Special Functions

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 2 November 2009

Special Functions

We define the exponential function by $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots \text{.}$$

We define the sine and cosine functions by $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\frac{x^9}{9!}-\frac{x^{11}}{11!}+\frac{x^{13}}{13!}-\cdots \text{, and}$$ $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\frac{x^{10}}{10!}+\frac{x^{12}}{12!}-\cdots \text{,}$$ respectively. We also define the tangent, cotangent, secant and cosecant functions by $$\tan x=\frac{\sin x}{\cos x}\text{,}\cot x=\frac{1}{\tan x}\text{,}\sec x=\frac{1}{\cos x}\text{,}\text{and}\csc x=\frac{1}{\sin x}\text{,}$$ respectively.

We define the hyperbolic sine and hyperbolic cosine functions by $$\sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\frac{x^9}{9!}+\frac{x^{11}}{11!}+\frac{x^{13}}{13!}+\cdots \text{, and}$$ $$\cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+\frac{x^{10}}{10!}+\frac{x^{12}}{12!}+\cdots \text{,}$$ respectively. We also define the hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant functions by $$\tanh x=\frac{\sinh x}{\cosh x}\text{,}\coth x=\frac{1}{\tanh x}\text{,}\mathrm{sech}x=\frac{1}{\cosh x}\text{,}\text{and}\mathrm{csch}x=\frac{1}{\sinh x}\text{,}$$ respectively.

We define the inverse functions to these functions in the following way. $\ln x$ is the inverse function to $e^x$.
${\sin }^{-1}x$ is the inverse function to $\sin x$.
${\cos }^{-1}x$ is the inverse function to $\cos x$.
${\tan }^{-1}x$ is the inverse function to $\tan x$.
${\cot }^{-1}x$ is the inverse function to $\cot x$.
${\sec }^{-1}x$ is the inverse function to $\sec x$.
${\csc }^{-1}x$ is the inverse function to $\csc x$.
${\sinh }^{-1}x$ is the inverse function to $\sinh x$.
${\cosh }^{-1}x$ is the inverse function to $\cosh x$.
${\tanh }^{-1}x$ is the inverse function to $\tanh x$.
${\coth }^{-1}x$ is the inverse function to $\coth x$.
${\mathrm{sech}}^{-1}x$ is the inverse function to $\mathrm{sech}x$.
${\mathrm{csch}}^{-1}x$ is the inverse function to $\mathrm{csch}x$.

Note. [???] I ADDED THIS, NOT SURE IF YOU WANT IT In this notation, negative powers of the trigonometric and hyperboloc functions will never be used; for example, instead of using a negative power of $\sin x$ a positive power of $\csc x$ will be used. Therefore the notation is unambiguous; for example ${\sin }^{-1}x$ is regarded as the inverse to $\sin x$ without ambiguity. The symbols $\arcsin$, $\arccos$, and so on, do not appear in this notation.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)