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both F and the element X. === Notion of a rational function on an algebraic variety === Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn). An extended version of the abstract idea of rationa... |
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f {\displaystyle f} at... |
reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions). === Fundamental theorem of algebra === The fundamental theorem of algebra states that e... |
manifolds. An important special case is the case that f {\displaystyle f} is a smooth function from R p {\displaystyle \mathbb {R} ^{p}} to R n {\displaystyle \mathbb {R} ^{n}} . If zero is a regular value of f {\displaystyle f} , then the zero set of f {\displaystyle f} is a smooth manifold of dimension m = p − n {\di... |
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions y(x) of Bessel's differential equation x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 0 {\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0} for an... |
of the second kind. Depending upon the circumstances, however, various formulations of these solutions are convenient. Different variations are summarized in the table below and described in the following sections.The subscript n is typically used in place of α {\displaystyle \alpha } when α {\displaystyle \alpha } is ... |
solutions are no longer linearly independent. In this case, the second linearly independent solution is then found to be the Bessel function of the second kind, as discussed below. ==== Bessel's integrals ==== Another definition of the Bessel function, for integer values of n, is possible using an integral representati... |
Yα(x) is related to Jα(x) by Y α ( x ) = J α ( x ) cos ( α π ) − J − α ( x ) sin ( α π ) . {\displaystyle Y_{\alpha }(x)={\frac {J_{\alpha }(x)\cos(\alpha \pi )-J_{-\alpha }(x)}{\sin(\alpha \pi )}}.} In the case of integer order n, the function is defined by taking the limit as a non-integer α tends to n: Y n ( x )... |
Y n ( x ) . {\displaystyle Y_{-n}(x)=(-1)^{n}Y_{n}(x).} Both Jα(x) and Yα(x) are holomorphic functions of x on the complex plane cut along the negative real axis. When α is an integer, the Bessel functions J are entire functions of x. If x is held fixed at a non-zero value, then the Bessel functions are entire function... |
i}J_{\alpha }(x)}{i\sin \alpha \pi }},\\[5pt]H_{\alpha }^{(2)}(x)&={\frac {J_{-\alpha }(x)-e^{\alpha \pi i}J_{\alpha }(x)}{-i\sin \alpha \pi }}.\end{aligned}}} If α is an integer, the limit has to be calculated. The following relationships are valid, whether α is an integer or not: H − α ( 1 ) ( x ) = e α π i H α ( 1 )... |
+1)}}\left({\frac {x}{2}}\right)^{2m+\alpha },\\[5pt]K_{\alpha }(x)&={\frac {\pi }{2}}{\frac {I_{-\alpha }(x)-I_{\alpha }(x)}{\sin \alpha \pi }},\end{aligned}}} when α is not an integer. When α is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments x. The series expans... |
x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y=0.} Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα are exponentially growing and decaying functions respectively. Like the ordinary Bessel function Jα, the function Iα goes to zero at ... |
