The exponential and trigonometric functions are simply functions that are solutions to very specific differential equation. The solution to the ordinary differential equation (ODE)
$y' - y = 0$
is $y(x) = e^x$, while the solutions to the ODE
$y'' + y = 0$
are $y_1(x) = sin(x)$ and $y_2(x) = cos(x)$. These are implemented as the functions exp(x), sin(x) and cos(x), respectively. There are, however, other differential equations, the solutions to which can be described as Taylor series, and these differential equations are sufficiently common that their solutions have been given special names. We'll look at some of these here:
The Bessel functions $J_\nu(x)$ and $Y_\nu(x)$ (also called cylinder functions) are solutions to homogenous Bessel's equation
$x^2 y'' + xy' + (x^2 - \nu^2)y = 0$
while the modified Bessel functions $I_\nu(x)$ and $K_\nu(x)$ are solutions to Bessel's modified equation
$x^2 y'' + xy' - (x^2 + \nu^2)y = 0$
These functions are implemented in Maple as BesselJ( nu, x ), BesselY( nu, x ), BesselI( nu, x ) and BesselK( nu, x ), respectively.
Another solution are Hankel's $H_\nu^{(1)}(x)$ and $H_\nu^{(2)}(x)$ functions, which are also solutions to Bessel's equation, and are defined as
$H_\nu^{(1)}(x) = J_\nu(x) + j Y_\nu(x)$
$H_\nu^{(2)}(x) = J_\nu(x) - j Y_\nu(x)$
These functions are implemented in Maple as HankelH1( nu, x ) and HankelH2( nu, x ), and these can be considered equivalent to saying that an alternative pair of solutions to the differential equation $y'' + y = 0$ is $y_1(x) = \cos(x) + j\sin(x)$ and $y_2(x) = \cos(x) - j\sin(x)$.
The Anger $\textrm{J}_\nu(x)$ function and the Weber $\textrm{E}_\nu(x)$ function are particular solutions to the non-homogeneous Bessel's equation
$x^2 y'' + xy' + (x^2 - \nu^2)y = \frac{(x - \nu)\sin(\pi\nu)}{\pi}$
and
$x^2 y'' + xy' + (x^2 - \nu^2)y = \frac{(x + \nu) + (x - \nu)\cos(\pi\nu)}{\pi}$,
respectively.
These functions are implemented in Maple as AngerJ( nu, x ) and WeberE( nu, x ), respectively.
The Lommel $s_{\mu, \nu}(x)$ and $S_{\mu, \nu}(x)$ are two particular solutions to the same non-homogeneous Bessel's equation
$x^2 y'' + xy' + (x^2 - \nu^2)y = x^{\mu + 1}$,
and these are implemented as LommelS1( mu, nu, x ) and LommelS2( mu, nu, x ), respectively.
The Struve $\textrm{H}_\nu(x)$ and $\textrm{L}_\nu(x)$ functions are particular solutions to the non-homogeneous Bessel's equation and modified Bessel's equation
$x^2 y'' + xy' + (x^2 - \nu^2)y = \frac{4\left(\frac{x}{2}\right)^{\nu + 1}}{\sqrt{\pi}\Gamma\left(\nu + \frac{1}{2}\right)}$
and
$x^2 y'' + xy' - (x^2 + \nu^2)y = \frac{4\left(\frac{x}{2}\right)^{\nu + 1}}{\sqrt{\pi}\Gamma\left(\nu + \frac{1}{2}\right)}$,
respectively. These are implemented as StruveH( nu, x ) and StruveL( nu, x ), respectively.
The Airy functions $\textrm{Ai}(x)$ and $\textrm{Bi}(x)$ are solutions to Airy's wave equation
$y'' - xy = 0$
and are implemented in Maple as AiryAi( x ) and AiryBi( x ), respectively.
Given $a$ and $q$, the Mathieu functions $\textrm{se}(x)$ and $\textrm{ce}(x)$ are solutions to Mathieu's equation
$y'' + (a - 2q \cos(2x))y = 0$
and are implemented in Maple as MathieuC( a, q, x ) and MathieuS( a, q, x ), respectively. These linearly independent solutions are chosen so that the first is even (cf. cosine) and the second is odd (cf. sine).
Given $q$, there are only countably many $a$ for which the solutions are either $\pi$- or $2\pi$-periodic. These are described as $\textrm{a}_n(q)$ for the even solutions and and $\textrm{b}_n(q)$ for the odd solutions, where $n$ is a positive integer.
