Tuple|coordinates]] $\tuple {\mathbf q, \mathbf p, t} = \tuple {q_1, \dots, q_n, p_1, \dots, p_n, t}$ is defined as:
:$\ds \sqbrk {f, g} = \sum_i \paren {\f... | Poisson Bracket satisfies Jacobi Identity/Proof 1 | https://proofwiki.org/wiki/Poisson_Bracket_satisfies_Jacobi_Identity/Proof_1 | https://proofwiki.org/wiki/Poisson_Bracket_satisfies_Jacobi_Identity/Proof_1 | [
"Poisson Bracket satisfies Jacobi Identity"
] | [
"Definition:Real Function",
"Definition:Canonical Variable",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Jacobi Identity"
] | [
"Definition:Poisson Bracket",
"Definition:Phase Space",
"Definition:Cartesian Product/Coordinate",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Poisson Bracket",
"Definition:Jacobi Identity"
] |
proofwiki-22589 | Laplace Transform of t^2 sinh a t | :$\map {\laptrans {t^2 \sinh a t} } s = 2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \sinh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {-2 a s} {\paren {s^2 - a^2}^2} }
| c = Laplace Transform of $t \sinh a t$
}}
{{eqn | ... | :$\map {\laptrans {t^2 \sinh a t} } s = 2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} }$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^2 \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t \sinh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {\dfrac {-2 a s} {\paren {s^2 - a^2}^2} }
| c = [[Laplace Transform of t sinh a t|Laplace ... | Laplace Transform of t^2 sinh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sinh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_sinh_a_t | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t sinh a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Hyperbolic Sine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Categ... |
proofwiki-22590 | Laplace Transform of t^2 cosh a t over 2 | :$\map {\laptrans {\dfrac {t^2 \cosh a t} 2} } s = \dfrac {s^3 + 3 a^2 s} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \cosh a t} 2} } s
| r = \dfrac 1 2 \paren {\map {\laptrans {t^2 \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} }
| c = Laplace Transform of... | :$\map {\laptrans {\dfrac {t^2 \cosh a t} 2} } s = \dfrac {s^3 + 3 a^2 s} {\paren {s^2 - a^2}^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^2 \cosh a t} 2} } s
| r = \dfrac 1 2 \paren {\map {\laptrans {t^2 \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 2 \paren {\dfrac {2 s \paren {s^2 + 3 a^2} } {\paren {s^2 - a^2}^3} }
| c = [[Laplace Transf... | Laplace Transform of t^2 cosh a t over 2 | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosh_a_t_over_2 | https://proofwiki.org/wiki/Laplace_Transform_of_t^2_cosh_a_t_over_2 | [
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^2 cosh a t"
] |
proofwiki-22591 | Laplace Transform of t^3 cosh a t over 6 | :$\map {\laptrans {\dfrac {t^3 \cosh a t} 6} } s = \dfrac {s^4 + 6 a^2 s^2 + a^4} {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cosh a t} 6} } s
| r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cosh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4} }
| c = Laplace Tran... | :$\map {\laptrans {\dfrac {t^3 \cosh a t} 6} } s = \dfrac {s^4 + 6 a^2 s^2 + a^4} {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \cosh a t} 6} } s
| r = \dfrac 1 6 \paren {\map {\laptrans {t^3 \cosh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 6 \paren {\dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4} }
| c = [[Laplac... | Laplace Transform of t^3 cosh a t over 6 | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosh_a_t_over_6 | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosh_a_t_over_6 | [
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^3 cosh a t"
] |
proofwiki-22592 | Laplace Transform of t^3 cosh a t | :$\map {\laptrans {t^3 \cosh a t} } s = \dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cosh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {2 s^3 + 6 a^2 s} {\paren {s^2 - a^2}^3} }
| c = Laplace Transform of $t^2 \cosh a ... | :$\map {\laptrans {t^3 \cosh a t} } s = \dfrac {6 \paren {s^4 + 6 a^2 s^2 + a^4} } {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \cosh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \cosh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-\dfrac {2 s^3 + 6 a^2 s} {\paren {s^2 - a^2}^3} }
| c = [[Laplace Transform of t^2 cos... | Laplace Transform of t^3 cosh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_cosh_a_t | [
"Laplace Transforms involving Hyperbolic Cosine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t^2 cosh a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Hyperbolic Cosine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"C... |
proofwiki-22593 | Laplace Transform of t^3 sinh a t over 24 a | :$\map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s = \dfrac {s^3 + a^2 s} {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s
| r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sinh a t} } s}
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r = \dfrac 1 {24 a} \paren {\dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4} }
| c = Lap... | :$\map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s = \dfrac {s^3 + a^2 s} {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {t^3 \sinh a t} {24 a} } } s
| r = \dfrac 1 {24 a} \paren {\map {\laptrans {t^3 \sinh a t} } s}
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn | r = \dfrac 1 {24 a} \paren {\dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4} }
| c =... | Laplace Transform of t^3 sinh a t over 24 a | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sinh_a_t_over_24_a | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sinh_a_t_over_24_a | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of t^3 sinh a t"
] |
proofwiki-22594 | Laplace Transform of t^3 sinh a t | :$\map {\laptrans {t^3 \sinh a t} } s = \dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sinh a t} } s}
| c = Derivative of Laplace Transform
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } }
| c = Laplace Transform of $t^... | :$\map {\laptrans {t^3 \sinh a t} } s = \dfrac {24 a s \paren {s^2 + a^2} } {\paren {s^2 - a^2}^4}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {t^3 \sinh a t} } s
| r = \map {\dfrac \d {\d s} } {-\map {\laptrans {t^2 \sinh a t} } s}