= ( 2 ξ / π ) − 1 / 2 exp ( − ξ ) {\displaystyle K_{\frac {1}{2}}(\xi )=(2\xi /\pi )^{-1/2}\exp(-\xi )} is useful to represent the Laplace distribution as an Exponential-scale mixture of normal distributions. The modified Bessel function of the second kind has also been called by the following names (now rare): Basse... |
− 6 x ) sin x x − ( 15 x 2 − 1 ) cos x x {\displaystyle {\begin{aligned}j_{0}(x)&={\frac {\sin x}{x}}.\\j_{1}(x)&={\frac {\sin x}{x^{2}}}-{\frac {\cos x}{x}},\\j_{2}(x)&=\left({\frac {3}{x^{2}}}-1\right){\frac {\sin x}{x}}-{\frac {3\cos x}{x^{2}}},\\j_{3}(x)&=\left({\frac {15}{x^{3}}}-{\frac {6}{x}}\right){\frac {\... |
r = 0 [ n − 1 2 ] ( − 1 ) r ( n + 2 r + 1 ) ! ( 2 r + 1 ) ! ( n − 2 r − 1 ) ! ( 2 x ) 2 r + 1 ] y n ( x ) = ( − 1 ) n + 1 j − n − 1 ( x ) = ( − 1 ) n + 1 π 2 x J − ( n + 1 2 ) ( x ) = = ( − 1 ) n + 1 2 x [ e i x ∑ r = 0 n i r + n ( n + r ) ! r ! ( n − r ) ! ( 2 x ) r + e − i x ∑ r = 0 n ( − i ) r + n ( n + r ) ! r ! ( ... |
) + i y n ( x ) , h n ( 2 ) ( x ) = j n ( x ) − i y n ( x ) . {\displaystyle {\begin{aligned}h_{n}^{(1)}(x)&=j_{n}(x)+iy_{n}(x),\\h_{n}^{(2)}(x)&=j_{n}(x)-iy_{n}(x).\end{aligned}}} There are simple closed-form expressions for the Bessel functions of half-integer order in terms of the standard trigonometric functions, a... |
asymptotic forms. For small arguments 0 < z ≪ α + 1 {\displaystyle 0<z\ll {\sqrt {\alpha +1}}} , one obtains, when α {\displaystyle \alpha } is not a negative integer: J α ( z ) ∼ 1 Γ ( α + 1 ) ( z 2 ) α . {\displaystyle J_{\alpha }(z)\sim {\frac {1}{\Gamma (\alpha +1)}}\left({\frac {z}{2}}\right)^{\alpha }.} When α is... |
1/2, the last terms in these formulas drop out completely; see the spherical Bessel functions above.) The asymptotic forms for the Hankel functions are: H α ( 1 ) ( z ) ∼ 2 π z e i ( z − α π 2 − π 4 ) for − π < arg z < 2 π , H α ( 2 ) ( z ) ∼ 2 π z e − i ( z − α π 2 − π 4 ) for − 2 π < arg z < π . {\displaystyle ... |
8 z + ( 4 α 2 − 1 ) ( 4 α 2 − 9 ) 2 ! ( 8 z ) 2 − ( 4 α 2 − 1 ) ( 4 α 2 − 9 ) ( 4 α 2 − 25 ) 3 ! ( 8 z ) 3 + ⋯ ) for | arg z | < π 2 , K α ( z ) ∼ π 2 z e − z ( 1 + 4 α 2 − 1 8 z + ( 4 α 2 − 1 ) ( 4 α 2 − 9 ) 2 ! ( 8 z ) 2 + ( 4 α 2 − 1 ) ( 4 α 2 − 9 ) ( 4 α 2 − 25 ) 3 ! ( 8 z ) 3 + ⋯ ) for | arg z | < 3 π 2 . {\di... |
For integer order α = n, Jn is often defined via a Laurent series for a generating function: e x 2 ( t − 1 t ) = ∑ n = − ∞ ∞ J n ( x ) t n {\displaystyle e^{{\frac {x}{2}}\left(t-{\frac {1}{t}}\right)}=\sum _{n=-\infty }^{\infty }J_{n}(x)t^{n}} an approach used by P. A. Hansen in 1843. (This can be generalized to non-i... |
) {\displaystyle e^{\pm iz\sin \phi }=J_{0}(z)+2\sum _{n=1}^{\infty }J_{2n}(z)\cos(2n\phi )\pm 2i\sum _{n=0}^{\infty }J_{2n+1}(z)\sin((2n+1)\phi )} which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal. More generally, a series f ( z ) = a 0 ν J ... |
2 ( z 2 ) ν ⋅ π ⋅ Γ ( 1 2 − ν ) ∫ 1 ∞ sin z u ( u 2 − 1 ) ν + 1 2 d u {\displaystyle {\begin{aligned}J_{\nu }(z)&={\frac {\left({\frac {z}{2}}\right)^{\nu }}{\Gamma \left(\nu +{\frac {1}{2}}\right){\sqrt {\pi }}}}\int _{-1}^{1}e^{izs}\left(1-s^{2}\right)^{\nu -{\frac {1}{2}}}\,ds\\[5px]&={\frac {2}{{\left({\frac {z}{... |
( k ) d k = δ ( x − 1 ) {\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)J_{\alpha }(k)\,dk=\delta (x-1)} A change of variables then yields the closure equation: ∫ 0 ∞ x J α ( u x ) J α ( v x ) d x = 1 u δ ( u − v ) {\displaystyle \int _{0}^{\infty }xJ_{\alpha }(ux)J_{\alpha }(vx)\,dx={\frac {1}{u}}\delta (u-v)} for α... |
