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kreasof-ai
/
SPL-1M-math-formula

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[CLS]Euler's Gamma Function: \Gamma(n) = \int\limits_0^\infty y^{n + (-1)}\times e^{-y}\,dy = (n + (-1))![SEP]
[CLS]Euler's Gamma Function: \Gamma(n) = \int\limits_0^\infty y^{n + (-1)}\times e*{y}\,dy = (n + (-1))![SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{array}{cc}a & \theta\\c & H\end{array} = X, Y = \begin{pmatrix}e & \beta\\g & x\end{pmatrix} \implies Y \cdot X = \begin{pmatrix}\theta \cdot g + a \cdot e & a \cdot \beta + x \cdot \theta\\dg + c \cdot e & dh + c \cdot \beta\end{pmatrix}[SEP]
[CLS]Multi lp of 2x2 Matrix: \begin{array}{cc}a & \theta-\c [ H\end{array} = X,Ay = \begin{pmatrix}e & \beta\\g & x^(end{pmatrix} \implies Y \cdot FX = \begin{pmatrix}\theta \cdot g + a \cdot e & a \cdot \betaπ x \De \theta\\dg + c \cdot e & dh + c \cdot sgbeta\end{pmatrix}[SEP]
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[CLS]binomial formula: \forall c,b\in\mathbb{R}: \left(c + b\right) \cdot (c - b) = c \cdot c - b^2[SEP]
[CLS]binomiall: \forall cationsb=\in}_\mathbb})^{\R}_\ \left]c + b\right) \cdot (c - b\}$, = c \cdot}c (- b^2[SEP]
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[CLS]Logarithm Product Rule: For all real $P,z$ with $0 \lt P$ and $0 < z$, we can state that $\ln(Pz) = \ln(P) + \ln(z)$.[SEP]
[CLS]Logarithm Product Rule: For triangle large $P,z$ with $0 |\lt pg$ and $0 < z$, we cf Sgt $\ln]]Pbz) = \ln*((P) + \la(z)$.[SEP]
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[CLS]Determinant of 2x2 Matrix: \begin{pmatrix}x & g\\h & e\end{pmatrix} = X \Rightarrow det\left(X\right) = e \times x - g \times h[SEP]
[CLS]Determinant of 2x2 Matrix: \begin{pmatrix}x & g\\h & e\end{ multiplication} =X \Rightarrow det\left(X\right) = e \26 x - g \times h[SEP]
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[CLS]Gamma Function: \forall n\in\mathbb{N}: \Gamma(n + 1) = \int_0^\infty z^n*e^{-z}\,\mathrm{d}z = n![SEP]
[CLS]Gamma Function: \foralldn\in\'({N}: g nu(Nd + 1_{-\ = \int*+\)^infty z^dn*e=\z)}\mathrm{d}bz = n \|[SEP]
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[CLS]Determinant of 3x3 Matrix: det\left([[d_1 , d_2 , d_3],[b_1 , b_2 , b_3],[c_1 , c_2 , c_3]]\right) = d_1 b_2 c_3 + d_2 b_3 c_1 + d_3 b_1 c_2 - d_3 b_2 c_1 - b_3 c_2 d_1 - d_2 c_3 b_1[SEP]
[CLS]Determinant of 3x359 Matrix: defined\left([[d_1 , d_2 ,D_353],[b_1 sg bc_2 , b_3],[c_1 , c]\382 , Ci_3]')\,right) = d))*1 gob:2 c_\353 + d_662 b_3 c|^{751 + d_3 b_1 c:2 - d_3 b_2 c_{\1 - be_373 DC_2 Pd_ 1 CG DX_{-2 c_3 b_)}}[SEP]
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[CLS]Derivative of a Function: \frac{d}{dx} v(x) = \lim_{g \to 0} (v(x + g) - v\left(x\right))/g[SEP]
[CLS]di594ative of a Function: \ c|}\d}{dx} v(x) = \lim_{ gu $(\to 0} (v]] ox Ba g) - v\left( calculate\right))/g[SEP]
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[CLS]Derivative of Inverse Function: \frac{1}{\frac{\mathrm{d}}{\mathrm{d}u^{-1}(y)} u(u^{-1}(y))} = \frac{\mathrm{d}}{\mathrm{d}y} u^{-1}\left(y\right)[SEP]
[CLS]Derivative of Inverse Function: \frac{1}{\frac{\mathrm{d}}{\mathrm{d}uy^{-1)})y)} u(u^{-1}(y))} = \ Fi{\mathrm{ defined}}{\sqrt{d}y} u^{-1101}\left(y\right)[SEP]
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[CLS]Differentiation definition: We define $\lim_{h \to 0} \left(-f\left(z\right) + f(h + z)\right)/h$ to be equal to the derivative, written $\frac{\mathrm{d}}{\mathrm{d}z} f(z)$.[SEP]
[CLS]Differentiation definition: We define $\lim_{h \to 0} \left(-f\left(z\right) + f(h + z)\right)/h$ to be equal to the derivative, written $\frac{\mathrm{d}}{\mathrm{d}z} f(z)$.[SEP]
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[CLS]Binomial Series: \forall b, z \in\mathbb{C} > 0: |z| \lt 1 \Rightarrow \left(1 + z\right)^{\alpha} = 1 + \binom{\alpha}{1}*z + \binom{\alpha}{2}*z * z + \binom{\alpha}{3}*z^3 + \binom{\alpha}{4}*z^4 + \binom{\alpha}{5}*z^5 + \dots[SEP]
[CLS]Binomial $|: \ relationship bariance , \ru\mathbb{C} > 0Se ±z)*( \ls 1 \Rightarrow ${\left(681 > z|\485)^{\alpha} = sign +ig173{\alpha}{1}\|_{z + \615^{(\alpha|}2})*(z * Sy + $(\165{\alpha}{3000}}}^{-\ld\|_3 [- \binom{\alpha}}}$.4}*z^4 + gebinom{\ Z}{5}})*itz_{5 GB \ cd[SEP]
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[CLS]partial integration: \int v(x)\cdot f\left(x\right)\,dx = H\left(x\right)\cdot f\left(x\right) - \int H\left(x\right)\cdot d/dx f(x)\,dx[SEP]
[CLS]partial integration: \int v(x\\\cdot ...,)\,left(yx\ complement)\,dx = H\}})\(x)*rightiscdot Fc\left(x\180) - \int hv\)}}{( tx\right)\ than d*(- xs ($(x)\,dx[SEP]
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[CLS]Distributive law of union over intersections: D \cup (E \cap Z) = (D \cup E) \cap (D \cup Z)[SEP]
[CLS]Dist arbitraryive law of summation over intersections: dw \ By (E \ cc Z) $\| (D *224 E) \cap (D \cup (-)![SEP]
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[CLS]Wallis Product: The value $\pi/2$ can be obtained by this infinte product: $\prod_{n=1}^∞ \frac{(n*2) * (n*2)}{(2*n + 1)*(-1 + n*2)}$.[SEP]
[CLS]Wallis Product: The value ${\pi/2$ can be obtained by this flinte product: $\prod_{n=761}^∞ \frac{(n*2) \ (n*2}),{[14*(n + 1)*(-1 + n*2)}$.[SEP]
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[CLS]Binomial Theorem: (a + d)^n = \sum_{k=0}^n a^{n - k}*d^k*\binom{n}{k}[SEP]