Given a positive integer, these specific values of $a$ may be found using MathieuA( n, q ) and MathieuB( n, q ), respectively.
The functions implemented as MathieuCE( n, q, x ) and MathieuSE( n, q, x ) are equivalent to scalar multiples of MathieuC( MathieuA( n, q ), q, x ) and MathieuS( MathieuB( n, q ), q, x ), respectively.
Given a value $\eta$, the Coulomb wave functions $F_\ell(x)$ is a solution to the ODE
$y'' + \left( 1 - \frac{2 \eta}{x} - \frac{\ell(\ell + 1)}{x^2}\right)y = 0$
that is $0$ at $x = 0$. This function is implemented in Maple as CoulombF( ell, eta, x ). You will note when $\eta = \ell = 0$, the differential equation simplifies to $y'' + y = 0$, and thus we see that:
[> CoulombF( 0, 0, x );
$\sin(x)$
This function is related to a solution to the Schrodinger equation with a Coulomb potential.
Given a degree $a$, the parabolic cylinder equation is
$y'' - \left( \frac{x^2}{4} + a \right)y = 0$
has two solutions $\textrm{U}(a, x)$ and $\textrm{V}(a, x)$ that have asymptotic expansions equal to
$e^{-\frac{x^2}{4}} z^{-a-\frac{1}{2}}$
and
$\sqrt{\frac{2}{\pi}} e^{\frac{x^2}{4}} z^{a-\frac{1}{2}}$,
respectively. These two functions are implemented in Maple as CylinderU( a, x ) and CylinderV( a, x ), respectively.The CylinderD( a, x ) function is a solution to a related ODE
$y'' - \left( \frac{x^2}{4} - a -\frac{1}{2} \right)y = 0$
and is related to the solution $\textrm{U}$.
The Kummer equation is
$xy'' + (\nu - x)y' - \mu y = 0$
oas two linearly independent solutions $M(\mu, \nu, x)$ and $U(\mu, \nu, x)$. These two functions are implemented in Maple as KummerM( mu, nu, x ) and KummerU( mu, nu, x ), respectively.
Solutions to Laplace's equations in spherical coordinates can be written to include a radial component and an angular component. The solutions to the angular component form an orthogonal sequence of functions denoted $Y_\lambda^\mu(\theta, \phi)$, being defined on the surface of a sphere. These are are implemented in Maple as SphericalY( lambda, mu, theta, phi ).
[> int( int(
conjugate(SphericalY( 2, 2, theta, phi ))
*SphericalY( 2, 2, theta, phi )
*sin(theta),
theta = 0..Pi
), phi = 0..2*Pi );
$1$
The Fresnel sine and cosine integrals are defined as
$C(x) = \int_0^x \cos\left( \frac{\pi \xi^2}{2} \right ) d \xi$
$S(x) = \int_0^x \sin\left( \frac{\pi \xi^2}{2} \right ) d \xi$
and are implemented in Maple as FresenlS( x ) and FresenlC( x ), respectively.
There are alternate definitions of the Fresnel sine and cosine integrals given by
$\tilde{C}(x) = \int_0^x \cos\left( \xi^2 \right ) d \xi$
$\tilde{S}(x) = \int_0^x \sin\left( \xi^2 \right ) d \xi$
The solutions to these integrals are related to FresnelS(...) and FresnelC(...) by
$\tilde{S}(x) = \sqrt{\frac{\pi}{2} } S\left(\sqrt{\frac{2}{\pi}} x\right)$
and
$\tilde{C}(x) = \sqrt{\frac{\pi}{2} } C\left(\sqrt{\frac{2}{\pi}} x\right)$
respectively, as is shown in Maple:
[> int( sin( xi^2 ), xi = 0..x );
$\frac{1}{2} \sqrt{2} \sqrt{\pi} \textrm{FresnelS}\left( \frac{ \sqrt{2} x}{\sqrt{\pi}} \right )$
[> int( cos( xi^2 ), xi = 0..x );
$\frac{1}{2} \sqrt{2} \sqrt{\pi} \textrm{FresnelC}\left( \frac{ \sqrt{2} x}{\sqrt{\pi}} \right )$
The anti-derivative of $\frac{\sin(x)}{x}$ that is $0$ when $x = 0$ is the function $\textrm{Si}(x)$, and the anti-derivative that is $0$ as $x \rightarrow \infty$ is $si(x)$. These are implemented in Maple as Si( x ) and Ssi( x ), respectively. It happens that $\textrm{si}(x) = \textrm{Si}(x) - \frac{\pi}{2}$.