+1}(x)} and 2 d Z α ( x ) d x = Z α − 1 ( x ) − Z α + 1 ( x ) , {\displaystyle 2{\frac {dZ_{\alpha }(x)}{dx}}=Z_{\alpha -1}(x)-Z_{\alpha +1}(x),} where Z denotes J, Y, H(1), or H(2). These two identities are often combined, e.g. added or subtracted, to yield various other relations. In this way, for example, one can co... |
=== In 1929, Carl Ludwig Siegel proved that Jν(x), J'ν(x), and the logarithmic derivative J'ν(x)/Jν(x) are transcendental numbers when ν is rational and x is algebraic and nonzero. The same proof also implies that Γ ( v + 1 ) ( 2 / x ) v J v ( x ) {\displaystyle \Gamma (v+1)(2/x)^{v}J_{v}(x)} is transcendental under ... |
+n}(z),} where λ and ν may be taken as arbitrary complex numbers. For |λ2 − 1| < 1, the above expression also holds if J is replaced by Y. The analogous identities for modified Bessel functions and |λ2 − 1| < 1 are λ − ν I ν ( λ z ) = ∑ n = 0 ∞ 1 n ! ( ( λ 2 − 1 ) z 2 ) n I ν + n ( z ) {\displaystyle \lambda ^{-\nu }I_... |
chain suspended from a fixed point above and free at its lower end. The solution of the differential equation led to the introduction of a function that is now considered J 0 ( x ) {\displaystyle J_{0}(x)} . Bernoulli also developed a method to find the zeros of the function. Leonhard Euler in 1736, found a link betwee... |
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the deriva... |
in 1757 by Vincenzo Riccati. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. As early as 1759, Daviet de Foncenex showed the interchangeability of the trigonometric and hyperbolic functions using the imaginar... |
make the solution unique; without them any pair of functions ( a e x + b e − x , a e x − b e − x ) {\displaystyle (ae^{x}+be^{-x},ae^{x}-be^{-x})} would be a solution. sinh(x) and cosh(x) are also the unique solution of the equation f ″(x) = f (x), such that f (0) = 1, f ′(0) = 0 for the hyperbolic cosine, and f (0) = ... |
) = cosh x {\displaystyle {\begin{aligned}\sinh(-x)&=-\sinh x\\\cosh(-x)&=\cosh x\end{aligned}}} Hence: tanh ( − x ) = − tanh x coth ( − x ) = − coth x sech ( − x ) = sech x csch ( − x ) = − csch x {\displaystyle {\begin{aligned}\tanh(-x)&=-\tanh x\\\coth(-x)&=-\coth x\\\operatorname {sech} (-x)&=\ope... |
= sinh x cosh y − cosh x sinh y cosh ( x − y ) = cosh x cosh y − sinh x sinh y tanh ( x − y ) = tanh x − tanh y 1 − tanh x tanh y {\displaystyle {\begin{aligned}\sinh(x-y)&=\sinh x\cosh y-\cosh x\sinh y\\\cosh(x-y)&=\cosh x\cosh y-\sinh x\sinh y\\\tanh(x-y)&={\frac {\tanh x-\tanh y}{1-\tanh ... |
1 ) | x | > 1 arsech ( x ) = ln ( 1 x + 1 x 2 − 1 ) = ln ( 1 + 1 − x 2 x ) 0 < x ≤ 1 arcsch ( x ) = ln ( 1 x + 1 x 2 + 1 ) x ≠ 0 {\displaystyle {\begin{aligned}\operatorname {arsinh} (x)&=\ln \left(x+{\sqrt {x^{2}+1}}\right)\\\operatorname {arcosh} (x)&=\ln \left(x+{\sqrt {x^{2}-1}}\right)&&x\geq 1\\\operator... |
sinh ( a x ) d x = a − 1 cosh ( a x ) + C ∫ cosh ( a x ) d x = a − 1 sinh ( a x ) + C ∫ tanh ( a x ) d x = a − 1 ln ( cosh ( a x ) ) + C ∫ coth ( a x ) d x = a − 1 ln | sinh ( a x ) | + C ∫ sech ( a x ) d x = a − 1 arctan ( sinh ( a x ) ) + C ∫ csch ( a x ) d x = a − 1 ln | tanh ( a x 2 ... |
( 2 n + 1 ) ! {\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}} This series is convergent for every complex value of x. Since the function sinh x is odd, only odd exponents for x occur in its Taylor series. cosh x = 1 + x 2 2... |