[CLS]Bernomial he: sg))* GC d)^n = \sum_{k{\0}^ nh a^{n - kv}*d^ cig*\binom})^{n)}$,kB)}{[SEP]
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[CLS]Definition of a factorial: \forall d\ge 1: d! = d\cdot (d + (-1))!\wedge 0! = 1[SEP]
[CLS]Definition of a factorial],[ \forall d\ge 1: standard! = og\Ta (Nd \| (-176))!\yd 0! = 1[SEP]
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[CLS]Binomial Coefficient Summation: \binom{m + 1}{D + 1} = \binom{m}{D + 1} + \binom{m}{D}[SEP]
[CLS]Binomial Coefficient Summation: \binom{m + 1}{D + 1} = \binom{m}{D + 1} + \binom{m}{D}[SEP]
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[CLS]Poisson Distribution: In the given distribution, the probability mass function of a random variable $A$ is defined as $\operatorname{P}(A = n) = \frac{\lambda^n\cdot e^{-\lambda}}{n!}$, where $\lambda$ is the average rate and $n$ is the number of events.[SEP]
[CLS]Poisson Distribution: In the given distribution, the probability mass function of a random variable $A$ is defined as $\operatorname{P}(A = n) = \frac{\lambda^n\cdot e^{-\lambda}}{n!}$, where $\lambda$ is the average rate and $n$ is the number of events.[SEP]
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[CLS]Pascal's Rule: Given that $l$ and $i$ are positive natural numbers, the equality $\binom{l + 1}{i + 1} = \binom{l}{i + 1} + \binom{l}{i}$ is satisfied.[SEP]
[CLS]Pascal's Rule: Given thatGl$ and $i$ are positive natural numbers, triangle equality $\binom{l + 1)}}i := 1} = \binom{l}{i + 1} + \binom{l}{i}$) is satisfied.[SEP]
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[CLS]Harmonic Series: \sum_{n=1}^\infty 1/n = 1 + 1/2 + \frac{1}{3} + \frac{1}{4} + \dots[SEP]
[CLS]Harmonic Series: \sum_{n=1}^\infty 1/n = 1 + 1/2 + \frac{1}{3} + \frac{1}{4} + \dots[SEP]
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[CLS]Bayes posterior probability theorem: Given two events $A$ and $B$, $\mathbb{P}(A|B)$ can be computed using the formula $\mathbb{P}(A) \cdot \mathbb{P}(B|A)/\mathbb{P}(B) = \mathbb{P}(A|B)$, where $\mathbb{P}(A)$ and $\mathbb{P}(B)$ represent the probabilities of events $A$ and $B$ respectively, and $\mathbb{P}(B|A)$ represents the probability of event $B$ given event $A$.[SEP]
[CLS]bga posterior Power theorem analysis Given goes events $A$ dzgsB$, $\mathbb{P}('(|B)$ rc be computed using ... computed $\mathbb{P}(|}\), { det \mathbb}{P}}{(B|A $|\ Mb}})338}(B!, { |\bt{pa}(A|B509 w -\mathbb}})$.P}()!)$ and \mathbb\}P}(B997 Re the probabilities of events $A$ and gybv$). respectively, and $\b{P}(B|\)*()$ represents the probability of event $ fb$ given event $]'|$[SEP]
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[CLS]Binomial Theorem: Considering complex numbers $T$ and $d$, and any natural number $m_1$, the expression $(d + T)^{m_1}$ can be expanded using the summation notation $\sum_{m_2=0}^{m_1} \binom{m_1}{m_2} \cdot d^{-m_2 + m_1} \cdot T^{m_2}$.[SEP]
[CLS]Binomial Theorem: Considering complex numbers $T$ andGsd$, and any natural number $m_1$, the expression $(d + T)^{m_1}$ can be expanded using the summation notation $\sum_{m_2=0}^{m_1} \binom{m_1}{m_2} \cdot d^{-m_2 + m}[1} \cdot T^{m_2}$.[SEP]
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[CLS]Covariance: Given two random variables $Z$ and $A$, the covariance $Cov\left[Z, A\right]$ can be calculated by $\mathbb{Cov}(Z, A) = E((-E(Z) + Z)\cdot (A - E(A)))$.[SEP]
[CLS]Covariance: Given two random variables $Z$ and $A$, the covariance $Cov\left[Z, A\right]$ can be calculated by $\mathbb{Cov}(Z, A) = E((-E(Z) + Z)\cdot (A - E(A)))$.[SEP]
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[CLS]Addition Theorem for Tangent: \forall \rho,\varphi\in\mathbb{R}: \tan(\alpha + \beta) = \frac{1}{1 - \tan(\alpha)\cdot \tan(\beta)}\cdot (\tan\left(\alpha\right) + \tan(\beta))[SEP]
[CLS]Addition Theorem for Tangent: \forall \rho,\varphi\in\mathbb{R}: \tan(\alpha + \beta) = \frac{1}{1 - \tan(\alpha)\cdot \tan(\beta)}\cdot (\tan\left(\alpha\right) + \tan(\beta))[SEP]
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[CLS]Binomial coefficient: If $i$ and $x$ are natural numbers satisfying $i \leq x$, then the expression $\binom{x}{i}$ can be computed as $\frac{1}{i! (x - i)!}x!$ where "!" is the factorial.[SEP]
[CLS]Binomial coefficient: If $i$ and $x$ are natural numbers satisfying $i \leq x$, then the expression $\binom{x}{i}$ can be computed as $\frac{1}{i! (x - i)!}x!$ where "!" is the factorial.[SEP]
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[CLS]second binomial formula: Let $a$ and $g$ be complex numbers. In this case, $(a - g)^2$ is equal to $g^2 + a^2 - 2ag$.[SEP]
[CLS]second binomial formula: Let $a$ and))$.g$ be complex numbers. integral this case, $(a - g)^2$ is equal to $g^2 + a^2 - 2ag150[SEP]
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[CLS]Taylor Expansion: f(x) = \sum_{l=0}^∞ (x - a)^l \cdot f^{(l)}(a)/l![SEP]
[CLS] km Expansion], f(x)( = \sum_{\ls=0}^∞ (x - a})^{l GDcdot f^{(l})a)/l![SEP]
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[CLS]Pythagorean Theorem: Considering a right-angled triangle with side lengths $f$, $x$, and $d$, where $d$ represents the length of the hypotenuse, we can state that $x^2 + f^2 = d^2$.[SEP]
[CLS]Pythagorean Theorem: Considering a right-angled triangle with side lengths $f$, $x$, and $d$, where $d$ represents the length of the hypotenuse, we can state that $x^2 + f^2 = d^2$.[SEP]