There is no anti-derivative of $\frac{\cos(x)}{x}$ that is $0$ when $x = 0$ (due to the singularity), but the anti-derivative that is $0$ as $x \rightarrow \infty$ is $\textrm{Ci}(x)$ and is implemented in Maple as Ci( x ).
Because $\cos(0) = 1$, we write $\frac{\cos(x)}{x} = \frac{1}{x} + \frac{\cos(x) - 1}{x}$, in which case, we have that $\textrm{Ci}(x) = \gamma + \ln(x) + \int_0^x \frac{\cos(\xi) - 1}{\xi} d\xi$.
The anti-derivative of $\frac{\sinh(x)}{x}$ that is $0$ when $x = 0$ is the function $Shi(x)$. This is implemented in Maple as Si( x ). It happens that $\textrm{Si}(x) = j\textrm{Shi}(x) $.
There is no anti-derivative of $\frac{\cosh(x)}{x}$ that is $0$ when $x = 0$ (due to the singularity) or $0$ as $x \rightarrow \infty$, as the value goes to infinity, and thus, the definition is just given to parallel that of $\textrm{Ci}(x)$:
$\textrm{Chi}(x) = \gamma + \ln(x) + \int_0^x \frac{\cosh(\xi) - 1}{\xi} d\xi$.
This function is implemented in Maple as Maple as Chi( x ).
The anti-derivative of $\frac{e^x}{x}$ that has a value of $0$ as $x \rightarrow -\infty$ and defined by analytic continuation for $x > 0$ is given by $\textrm{Ei}(x)$ and is implemented in Maple as Ei( x ).
The anti-derivative of $\frac{e^{-x}}{x}$ that has a value of $0$ as $x \rightarrow \infty$ and defined by analytic continuation for $x < 0$ is given by $\textrm{Ei}_1(x)$ and is implemented in Maple as Ei( 1, x ); that is
$\textrm{Ei}_1(x) = \int_x^\infty \frac{e^{-\xi}}{\xi} d\xi$.
After this, we have that
$\textrm{Ei}_{k + 1}(x) = \int_x^\infty \textrm{Ei}_k(x) d\xi$
for integer values of $x$, and consequently, we have that
[> Ei( 0, x );
$\frac{e^{-x}}{x}$
The anti-derivative of $e^{-x}$ that has a value of $0$ at $x = 0$ multiplied by the scalar $\frac{2}{\sqrt{\pi}}$ is given by the function $\textrm{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-\xi^2} d\xi$ and is implemented in maple as erf( x ).
The anti-derivative of $e^{x^2}$ that has a value of $0$ at $x = 0$ multiplied by the function $e^{-x^2}$ is given by the function $\textrm{dawson}(x) = e^{-x^2} \int_0^x e^{\xi^2} d\xi$ and is implemented in maple as dawson( x ).
We know that $(\cos(\theta), \sin(\theta))$ defines a point on a circle of radius $r = 1$ with an angle of $\theta$ relative to the positive $x$-axis.
Suppose we have an ellipse with the major axis along the $x$-axis defined by $\frac{x^2}{a^2} + y^2 = 1$, so $a \ge 1$. In this case, the eccentricity is $e = \sqrt{ 1 - \frac{1}{a^2} }$ and thus $0 \le e < 1$.
In this case, the ellipse can be traced out in a similar manner as the circle by plotting $(a \textrm{cn}(u, e), \textrm{sn}(u, e))$ where $\textrm{cn}(u, e)$ and $\textrm{sn}(u, e)$ are two of the Jacobi elliptic functions implemented in Maple as JacobiCN( x, e ) and JacobiSN( x, e ), respectively.
The distance from the origin to the ellipse is given by the function $a \textrm{dn}(u, e)$, and this is equal to $\sqrt{a^2\textrm{cn}^2(u, e) + \textrm{sn}^2(u, e)}$. This distance function is implemented as JacobiDN(u, e).
The relationship between $u$ and $\theta$ is given by $du = r d\theta$.
For further investigations into the Jacobi elliptic functions, see this playlist.
As with trigonometric functions, there are additional Jacobi elliptic functions that parallel tangent, cotangent, secant and cosecant, and these are implemented as JacobiSC( u, e ), JacobiCS( u, e ), JacobiNC( u, e ), JacobiNS( u, e ), as well as a few others that you can investigate on your own time.