n = 1 ∞ ( 1 + x 2 n 2 π 2 ) = x 1 − x 2 2 ⋅ 3 + x 2 − 2 ⋅ 3 x 2 4 ⋅ 5 + x 2 − 4 ⋅ 5 x 2 6 ⋅ 7 + x 2 − ⋱ {\displaystyle \sinh x=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{n^{2}\pi ^{2}}}\right)={\cfrac {x}{1-{\cfrac {x^{2}}{2\cdot 3+x^{2}-{\cfrac {2\cdot 3x^{2}}{4\cdot 5+x^{2}-{\cfrac {4\cdot 5x^{2}}{6\cdot 7+x^{2}-\... |
cosh x + sinh x ) ( cos y + i sin y ) {\displaystyle e^{x+iy}=(\cosh x+\sinh x)(\cos y+i\sin y)} for the general complex exponential function. Additionally, e x = 1 + tanh x 1 − tanh x = 1 + tanh x 2 1 − tanh x 2 {\displaystyle e^{x}={\sqrt {\frac {1+\tanh x}{1-\tanh x}}}={\frac {1+\tanh {\frac {x}{2}}}... |
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, ( 1 + 5 ) / 2 {\displaystyle (1+{\sqrt {5}})/2} , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a val... |
when undefined): for example, cos π/7, cos 3π/7, and cos 5π/7 satisfy 8x3 − 4x2 − 4x + 1 = 0. This polynomial is irreducible over the rationals and so the three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, and tan 15π/16 satisfy the irreducible polynomial x4 − 4x3 −... |
set { a i | 1 ≤ i ≤ k } {\displaystyle \{a_{i}|1\leq i\leq k\}} in Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} such that Q ( α ) = ∑ i = 1 k a i Q {\displaystyle \mathbb {Q} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q} } ; that is, every member in Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} can be written as ∑ i = 1 k... |
} or (for β ≠ 0 {\displaystyle \beta \neq 0} ) α / β {\displaystyle \alpha /\beta } , is a linear subspace of the finite-degree field extension Q ( α , β ) {\displaystyle \mathbb {Q} (\alpha ,\beta )} , and therefore has a finite degree itself, from which it follows (as shown above) that γ {\displaystyle \gamma } is al... |
complex) nth roots where n is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of Galois theory (see Quintic equations and the Abel–Ruffini theorem). For exampl... |
introduction to the theory of numbers (5th ed.), Oxford: Clarendon, ISBN 0-19-853171-0 Ireland, Kenneth; Rosen, Michael (1990) [1st ed. 1982], A Classical Introduction to Modern Number Theory (2nd ed.), Berlin: Springer, doi:10.1007/978-1-4757-2103-4, ISBN 0-387-97329-X, MR 1070716 Lang, Serge (2002) [1st ed. 1965], Al... |
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable x {\displaystyle x} is denoted exp x {\displaystyle \exp x} or e x {\displaystyle e^{x}} , with the two notations used interchangeabl... |
d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its height (its y-coordinate) at that point. == Definitions and fundamental properties == There are several equivalent definitions of the exponential function, although of very differen... |
results from the uniqueness and the fact that the function f ( x ) = exp ( x + y ) / exp ( y ) {\displaystyle f(x)=\exp(x+y)/\exp(y)} satisfies the above definition. It can be proved that a function that satisfies this functional equation has the form x ↦ exp ( c x ) {\displaystyle x\mapsto \exp(cx)} if it is... |
) . {\displaystyle b^{x}=\exp(x\ln b).} In particular, if b is the Euler's number e = exp ( 1 ) , {\displaystyle e=\exp(1),} one has ln e = 1 {\displaystyle \ln e=1} (inverse function) and thus e x = exp ( x ) . {\displaystyle e^{x}=\exp(x).} This shows the equivalence of the two notations for the exponential fun... |