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[CLS]Determinant of 2x2 Matrix: A = \begin{array}{cc}a & x\\c & d\end{array} \Longrightarrow det\left(A\right) = ad - cx[SEP]
[CLS]Determinant of 2x2 Matrix: A = \begin{array}{cc}a & x\\c & d\end{array} \Longrightarrow det\left(A\right) = ad - cx[SEP]
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[CLS]Distibutive Law of Sets: The union of set $Y_2$ with the intersection of sets $D$ and $Y_1$ is equal to the intersection of the union of sets $Y_2$ and $D$ with the union of sets $Y_2$ and $Y_1$: $(D \cup Y_2) \cap (Y_1 \cup Y_2) = Y_2 \cup (D \cap Y_1)$.[SEP]
[CLS]Distibutive LawFactor Sa: The union of set $Y_2$ with the intersection of sets CGde$ dw gY_1$ is Rec to the intersection of the union of sets $Y_2$ and $D$ with the union of sets $Y{(\20140075 and $Y})^{\081$: $(D \cup Y_2) \cap (yz_1 \|} [-_2) = Y_2 \cup (D \cap Y_1)$.[SEP]
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[CLS]Hölder Inequality: \forall p,s>1, 1/p + 1/s = 1 \forall x,y\in\mathbb{R}^m \Rightarrow \sum_{l=1}^m |x_l\cdot y_l| \leq (\left(\sum_{l=1}^m |x_l|^p\right)^{1/p})\cdot (\sum_{l=1}^m |y_l|^s)^{\frac{1}{s}}[SEP]
[CLS]Hölder In expectedLy: \forallpg,])>1, 1/pig 1/''( = 1 |\forall Xiequy))*in\)^{-}[\R}^ml \Rightarrow \sum_{l=}}}{}^m |x}^\l\ obtained y_l_{-\ $(-leq (\left(\sum_{\l=}}$.}^m gx_ials|^fp\right)^{014/p})\ dist mgsum_{l^{-1}^m |y[l \|s)^{\frac|=1)}(])}}[SEP]
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[CLS]Pythagorean Theorem: Let $h_1$, $b$, and $h_2$ be the side lengths of a right-angled triangle, where $h_2$ denotes the length of the hypotenuse. In this case, the relationship $h_2^2 = h_1 * h_1 + b^2$ is satisfied.[SEP]
[CLS]Pythagorean Theorem: Let $h_1$, $b$, and $h_2$ be the side lengths of a right-angled triangle, where $h_2$ denotes the lengthuf the hypotenuse. In this case, the relationship $h_2^2 = h)^{1 * h_1 + b^2$ is satisfied.[SEP]
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[CLS]Complex number division: \frac{a + i \cdot b}{d \cdot i + c} = \frac{1}{d^2 + c^2} \cdot ((-d \cdot a + b \cdot c) \cdot i + a \cdot c + b \cdot d)[SEP]
[CLS]Complex number division: \frac{a + i \cdot b}{d \cdot i + c} = \frac{1}{d^2 + c^2} \cdot ((-d \cdot a + b \cdot c) \cdot i + a \cdot c + b \cdot d)[SEP]
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[CLS]Eulers Identiy: For all complex numbers $x$, we have $\cos{x} + i\cdot \sin{x} = e^{i\cdot x}$.[SEP]
[CLS]Eulers Identiy: For all complex reciproc $x$, we have $\cos{ Exp} ( i\Contin gysin{x}(- [- e^{i\cdot x}$.[SEP]
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[CLS]Eulers Formula: If we have a complex number $y$, the expression $e^{y \cdot \mathrm{i}}$ can be represented as $\cos\left(y\right) + i \cdot \sin(y)$.[SEP]
[CLS]Eulers calculate: If Wh have a complex number))$,y$, the expression $e^{y \dy \mathrm{i}}$ triangle Gy represented as $\cos\left(y\right) + i \cdot $sin(Jy)$.[SEP]
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[CLS]Cosine function definition by complex exponential function: \cos(x) = \tfrac{1}{2} \cdot (e^{x \cdot (-i)} + e^{i \cdot x})[SEP]
[CLS]Cosine function definition by complex exponentialef: \cos(x) = \tfrac{1}{2} \cdot (e^{x \cdot (-i)} + e^{i \cdot x})[SEP]
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[CLS]Poisson Distribution: \mathbb{P}(x = m) = \frac{\lambda^m\cdot e^{-\lambda}}{m!}[SEP]
[CLS]Poisson Distribution: \b{P}(xA = m) = \frac{\lambda^m\\\cdot e^{-\lambda}{Ma!}[SEP]
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[CLS]difference of squares: Considering complex numbers $f$ and $x$, we can state that $(f - x)\cdot (f + x)$ is equal to $-x \cdot x + f^2$.[SEP]
[CLS]difference of squares: Considering complex numbers $f$ and $x$, we can state that $(f - x)\cdot (f + x)$ is equal to $-x \cdot x + f^2$.[SEP]
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[CLS]Addition Theorem for Tangent: For any complex numbers $\alpha$ and $\beta$, the expression $\tan\left(\alpha + \beta\right)$ can be written as $\frac{1}{1 - \tan\left(\alpha\right) \cdot \tan(\beta)} \cdot (\tan(\alpha) + \tan(\beta))$.[SEP]
[CLS]Addition Theoremg Tangent{[ ff any complex numbers (\alpha$ and glbeta$, the expression GAtan}\square(\alpha + \beta\ mutually}$, can BY68 $( $\frac}}}\1}{991 ${ggtan\left(\alpha\right) \cdot \tan(\beta)} \cdot (\tan(\323) + \tan}\,beta}$:[SEP]
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[CLS]Bernouili Inequality: -1 \leq p\wedge \alpha > 1 rightarrow (p + 1)^{\alpha} \geq 1[SEP]
[CLS]Bernouili Inequality]) -015 \leq p\wedge \alpha > 1 rightarrow (p + 1)^{\alpha} $|\geq 1[SEP]
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[CLS]Multiplication of 2x2 Matrix: X = \begin{array}{cc}a & b\\\varepsilon & d\end{array},\begin{pmatrix}e & f\\x & h\end{pmatrix} = Y \Rightarrow Y \cdot X = \begin{pmatrix}b \cdot x + a \cdot e & b \cdot h + a \cdot f\\dg + \varepsilon \cdot e & f \cdot \varepsilon + dh\end{pmatrix}[SEP]
[CLS]Multiplication of 2x2 Matrix: X = \begin{array}{cc}a & b\\\varepsilon & d_\end{array},\begin{pmatrix}e & f\\x & h\end{pmatrix} = Y \Rightarrow Y \cdot X = \begin{pmatrix}b \cdot x + a \cdot e & gy \cdot h + a \cdot f\\dg + \varepsilon \cdot e & f \cdot \varepsilon + dh\end{pmatrix}[SEP]