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = \map {\dfrac \d {\d s} } {-2 a \paren {\dfrac {3 s^2 + a^2} {\paren {s^2 - a^2}^3} } }
| c = [[Laplace Transform ... | Laplace Transform of t^3 sinh a t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sinh_a_t | https://proofwiki.org/wiki/Laplace_Transform_of_t^3_sinh_a_t | [
"Laplace Transforms involving Hyperbolic Sine Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Laplace Transforms"
] | [] | [
"Derivative of Laplace Transform",
"Laplace Transform of t^2 sinh a t",
"Quotient Rule for Derivatives",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Category:Laplace Transforms involving Hyperbolic Sine Function",
"Category:Inverse Laplace Transforms of Rational Functions",
"Cat... |
proofwiki-22595 | Inverse Laplace Transform of 1 over s^3 + a^3 | :$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s = \dfrac 1 {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s
| r = \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{a t / 2} \... | :$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s = \dfrac 1 {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a^2} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{-3 a t / 2} } } } s
| r = \dfrac {\sqrt 3} {3 a^2} \map {\laptrans {e^{a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{a t / 2} \... | Inverse Laplace Transform of 1 over s^3 + a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^3_+_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^3_+_a^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Sine",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential",
"Definition:Common Denominator",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-22596 | Inverse Laplace Transform of s over s^3 + a^3 | :$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s = \dfrac s {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s
| r = \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{a t / 2} \sin \d... | :$\map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s = \dfrac s {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{a t / 2} } {3 a} \paren {\cos \dfrac {\sqrt 3} 2 a t + \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - e^{-3 a t / 2} } } } s
| r = \dfrac 1 {3 a} \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s + \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{a t / 2} \sin \d... | Inverse Laplace Transform of s over s^3 + a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^3_+_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^3_+_a^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential times Sine",
"Laplace Transform of Exponential",
"Definition:Common Denominator",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-22597 | Inverse Laplace Transform of s^2 over s^3 + a^3 | :$\map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s
| r = \dfrac 1 3 \map {\laptrans {e^{-a t} } } s + \dfrac 2 3 \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
| c = Linear Combination of Laplace Transforms
}}
{{eqn | r =... | :$\map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s = \dfrac {s^2} {s^3 + a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac 1 3 \paren {e^{-a t} + 2 e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } } s
| r = \dfrac 1 3 \map {\laptrans {e^{-a t} } } s + \dfrac 2 3 \map {\laptrans {e^{a t / 2} \cos \dfrac {\sqrt 3 a t} 2} } s
| c = [[Linear Combination of Laplace Transforms]]
}}
{{eqn |... | Inverse Laplace Transform of s^2 over s^3 + a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^3_+_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s^2_over_s^3_+_a^3 | [
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential",
"Definition:Common Denominator",
"Sum of Two Odd Powers/Examples/Sum of Two Cubes"
] |
proofwiki-22598 | Inverse Laplace Transform of 1 over s^3 - a^3 | :$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s = \dfrac 1 {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s
| r = \dfrac 1 {3 a^2} \map {\laptrans {e^{a t} } } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s - \dfra... | :$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s = \dfrac 1 {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a^2} \paren {e^{3 a t / 2} - \cos \dfrac {\sqrt 3} 2 a t - \sqrt 3 \sin \dfrac {\sqrt 3} 2 a t} } } s
| r = \dfrac 1 {3 a^2} \map {\laptrans {e^{a t} } } s - \dfrac 1 {3 a^2} \map {\laptrans {e^{-a t / 2} \cos \dfrac {\sqrt 3} 2 a t} } s - \dfra... | Inverse Laplace Transform of 1 over s^3 - a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^3_-_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_1_over_s^3_-_a^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential times Sine",
"Definition:Common Denominator",
"Difference of Two Powers/Examples/Difference of Two Cubes"
] |
proofwiki-22599 | Inverse Laplace Transform of s over s^3 - a^3 | :$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s = \dfrac s {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s
| r = \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{-a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} \cos ... | :$\map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s = \dfrac s {s^3 - a^3}$ | {{begin-eqn}}
{{eqn | l = \map {\laptrans {\dfrac {e^{-a t / 2} } {3 a} \paren {\sqrt 3 \sin \dfrac {\sqrt 3} 2 a t - \cos \dfrac {\sqrt 3} 2 a t + e^{3 a t / 2} } } } s
| r = \dfrac {\sqrt 3} {3 a} \map {\laptrans {e^{-a t / 2} \sin \dfrac {\sqrt 3} 2 a t} } s - \dfrac 1 {3 a} \map {\laptrans {e^{-a t / 2} \cos ... | Inverse Laplace Transform of s over s^3 - a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^3_-_a^3 | https://proofwiki.org/wiki/Inverse_Laplace_Transform_of_s_over_s^3_-_a^3 | [
"Laplace Transforms involving Sine Function",
"Laplace Transforms involving Cosine Function",
"Laplace Transforms involving Exponential Function",
"Inverse Laplace Transforms of Rational Functions",
"Examples of Inverse Laplace Transforms"
] | [] | [
"Linear Combination of Laplace Transforms",
"Laplace Transform of Exponential times Cosine",
"Laplace Transform of Exponential times Sine",
"Laplace Transform of Exponential",
"Definition:Common Denominator",
"Difference of Two Powers/Examples/Difference of Two Cubes"
] |
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