function. Exponential growth or exponential decay—where the variable change is proportional to the variable value—are thus modeled with exponential functions. Examples are unlimited population growth leading to Malthusian catastrophe, continuously compounded interest, and radioactive decay. If the modeling function has... |
is independent of both x {\displaystyle x} and d {\displaystyle d} . == Compound interest == The earliest occurrence of the exponential function was in Jacob Bernoulli's study of compound interests in 1683. This is this study that led Bernoulli to consider the number lim n → ∞ ( 1 + 1 n ) n {\displaystyle \lim _... |
function, and also denoted e z {\displaystyle e^{z}} or exp ( z ) {\displaystyle \exp(z)} . For distinguishing the complex case from the real one, the extended function is also called complex exponential function or simply complex exponential. Most of the definitions of the exponential function can be used ver... |
the functional identity. The complex conjugate of the complex exponential is e z ¯ = e z ¯ . {\displaystyle {\overline {e^{z}}}=e^{\overline {z}}.} Its modulus is | e z | = e | ℜ ( z ) | , {\displaystyle |e^{z}|=e^{|\Re (z)|},} where ℜ ( z ) {\displaystyle \Re (z)} denotes the real part of z {\displaystyle z} . ... |
third image shows the graph extended along the real x {\displaystyle x} axis. It shows the graph is a surface of revolution about the x {\displaystyle x} axis of the graph of the real exponential function, producing a horn or funnel shape. The fourth image shows the graph extended along the imaginary y {\displaystyle y... |
an are distinct complex numbers, then ea1z, ..., eanz are linearly independent over C ( z ) {\displaystyle \mathbb {C} (z)} , and hence ez is transcendental over C ( z ) {\displaystyle \mathbb {C} (z)} . == Computation == The Taylor series definition above is generally efficient for computing (an approximation of) e x ... |
}}}}}}}}} with a special case for z = 2: e 2 = 1 + 4 0 + 2 2 6 + 2 2 10 + 2 2 14 + ⋱ = 7 + 2 5 + 1 7 + 1 9 + 1 11 + ⋱ {\displaystyle e^{2}=1+{\cfrac {4}{0+{\cfrac {2^{2}}{6+{\cfrac {2^{2}}{10+{\cfrac {2^{2}}{14+\ddots }}}}}}}}=7+{\cfrac {2}{5+{\cfrac {1}{7+{\cfrac {1}{9+{\cfrac {1}{11+\ddots }}}}}}}}} This formula also... |
In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant... |
ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA) use those very same capitalised representations for the standard trig functions, whereas others (Python, SymPy, NumPy, Matlab, MAPLE, etc.) use lower-case. Hence, since 2009, the ISO 80000-2 standard has specified sol... |
{\textstyle {\frac {\pi }{2}}<y\leq \pi .} For a similar reason, the same authors define the range of arccosecant to be ( − π < y ≤ − π 2 {\textstyle (-\pi <y\leq -{\frac {\pi }{2}}} or 0 < y ≤ π 2 ) . {\textstyle 0<y\leq {\frac {\pi }{2}}).} ==== Domains ==== If x is allowed to be a complex number, then the range of y... |
∪ ( π , 2 π ) ∪ ⋯ = R ∖ π Z {\displaystyle {\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}} Domain of tangent tan {\displaystyle \tan } and secant sec {\displaystyle \sec } : The domains... |
.} This periodicity is reflected in the general inverses, where k {\displaystyle k} is some integer. The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values θ , {\displaystyle \theta ,} r , {\di... |