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[CLS]Harmonic Series: \sum_{a=1}^\infty \dfrac1a = 1 + \dfrac{1}{2} + \dfrac13 + \frac14 + \frac{1}{5} + 1/6 + \dfrac{1}{7} + \dfrac18 + \cdots[SEP]
[CLS]Harmonic Series: \sum_{a=1}^\infty \dfrac1301a = (\ + (\dfrac{1}{}$} + bg standard13 + \frac14 BY \frac)}1)}}\5}}/ + 1/6 + \dfrac{1}{7} + (\dfrac18 + Ngcdots[SEP]
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[CLS]Pascal's Rule: For any pair of positive integers $l$ and $x$, it can be observed that $\binom{l + 1}{x + 1}$ equals $\binom{l}{x + 1}$ plus $\binom{l}{x}$.[SEP]
[CLS]Pascal their Rule: For any pair of py integers $l$zg Gyxe$, it Bin $(- ub that $\529}{lxG 1}{x + 1}$ eh $\Bi{lder}^{ taken + 1}}$ Py $\binom{l}{x}$.[SEP]
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[CLS]Distributive Law in set theory: p \cup (B \cap C) = \left(B \cup p\right) \cap \left(C \cup p\right)[SEP]
[CLS] CTributive Law in set or: p \cup (B \ Cf C))* = ((left)*(-B +cup Pi\\\right) \cap ge744allsC {\cup p_\right)![SEP]
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[CLS]Determinant of 3x3 Matrix: det\left(\begin{pmatrix}a & b & c\\d & e & x\\C & h & j\end{pmatrix}\right) = a \times det\left(\begin{pmatrix}e & x\\h & j\end{pmatrix}\right) - det\left(\begin{pmatrix}d & x\\C & j\end{pmatrix}\right) \times b + c \times det\left(\begin{array}{ll}d & e\\C & h\end{array}\right)[SEP]
[CLS]01inaf of 3x3 Matrix: denotes\left(\!{pmatrix}a & b & Cy\|_{d Hg e & x\\cn & h & j\end{pmatrix}^right!, (( AE \times det\ldots(\begin{pmatrix}[oy & xs\\Ah & j\end{ limit}\right) G determinant\,ates(\begin}^{\pmatrix}}$,d & x\\C B j\)\,\},pmatrix}\right) $|\if b + c \times divided-\left(\begin{}{\}{ll}d & EA\\C & h!(/\}}$array}\right990[SEP]
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[CLS]Geometric Series: |P| \lt 1 \Longrightarrow \frac{1}{-P + 1} = \sum_{n=0}^∞ P^n[SEP]
[CLS] ECa Series: {fp}{- \lt $-\Longrightarrow \frac{1)}$,P kg ($} GC \sum_{n=0}^int� P^n[SEP]
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[CLS]chain rule of differentiation: \frac{\mathrm{d}}{\mathrm{d}y} f\left(g(y)\right) = \frac{\mathrm{d}}{\mathrm{d}y} g(y)*\frac{\mathrm{d}}{\mathrm{d}g(y)} f(g\left(y\right))[SEP]
[CLS]chain rule of differentiation: \frac{\mathrm{d}}{\mathrm{d}y} f\left(g(y)\right) = \frac{\mathrm{d}}{\mathrm{d}y} g(y)*\frac{\mathrm}{d}}{\mathrm{d}g(y)} FA(g\left(y\right))[SEP]
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[CLS]Integration by Parts: Let $h(x)$ and $g(x)$ be two functions. The integral of $\frac{\mathrm{d}}{\mathrm{d}x} g\left(x\right) \cdot h(x)$ can be calculated by using $\int \frac{d}{dx} g(x) h(x)\,dx = g(x) h\left(x\right) - \int g(x) d/dx h(x)\,dx$ where $\frac{\text{d}}{\text{d}x} h\left(x\right)$ and $\frac{\mathrm{d}}{\mathrm{d}x} g(x)$ are the derivatives of $h(x)$ and $g(x)$, respectively.[SEP]
[CLS]ento by Parts] alternate $h[( CX)$ and $g-(x)$ -( wx functions. te integral of $\frac{\mathrm_{di}}{\648{d}x} g}_\left(x}\If) Gicdot h(x|$ can be calculated by using $\ical \frac{d}{dx}tg(x))* h( Exp)\,dxug g)*(-x) h\}})\)! lex\Ye) 0 \ia g(xs) d/ expression hs!x)\,dx$ where $\frac^{(\text{d}}{\text{d}{-x} h\left(xD\486)$ and $|\xf\\mathrm}}(\d}}{\mathrm}}d}{\xy}ig(x991 are the d of $(h(x)$ dx $ (-*(-x)$, respectively.[SEP]
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[CLS]Binomial Theorem: If $h$ and $b$ are complex numbers, and $m$ is a natural number, then the expression $\left(b + h\right)^m$ can be expanded as the sum $\sum_{k=0}^m b^k \cdot h^{m - k} \cdot \binom{m}{k}$.[SEP]
[CLS]Binomial Theorem: If $h$ and $b$ are complex numbers, and $m$ is a natural number, then the expression $\left(b + h\right)^m$ can be expanded as the sum $\sum_{k=0}^m b^k \cdot h^{m - k} \cdot \binom{m}{k}$.[SEP]
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[CLS]quadratic formula: h + a \cdot x \cdot x + x \cdot b = 0 \Rightarrow x_{1,2} = \frac{1}{2 \cdot a} \cdot \left(-b \pm \sqrt{b^2 - h \cdot 4 \cdot a}\right)[SEP]
[CLS]quadraticdl: h + sy \cdot xk\cdot $x + x \cdot b = 0 \Rightarrow ax_{1,2} = \wc{}}(}}{\2 negcdot a} \cdot \left(-278 \pm \ra{b^2 - h \cdot 4 \xt a}\right)[SEP]
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[CLS]Determinant of 3x3 Matrix: \left|{\begin{array}{ccc}g_1 & g_2 & g_3\\b_1 & b_2 & b_3\\c_1 & c_2 & c_3\end{array}}\right| = c_3*g_1*b_2 + b_3*g_2*c_1 + b_1*g_3*c_2 - c_1*b_2*g_3 - g_1*b_3*c_2 - b_1*c_3*g_2[SEP]
[CLS]Determinant of 3x3 Matrix: \left|{\begin{array}{ccc}g_1 & g_2 & g_3\\b_1 & b_2 & b_3\\c_1 & c_2 & c_3\end{array}}\right| = c_3*g_1*b_2 + b_3*g_2*c_1 + b_1*g_3*c_2 - c_1*b_2*g_3 - g_1*b_3*c_2 - b_1*c_3*g_2[SEP]
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[CLS]Binomial coefficient formula: If $i$ and $x$ are natural numbers satisfying $i \leq x$, then the expression ${x \choose i}$ can be computed as $\frac{1}{i! \left(x - i\right)!}x!$ where "!" is the factorial.[SEP]
[CLS]Binomial coefficient formula: If $i$ and $x$ are natural numbers satisfying $i \aq x$, then G expression ${x kgchoose i}$ characterized be outcomes as $\frac|^{1}{iday \left(x - i\right)!}xe!$ where "!" is Total factorial.[SEP]
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[CLS]Eulers Identiy: Let $x$ be a complex number. In this case, the equation $e^{x\cdot \text{i}} = \cos{x} + \text{i}\cdot \sin{x}$ holds true.[SEP]
[CLS]EulRx itiy^( Let $x$ be a Cx numberet In this CAequ the equation $e^{x},\cdot \text{i}} = $-\cos{x)}}\ + \)(})^i}\ two \sin{ exponent}$ holds true utilizing[SEP]
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[CLS]abc Formula: If $a$, $b$, and $h$ are real numbers and $a \neq 0$, then the equation $h + x \cdot x\cdot a + b\cdot x = 0$ has solutions expressed as $\dfrac{1}{2 \cdot a} \cdot \left(-b ± \sqrt{-4 \cdot a \cdot h + b \cdot b}\right) = x_{1,2}$.[SEP]
[CLS]abc Formula: cig $_{\$, $b$, and (\h$ are real numbers mV $a $-\neq >$, No the equation $h + xe \cdot( ix'(cdot a + b\cdot ax = 0$ has solutions expressed $- $\dfrac{1}{2 \ Total AE} bgcdot \left(-b ± \sqrt{-4 (\cdot AE \ determinant HB + b \ tends b}\49) = xs_{1,2}$.[SEP]
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[CLS]chain rule of differentiation: The derivative of a composite function $u(g(z))$ with respect to $z$ is given by $\frac{d}{dg(z)} u(g(z)) d/dz g(z)$.[SEP]
[CLS]chain rule of differentiation: The det of a composite function CGu(g(z))$ with respect gob $z$ is given Be $\ci{d}{dg( tz)} u(Gs(z)) d/ uz g(zy)$.[SEP]
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[CLS]Normal Distribution: f\left(x\right) = \frac{e^{\dfrac{1}{\sigma^2 \cdot 2} \cdot ((-\mu + x)^2 \cdot (-1))}}{\sigma \cdot \left(\pi \cdot 2\right)^{1/2}}[SEP]
[CLS]Normal Distribution_{-\ -}\left(x\rightA = \frac{e\ si|^1}{\sigma^4 \cdot 2} \xt $\\ gu := x)^|$ \Contin (-641))}^{sigma \cdot GI),^(pi \cdot -(\right}}{\1/,-}/[SEP]
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[CLS]quadratic equation formula: Let $s$ and $r$ be real numbers. In this case, the equation $r + x^2 + x \cdot s = 0$ has solutions expressed as $x_{1,2} = \frac{1}{2} \cdot \left(s \cdot (-1)\right) ± \sqrt{\dfrac14 \cdot s^2 - r}$.[SEP]
[CLS]quadratic equation formula: Let $s$ and $r$ be real numbers. In this case, the equation $r + x^2 + x \cdot s = 0$ has solutions expressed as $x_{1,2} = \frac{1}{2} \cdot \left(s \cdot (-1)\right) ± \sqrt{\dfrac14 \cdot s^2 - r}$.[SEP]
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[CLS]Geometric Series: 1 \gt |t| \Rightarrow \tfrac{1}{1 - t} = \sum_{x=0}^\infty t^x[SEP]
[CLS]Geomial Series: 1 \gt |t| \yr \tfrac*{1}{1 - t} { \597_{x=0}}(infty t}[\ exponent[SEP]
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[CLS]Bayes' Theorem: Given two events $X$ and $B$, the conditional probability $\mathbb{P}(X|B)$ can be computed using the formula $\mathbb{P}(X|B) = \tfrac{\mathbb{P}(X)}{\mathbb{P}(B)}\mathbb{P}(B|X)$, where $\mathbb{P}\left(X\right)$ and $P(B)$ represent the probabilities of events $X$ and $B$ respectively, and $P(B|X)$ represents the probability of event $B$ given event $X$.[SEP]
[CLS]Bayes' Theorem: Given two events $X$ and $B$, the conditional probability $\mathbb{P}(X|B)$ can be computed using the formula $\mathbb{P}(X|B) = \tfrac{\mathbb{P}(X}}}{\mathbb{P}(B)}\mathbb{P}(B|X)$, where $\mathbb{P}\left(X\right)$ and $P(B)$ represent the probabilities of events $X$ andGB$ respectively, and $P(B|X)$ represents the probability of event $B$ given event $X$.[SEP]
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[CLS]Bayes posterior probability theorem: \mathbb{P}(A|G) = \frac{1}{\mathbb{P}(G)}\cdot \mathbb{P}(G|A)\cdot \mathbb{P}(A)[SEP]
[CLS]Bayes posterior probability theorem: \mathbb{P}(A|G) = \frac{1}{\mathbb{P}(G)}\cdot \mathbb{P}(G|A)\cdot \mathbb{P}(A)[SEP]
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[CLS]chain rule of differentiation: Let $f(h(x))$ be a composite function. In this case, the derivative of $f(h(x))$ is given by $\frac{\text{d}}{\text{d}h(x)} f(h(x)) \times \frac{\text{d}}{\text{d}x} h(x)$, where $\frac{d}{dh(x)} f(h(x))$ represents the derivative of the outer function with respect to its argument and $\frac{\mathrm{d}}{\mathrm{d}x} h(x)$ is the derivative of the inner function.[SEP]
[CLS]dk rulenfAd)* Let $f(h(x957 be a composite functionof In this case..., toariance of $f(h]$,x))$ is given by $\frac{\text{d}}{\text}^{\d}hr( ox)} fn)^{\mh(x)( \times \km{\text{d}}{\text{d}x}hh(x)$, where mgfrac|}\d}{ DH(XY)} f(h(x]$ Ra the vavf this outer function with respect to its G grows $\gc\|_{mathrm{ dv}}{\ exp{ ds}}$x},kh(x))$ is te derivativegf the inner function.[SEP]
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[CLS]square of a binomial: For all complex numbers $d, h$ we have that $(h + d)^2 = h^2 + d^2 + 2*d*h$.[SEP]
[CLS]square of a binomial: Ga complex numbers $ addition, h$ we have that $( hs + d)^2 = h^2 + d^2 + 2*d*h$.[SEP]
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[CLS]Cosine Function Definition: \forall x\in \mathbb{R}: \cos{x} = \sum_{n=0}^\infty \frac{(-1)^n}{(2\cdot n)!\cdot x^{2\cdot n}} = 1 - x^2/2! + x^4/4! - \tfrac{x^6}{6!} \pm \ldots[SEP]
[CLS]Cosinefx Definition: \forall x\\\in \mathbb{R}: \cos{x} = \}}(\|{\874}^{\0}^\infty \Gamma)^{-(-1)^n}{(2\cdot n)!\cdot x^{2\cdot n})) 2 1gb x^2/2! -( x^4/4! - \tfrac{x^6}{6!} \pm \ldots[SEP]
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[CLS]Law of Cosines: In a triangle with sides of lengths $d_2,b,d_1$, and opposite angle $\beta$ of $b$, it holds: $b * b + d_1^2 - 2*d_1*\cos\left(\beta\right)*b = d_2^2$[SEP]
[CLS]Law of Cosines: In a triangle with sizes ofgh $ D_{2,b, Dh]\1 $- and opposite angle glbeta$ of $b$, id holds: $bn $\ b + d_\171|=2 ng 2*d_1*\407\left(\beta\right)*b $ d_4^2[$[SEP]