(h)={\begin{cases}0&{\text{if }}h{\text{ is even }}\\1&{\text{if }}h{\text{ is odd }}\\\end{cases}}} it is possible to write a solution to cos θ = x {\displaystyle \cos \theta =x} that doesn't involve the "plus or minus" ± {\displaystyle \,\pm \,} symbol: c o s θ = x {\displaystyle cos\;\theta =x\quad } if and only i... |
{\displaystyle x=\cos \pi =-1} ) then both statements (1) and (2) hold, although with different values for the integer k {\displaystyle k} : if K {\displaystyle K} is the integer from statement (1), meaning that θ = π + 2 π K {\displaystyle \theta =\pi +2\pi K} holds, then the integer k {\displaystyle k} for statement ... |
case) and so consequently, θ = ± arccos x + 2 π k = ± ( π 2 ) + 2 π ( 0 ) = ± π 2 . {\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.} This means that θ {\displaystyle \theta } could be either π / 2 {\displaystyle \,\pi /2\,} or − π / 2. {\displaystyl... |
( − θ ) = − cos ( π + θ ) = − cos ( π − θ ) = − sin ( π 2 + θ ) = − sin ( π 2 − θ ) = − sin ( − π 2 − θ ) = − sin ( − π 2 + θ ) = − sin ( 3 π 2 − θ ) = − sin ( − 3 π 2 + θ ) tan θ = − tan ( − θ ) = − tan ( π + θ ) = − tan ( π − θ ) = − cot ( π 2 + θ ) = − cot ( π 2 − θ ) = − cot ( − π 2 − ... |
some }}k\in \mathbb {Z} ,} which becomes: π 2 − θ = ( − 1 ) k arcsin ( x ) + π k for some k ∈ Z {\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} } where using the fact that ( − 1 ) k = ( − 1 ) − k {\displaystyle (-1)^{k}=(-1)^{-k}} and substituting h := − k ... |
( x ) arcsec ( 1 x ) = arccos ( x ) arctan ( 1 x ) = arccot ( x ) = π 2 − arctan ( x ) , if x > 0 arctan ( 1 x ) = arccot ( x ) − π = − π 2 − arctan ( x ) , if x < 0 arccot ( 1 x ) = arctan ( x ) = π 2 − arccot ( x ) , if x > 0 arccot ( 1 x ) = arctan ( x ) + π = 3 π 2 − arccot ( x ) , if x ... |
if }}0\leq x\leq 1\\\arcsin &\left({\sqrt {1-x^{2}}}\right)={\frac {\pi }{2}}-\operatorname {sgn}(x)\arcsin(x)\\\arctan(x)&=\arcsin \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\\\operatorname {arccot}(x)&=\arccos \left({\frac {x}{\sqrt {1+x^{2}}}}\right)\end{aligned}}} Whenever the square root of a complex number is used ... |
) = − 1 1 + z 2 ; z ≠ − i , + i d d z arcsec ( z ) = 1 z 2 1 − 1 z 2 ; z ≠ − 1 , 0 , + 1 d d z arccsc ( z ) = − 1 z 2 1 − 1 z 2 ; z ≠ − 1 , 0 , + 1 {\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin(z)&{}={\frac {1}{\sqrt {1-z^{2}}}}\;;&z&{}\neq -1,+1\\{\frac {d}{dz}}\arccos(z)&{}=-{\frac {1}{\sqrt {1-z^{2}}}}\;... |
When x equals 1, the integrals with limited domains are improper integrals, but still well-defined. === Infinite series === Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivat... |
{\displaystyle \arctan(z)={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac {2kz^{2}}{(2k+1)(1+z^{2})}}.} (The term in the sum for n = 0 is the empty product, so is 1.) Alternatively, this can be expressed as arctan ( z ) = ∑ n = 0 ∞ 2 2 n ( n ! ) 2 ( 2 n + 1 ) ! z 2 n + 1 ( 1 + z 2 ) n + 1 . {\displays... |
arccot ( z ) + 1 2 ln ( 1 + z 2 ) + C ∫ arcsec ( z ) d z = z arcsec ( z ) − ln [ z ( 1 + z 2 − 1 z 2 ) ] + C ∫ arccsc ( z ) d z = z arccsc ( z ) + ln [ z ( 1 + z 2 − 1 z 2 ) ] + C {\displaystyle {\begin{aligned}\int \arcsin(z)\,dz&{}=z\,\arcsin(z)+{\sqrt {1-z^{2}}}+C\\\int \arccos(z)\,dz&{}=z\,\arccos(z... |