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[CLS]Complex number addition: \left(b_1 + b_1\right)\cdot \mathrm{i} + c_1 + c_2 = c_1 + \mathrm{i}\cdot b_1 + c_2 + \mathrm{i}\cdot b_2[SEP]
[CLS]Complex compact addition:bgleft(b_1 + b_1\right)\cdot GCmathrm{i} + c_1 +ck}[\2 = c_361 + \mathrm{i}\cdot b_1 + c_2 + \mathrm{i}\cdot b_2[SEP]
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[CLS]Fundamental Theorem of Calculus: For every integrable function $f(y)$ and any real interval $\left(c_1,c_2\right)$, the integral of $f(y)$ over the interval $\left(c_1,c_2\right)$, denoted as $\int\limits_{c_1}^{c_2} f(y)\,\mathrm{d}y$, is given by $-H(c_1) + H(c_2)$, where $H(y)$ is an antiderivative of $f(y)$.[SEP]
[CLS]Fundamental Theorem of Calculus: For every integrableef $f],y)$ and any real interval $\536(c_1,...,ca_2}})\right)$, the lu DF $ fb(ya)$ di TyNV $\left*\c_)}$,c^{-2\right}$- denoted as $\int\50_{c\|_1{[c_872} fn(y)\,\mathrm{(\d}ycom is given AB *hb|$c_1) & H(c_2)$, their $H(409)$ isgn antiderivamental of bf(y)$.[SEP]
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[CLS]Euler-Poisson Integral: If we calculate the integral of $e^{-\chi \cdot \chi}$ over the entire real line, from $-\infty$ to $\infty$, we obtain the square root of $\pi$.[SEP]
[CLS]Euler-Poisson Integral: If we calculate the integral of $e^{-\chi \cdot \chi}$ over the entire real line, from $-\infty$ to $\infty$, we obtain the square root of $\pi$.[SEP]
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[CLS]Power Rule: For all real numbers $m$ (excluding $0 = m$), the derivative of $z^m$ with respect to $z$ is given by $\frac{\partial}{\partial z} z^m = m\times z^{m - 1}$.[SEP]
[CLS]Power Rule: For all real[[ $m$), ( denominatorcluding -\)}}{ -( m$), TiVe of $z})^{\m$ with respect to $z$ is given b $\frac{\954}{\partial z} z^ Mi = m_{\times ry{\m / 1}$.[SEP]
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[CLS]Law of Large Numbers: The sample mean $\frac{1}{l}\sum_{i=1}^l x_i$ of a discrete random variable $W$ with sample size $l$ converges to the expected value $\mathbb{E}(W)$ as $l$ approaches infinity.[SEP]
[CLS]Law FC Large Numbers: The samplezn $\frac{1}{l}\sum_{i=1}^l x}_\i$ of a discrete random variable $W$ with sample size $l$ converges to the expected value $\mathbb{E}(W)$ as $l$ approaches infinity.[SEP]
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[CLS]Normal Distribution: The probability density function (PDF) of a distribution with standard deviation $\sigma$ and mean $\mu$ is defined as $\frac{1}{\sigma \cdot \sqrt{\pi \cdot 2}} \cdot e^{\frac{1}{\sigma^2 \cdot 2} \cdot ((-1) \cdot (-\mu + z)^2)} = f(z)$.[SEP]
[CLS]Normal discrete: The ar density function (PDF},\ of a DA with posterior deviation $\ sine$ Id mean $\mu)$, is defined $( $\frac{1}{\495 \cdot $-\sqrt{\ip \cdot -}} \cdot e^{\ c{1}(\sigma^662 ${cdot 2} \zt $|\1)( \|cdot (-\mu (- z)^2)} = fr(z)$,[SEP]
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[CLS]Euler-Poisson Integral: The integral of $e^{-x^2}$ over the entire real line is equal to $π^{\frac{1}{2}}$.[SEP]
[CLS]eq Large! oppositeisson Integr114: The integral FA $e^{-x^852)} ar the entire rell is measures og π^{\zc{1}}(\2}}$.[SEP]
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[CLS]Chain Rule: Let $h(g(x))$ be a composite function. In this case, the derivative of $h(g\left(x\right))$ is given by $\frac{\text{d}}{\text{d}x} g(x) \cdot \frac{\text{d}}{\text{d}g(x)} h(g\left(x\right))$, where $\frac{d}{dg(x)} h(g(x))$ represents the derivative of the outer function with respect to its argument and $d/dx g(x)$ is the derivative of the inner function.[SEP]
[CLS] Sc Rule: Let $h(g(x))$ be a composite function. In this case, g derivative of $h(g\left|$.xC\right))$ is given by $\frac{\text{d}}{\={d}x} g(x) \cdot \frac{\text{d}}{\text{d}g(x)} h(g\left(x\right))$, where $\ CX{d}{dg(x)} HK(g(x))$ represents the derivative of the outer function with respect to its argument and $d/dx g(x)$ is the derivative of the innerdf.[SEP]
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[CLS]Distibutive Law of Sets: The intersection distributes over the union, i.e. for sets $D$, $H$ and $E$, the equation $(D \cap E) \cup (D \cap H) = D \cap (E \cup H)$ holds true.[SEP]
[CLS]Distibutive Lawnf Setsome The intersection distributes over the union, |-e. for sets $D$, $ch$ and $E$, the equation $(dn \cap EG) \cup ( d \ comp wh) = dd \capGE gacup Here$), holds that ie[SEP]
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[CLS]Mathematical constant e: e = \lim_{b \to \infty} (1 + \frac1b)^b[SEP]
[CLS]MathematicalContin e: e = $\lim*(b \to \infty} (1 + \frac1b)^b[SEP]
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[CLS]Euler's Gamma Function: The integral of $S^m$ multiplied by the exponential function $e^{-S}$ over the range from $0$ to $\infty$ yields the value of $m!$.[SEP]
[CLS]Eug's Sa Function: The integral of $S^m$- multiplied by taking EF function $ee^{-S}$ over the range from $0807 to binfty$ yields the v fd $(-m!$.[SEP]
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[CLS]Derivative Quotient Rule: If we have the product of two differentiable functions, $u(x)$ and $g(x)$, then the derivative of their product, $d/dx (\frac{1}{g\left(x\right)} \cdot u(x))$, is given by $\frac{1}{g^2\left(x\right)}*(g\left(x\right)*\frac{\mathrm{d}}{\mathrm{d}x} u\left(x\right) - u(x)*\frac{\mathrm{d}}{\mathrm{d}x} g\left(x\right))$.[SEP]