v\,du} (i.e. integration by parts), set u = arcsin ( x ) d v = d x d u = d x 1 − x 2 v = x {\displaystyle {\begin{aligned}u&=\arcsin(x)&dv&=dx\\du&={\frac {dx}{\sqrt {1-x^{2}}}}&v&=x\end{aligned}}} Then ∫ arcsin ( x ) d x = x arcsin ( x ) − ∫ x 1 − x 2 d x , {\displaystyle \int \arcsin(x)\,dx=x\arcsin(x)-\int {\f... |
axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets; arccsc ( z ) = arcsin ( 1 z ) z ≠ − 1 , 0 , + 1 {\displaystyle \operatorname {arccsc}(z)=\arcsin \left({\frac {1}{z}}\right)\quad z\neq -1,0,+1} which has the same cut as arcsec. === Logarithmic forms === These funct... |
{\displaystyle ce^{i\theta }=a+ib} where a {\displaystyle a} is the adjacent side, b {\displaystyle b} is the opposite side, and c {\displaystyle c} is the hypotenuse. From here, we can solve for θ {\displaystyle \theta } . e ln ( c ) + i θ = a + i b ln c + i θ = ln ( a + i b ) θ = Im ( ln ( a + i b ) ) {\dis... |
( ln ( z + z 2 − 1 ) ) arctan ( z ) 1 z 1 + z 2 − i ln ( 1 + i z 1 + z 2 ) = − i 2 ln ( i − z i + z ) Im ( ln ( 1 + i z ) ) arccot ( z ) z 1 z 2 + 1 − i ln ( z + i z 2 + 1 ) = − i 2 ln ( z + i z − i ) Im ( ln ( z + i ) ) arcsec ( z ) 1 z 2 − 1 z − i ln ( 1 + i z 2 − 1 z ) = − i ln ( 1 z + ... |
of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z {\displaystyle z} , the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. It's possible to algeb... |
x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many computer programming languages, but it is now also common in oth... |
π/2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods. == See also == == Notes == == References == Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Ma... |
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbo... |
x.} Especially inconsistent is the conventional use of positive integer superscripts to indicate an exponent rather than function composition, e.g. sinh 2 x {\displaystyle \sinh ^{2}x} conventionally means ( sinh x ) 2 {\displaystyle (\sinh x)^{2}} and not sinh ( sinh x ) . {\displaystyle \sinh(\sinh x).} Becau... |
v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)} artanh u ± artanh v = artanh ( u ± v 1 ± u v ) {\displaystyle \operatorname {artanh} u\pm \operatorname {artanh} v=\operatorname {artanh} \left({\frac {u\pm v}{1\pm uv}}\right)} arcoth u ± arcoth v = arcoth ( 1 ± u v u ± v ) {\display... |
α cos α ) = ± arcosh ( 1 cos α ) {\displaystyle \operatorname {arsinh} \left(\tan \alpha \right)=\operatorname {artanh} \left(\sin \alpha \right)=\ln \left({\frac {1+\sin \alpha }{\cos \alpha }}\right)=\pm \operatorname {arcosh} \left({\frac {1}{\cos \alpha }}\right)} ln ( | tan α | ) = − artanh ( cos 2 α... |
\theta ={\sqrt {1+x^{2}}},} so d d x arsinh ( x ) = d θ d x = 1 d x / d θ = 1 1 + x 2 . {\displaystyle {\frac {d}{dx}}\operatorname {arsinh} (x)={\frac {d\theta }{dx}}={\frac {1}{dx/d\theta }}={\frac {1}{\sqrt {1+x^{2}}}}.} == Series expansions == Expansion series can be obtained for the above functions: arsinh x =... |
ln 2 x − ( ( 1 2 ) x 2 2 + ( 1 ⋅ 3 2 ⋅ 4 ) x 4 4 + ( 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ) x 6 6 + ⋯ ) = ln 2 x − ∑ n = 1 ∞ ( ( 2 n ) ! 2 2 n ( n ! ) 2 ) x 2 n 2 n , 0 < x ≤ 1 {\displaystyle {\begin{aligned}\operatorname {arsech} x=\operatorname {arcosh} {\frac {1}{x}}&=\ln {\frac {2}{x}}-\left(\left({\frac {1}{2}}\right){\frac {x... |
for non-positive real values of the variable, for which two different values of the logarithm reach the minimum. For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of § Definitions in t... |