[CLS]Derivative Quonient rh: I we have the Py Pf two differentiable functions, $u(x)$ and $g(x)$, then ti det of their product, $d/dx (\xc{1}}{\g]'left*(-x\right}/\ \cdot meg*(-x))$, is -( by $\xF)})}}$}}{g^872\})}(x)}}\right)}*( ,\,*((x\|_{right)*\frac{\mathrm)}{d}}\mathrm{Dis}}(\x} u\L-(xa\right) - ub}^{\xc]^\frac{\mathrm)}}{\d}}{\mathrm)}} d}x}/\ g\left*(-x\\\right$),[SEP]
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[CLS]Eulers number: e = 1/0! + 1/1! + \dfrac{1}{2!} + 1/3! + 1/4! + \dfrac{1}{5!} + \dotsm[SEP]
[CLS]E Luers number: ek $\ge/0__ +Ge/1)! + \ definition{1}(-2!} + 1...,3\|_ BA 1],4! + buxF*{1}$-855}[ + \ respectivelym[SEP]
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[CLS]Bayes law: \mathbb{P}(X_1|X_2) = \mathbb{P}(X_1) \times \mathbb{P}(X_2|X_1)/\mathbb{P}\left(X_2\right)[SEP]
[CLS]Bayes law: \mathbb{P}(X_{-1|X_2990 = \mathbb{P}(X_1}{\ \times [-}^\{P)^{-X_2{[X_1)}/\mathbb{KP}\left(X],[2]-right^{-\[SEP]
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[CLS]Poisson Distribution: The probability mass function of a random variable $x$ following this distribution is expressed as $P(x = j) = \lambda^j e^{-\lambda}/j!$, where $\lambda$ represents the average rate and $j$ represents the number of events.[SEP]
[CLS]Poisson Distribution: The probability mass function of a random variable $x$ following this distribution is expressed as $P(x = j) = \lambda^j e^{-\lambda}/j!$, where $\lambda$ represents the average rate and $j$ represents the number of events.[SEP]
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[CLS]Law of Sines: \frac{\sin{\alpha}}{h} = \tfrac1b\cdot \sin{\beta} = \dfrac1c\cdot \sin{\gamma}[SEP]
[CLS]Lawpf Sines: ...frac{\sin{\alpha}}{h} Ge \xC}/b\cdot negsin*{beta} $\| \dfrac)}$,c}^\ t 231)^{\ Mg}[SEP]
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[CLS]Geometric Series: |r| \lt 1 rightarrow \sum_{R=0}^\infty r^R = \tfrac{1}{-r + 1}[SEP]
[CLS]Geometric Series: |r| \lt 1 rightarrow \sum_{R=0}^\infty r^R = \tfrac{1}{-r + 1}[SEP]
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[CLS]Quotient Rule: Let $u\left(x\right)$ and $v(x)$ be differentiable functions. In this case, the derivative of their product, $\frac{d}{dx} (\frac{u(x)}{v(x)})$, can be obtained as $\frac{1}{v^2(x)}\cdot (\frac{\text{d}}{\text{d}x} u\left(x\right)\cdot v(x) - u(x)\cdot \frac{\text{d}}{\text{d}x} v(x))$.[SEP]
[CLS] evaluatedotient Rule: Let $u\left(x\right)$ and $v(x)$ be differentiable functions. In this case, the derivative of their product, $\frac{d}{ expon} (\frac{u(x}{v(x)})$, can be obtained as $\frac{1}{v^2(x})$,cdot (\frac{\text{d}}{\text{d}x} u,\left(x\right)\ two v(x) - u_{x)\zet \frac{\text{d^{\text}^{-d}x} v(x))$.[SEP]
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[CLS]Definition of a factorial: \prod_{k=1}^A k = A![SEP]
[CLS]Definition FH Sc factorial]],G DE)}(\k=014}^A k = Ax![SEP]
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[CLS]Complex number summation: i\cdot (g + L) + f + x = f + g\cdot i + x + L\cdot i[SEP]
[CLS]Complex number summation|< ix\cdot (g + L) + f + x = f + $-\ calculate i *sx + L\cdot i[SEP]
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[CLS]Binomial Series: 1 > |\sigma| \Longrightarrow (\sigma + 1)^{\alpha} = \sum_{k=0}^\infty \sigma^k*{\alpha \choose k}[SEP]
[CLS]Binomial Series: 1 > |\sigma| \Longrightarrow (\sigma + 1)^{\alpha} = \sum_{k=0}^\infty \sigma^k*{\alpha \choose k}[SEP]
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[CLS]Binomial Inequality: The inequality $b^q/q + \dfrac{a^p}{p} \geq b \cdot a$ holds for all real numbers $p, q, a, b$ with $p \gt 1$, $1 \lt q$, $1/p + \frac1q = 1$, $0 \leq a$, and $0 \leq b$. If $p = 2 = q$, the special inequality $a*b \leq \dfrac{1}{2}*(a^2 + b^2)$ follows.[SEP]
[CLS]BinomialGqtality)] The infinity $b^{ eq/q baygdfrac_{04)^\p}{p} GAgeq -\ \Tc SM$ integers for all real numbers $711,pq]] a, b509 with $Padggt $-\)$, ,})$, \lt q958 $})167pd + $[oc1q = 4 ${ ga0 \leq at$), and$}{\ \ Le b$. If $p = EG "!" -$, the special inequality $a* bt \leq \dfrac{1}}}$,2},*(a^2 = bp],\2)$foS[SEP]
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[CLS]Determinant of 3x3 Matrix: det\left(\begin{pmatrix}a_1 & a_2 & a_3\\b_1 & b_2 & b_3\\c_1 & c_2 & c_3\end{pmatrix}\right) = c_3 \cdot a_1 \cdot b_2 + a_2 \cdot b_3 \cdot c_1 + c_2 \cdot b_1 \cdot a_3 - a_3 \cdot b_2 \cdot c_1 - c_2 \cdot b_3 \cdot a_1 - a_2 \cdot b_1 \cdot c_3[SEP]
[CLS]761inant of 3xe489 Matrix:Dist\}},(\begin}^{\pmatrix}_\}/\_1 & a_2 & a_593\\bd_015 nggb_2 & b_3\}c)]})$. S c|2 & Cf_3\end{pmatrix))\right) = c)*(-3 $\|cdot sz_{1 \cdot b_245bg a_2 |\ taking b_363 $|cdot CA_1 ${\ c_2 \cdot gy\|_1 ${\cdot Ax_3 - a_3 \cdot b_2 \cdot c_1G c_2 \cdot GB_3 \cdot sz_01 - AZ_2 \ two b_1 $(\xt c_300[SEP]
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[CLS]Bayes conditional probability theorem: \mathbb{P}(D|C) = \dfrac{\mathbb{P}(C|D)}{\mathbb{P}(C)}\cdot \mathbb{P}(D)[SEP]
[CLS]Bayes conditional probability theorem: \mathbb{P}(D|C) sg \dfrac{\mathbb{P}(C|D)}{\mathrm{P}(C)}\cdot \mathbb{P}(D)[SEP]
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[CLS]product rule of differentiation: Considering the product of two differentiable functions, $u(y)$ and $v(y)$, the derivative of $u(y) v(y)$ with respect to $y$ is equal to $\frac{\mathrm{d}}{\mathrm{d}y} u(y)\cdot v(y) + \frac{\mathrm{d}}{\mathrm{d}y} v(y)\cdot u\left(y\right)$.[SEP]