∞). For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1]. Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and [1, ∞) for the inverse hyperbolic tangent, and [−1, 1] for the inverse hyperbolic cotangent. In vie... |
If the argument of the logarithm is real and negative, then z is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two branch cuts, the real intervals (−∞, 0] and [1, +∞). For z = 0, there is a singular point that is included in one of the branch cuts. =... |
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multip... |
quantities in the form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra, which is not limited to a particular domain and examines algebraic structures such as groups and rings. It extends beyond typical arithmetic operations by also covering other types of operations. Univ... |
equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplicatio... |
transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the righ... |
each other, like x 4 + 3 x y 2 + 5 x 3 − 1 {\displaystyle x^{4}+3xy^{2}+5x^{3}-1} . Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive integer power. A monomial is a polynomial with one term while two- and three-term polynomials are called ... |
+ . . . + a n x n = b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b} , where a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , ..., a n {\displaystyle a_{n}} and b {\displaystyle b} are constants. Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and 1 4 x − y = 4 {\displ... |
vectors and linear maps can be represented by matrices. It follows that the theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representatio... |
operation. The underlying set can contain mathematical objects other than numbers, and the operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations, such as rotations, under which the object remai... |
and are named and generally denoted similarly. A ring is a commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and distributive with respect to addition; that is, a ( b + c ) = a b + a c {\displaystyl... |
( y ) {\displaystyle h(x\circ y)=h(x)\star h(y)} . The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in the second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism t... |
exists a morphism from object a {\displaystyle a} to object b {\displaystyle b} , and another morphism from object b {\displaystyle b} to object c {\displaystyle c} , then there must also exist one from object a {\displaystyle a} to object c {\displaystyle c} . Composition of morphisms is required to be associ... |
methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides. Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra also in the 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries. In India, Brahma... |
basis of arbitrary algebraic operations. The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra, vector algebra, and matrix algebra. Influential early developments in abstract algebra were made by the German mathematicians David Hilbert, E... |
on algebraic theories such as group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on the existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and pr... |
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