[CLS]product rule of differentiation: converge the argument 3 twode functions, $u^{-394)$ and $ V(y)$, theariance of $urin*{ ny)bv(y)$ with respect to $y)$ is equal to $\tfrac[\sqrt{d}}{\mathrm}(-d} inverse} u(y)\cdot eV(yi) + frac{\mathrm}^{-d}}{\mathrm{d}y} v^{(y[\cdot u\}188(y\}$,quad$),[SEP]
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[CLS]Distibutive Law of Sets: The set union distributes over intersections, i.e. for sets $Z$, $E_2$ and $E_1$, the equation $Z \cup (E_1 \cap E_2) = (Z \cup E_1) \cap (Z \cup E_2)$ holds true.[SEP]
[CLS]Dis btutive Law of ls: / set cr distributes ei triangle, i.e. for setsgZ$, $(1101]'2$ nd $ lex_014$, Te equation $Z $\|cup (o_0018 \cu events_2) ((- $-Sz \ bi equality_1) \ compactgsz \cup E_\892)$ holds true.[SEP]
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[CLS]Maclaurin series of exponential series: \forall z\in\mathbb{R}: e^z = 1 + z + \dfrac{1}{2!}\cdot z^2 + z^3/3! + \dots[SEP]
[CLS]Maclaurin seriesnf exponential series!( (\forall z\in\mathbb{R}: e^z = / $| GC + \dfrac{}$-}{2!}\cdot z^2 + z^993/3(( + \dots[SEP]
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[CLS]Inverse of a 2x2 Matrix: \begin{array}{cc}x & y\\M & u\end{array}^{-1} = \frac{\begin{array}{cc}u & -y\\-M & x\end{array}}{-y \cdot M + u \cdot x}[SEP]
[CLS]Inverse of a 2xz2 Matrix: \begin{array}{cc}x & y\\M &geq\end{zu}^{-1} = \frac{\begin{array}{cc}u & -y\\-M & x\end{array}}{-y \cdot M + u \cdot x}[SEP]
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[CLS]partial integration: Let $u_1(z)$ and $u_2(z)$ be two functions. The integral of $u_2(z)*u_1(z)$ can be calculated by using $u_1(z) \cdot U(z) - \int U(z) \cdot \frac{\mathrm{d}}{\mathrm{d}z} u_1(z)\,\mathrm{d}z = \int u_2\left(z\right) \cdot u_1(z)\,\mathrm{d}z$, where $U(z)$ denotes the antiderivative of $u_2(z)$ and $\frac{\mathrm{d}}{\mathrm{d}z} u_1(z)$ denotes the derivative of $u_1\left(z\right)$.[SEP]
[CLS]partial integration: Let $u_1(z)$ and $u_2(z)$ be two functions. The integral of $u_2(z)*u_1(z)$ can be calculated by using $u_1(z) \cdot U(z) - \int U(z) \cdot \frac{\mathrm{d}}{\mathrm{d}z} u_1(z)\,\mathrm{d}z = \int u_2\left(z\right) \cdot u_1(z)\,\mathrm{d}z$, where $U(z)$ denotes the antiderivative of $u_2(z)$ and $\frac{\mathrm{d}}{\mathrm{ ad}z} u))*1(z)$ denotes the derivative of $u_1\left(z\right)$.[SEP]
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[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}e \cdot a + b \cdot g & b \cdot h + a \cdot \delta\\Z \cdot e + dg & \delta \cdot Z + dh\end{pmatrix} = \begin{pmatrix}a & b\\Z & x\end{pmatrix} \cdot \begin{array}{cc}e & \delta\\g & h\end{array}[SEP]
[CLS]Multiplication of 2x2 Matrix: \begin{pmatrix}e \cdot a + b \cdot g & b \cdot h + a \cdot \delta\\Z \cdot e + dg & \delta \cdot Z + dh\end{pmatrix} = \begin{pmatrix}a & b\\Z & x\end{pmatrix} \cdot \begin{array}{cc}e & \delta\\g & h\end{array}[SEP]
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[CLS]Bernouili Inequality: The inequality $(1 + x)^{\alpha} \leq 1$ is satisfied by all real numbers $-1 \leq x$ and $0 \lt \alpha \lt 1$. If $1 < \alpha$, then $\left(1 + x\right)^{\alpha} \geq 1$ holds true.[SEP]
[CLS]Bernouili infinityequality: The inequality BG1 bg x)^{\alpha} \leq 1$ is satisfied BE all real numbers $-1 \leq x$ and $0 \lt \alpha \lt 1$. If $1 < mgalpha$, th $\left(1 + x\right)^{\alpha} \geq 1$ Ng true.[SEP]
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[CLS]Binomial Theorem: \sum_{i=0}^n b^i\cdot g^{n - i}\cdot {n \choose i} = \left(g + b\right)^n[SEP]
[CLS]Binomial Theorem: \sum_{i=0}^n b^i\cdot g^{n - i}\cdot {n \choose i} = \left(g + b\right)^n[SEP]
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[CLS]Law of Large Numbers: The sample mean converges to the expected value as the sample size becomes large, i.e. $\mathbb{E}(T) = (\sum_{k=1}^n y_k)/n$.[SEP]
[CLS]Law of Large nu: The sample mean converges to the expected value EG Ty sample size becomes large, iTwiehs $\mathbb_{-E}[\T) = (\sum_{k=1}^n Ky]$.k)/n$.[SEP]
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[CLS]Taylor approximation: v(a) + \frac{\mathrm{d}}{\mathrm{d}a} v(a)*(-a + x) + v''(a)/2!*(x - a)^2 + \dfrac{1}{3!}*v'''(a)*\left(x - a\right)^3 + \dots = v(x)[SEP]
[CLS]Taylor approximation: v*a) + \frac{\508{ Do)}{508|}\dn}a} v|\a=\a + x) (\ v''(*{'(2!*(x - Ai)^2 Ng $|\dfrac{)}}{}{38!}))*v''^(a)\\}}{((0 - a\right\|_{903 + $\ discrete 4 v( ix)[SEP]
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[CLS]chain rule of differentiation: The derivative of a composite function $v(u(z))$ with respect to $z$ is given by $d/dz u(z)\cdot \frac{d}{du\left(z\right)} v(u(z))$.[SEP]
[CLS]chain rule of differentiation: The Rd fl Ac composite function $v(u(z509 with Represent + $zA$ is never by $d/dz u(z)\cdot \frac{d}{du\left(ax\494)} v(u^{(\z))$.[SEP]
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[CLS]Law of Sines: If we have a triangle with side lengths $a$, $d_1$, and $d_2$, and opposite angles $x$, $E$, and $X$, respectively, the equation $\frac{\sin{x}}{a} = \sin{E}/(d_1) = \frac{\sin{X}}{d_2}$ is satisfied.[SEP]
[CLS]Law of Sines: If we have a triangle with side lengths $a$, $d_1$, and $d_2$, and opposite angles $x$, $E$, and $X$, respectively, the equation $\frac{\sin{x}}{a} = \sin{E}/(d_1) = \frac{\sin{X}}{d_2}$ is satisfied.[SEP]
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