Some special functions and their applications

December 19, 2016

Some special functions and their applications

Filed under: mathematics — ckrao @ 9:55 am

Here are some notes on special functions and where they may arise. We consider functions in applied mathematics beyond field (four arithmetic operations), composition and inverse operations applied to the power and exponential functions.

1. Bessel and related functions

Bessel functions of the first ( $J_{\alpha}(x)$ ) and second ( $Y_{\alpha}(x)$ ) kind of order $\alpha$ satisfy:

$\displaystyle x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - \alpha^2)y = 0$ .

Solutions for integer $\alpha$ arise in solving Laplace’s equation in cylindrical coordinates while solutions for half-integer $\alpha$ arise in solving the Helmholtz equation in spherical coordinates. Hence they come about in wave propagation, heat diffusion and electrostatic potential problems. The functions oscillate roughly periodically with amplitude decaying proportional to $1/\sqrt{x}$ . Note that $Y_{\alpha}(x)$ is the second linearly independent solution when $\alpha$ is an integer (for integer $n$ , $J_{-n}(x) = (-1)^n J_n(x)$ ). Also, for integer $n$ , $J_n$ has the generating function

$\displaystyle \sum_{n=-\infty}^\infty J_n(x) t^n = e^{(\frac{x}{2})(t-1/t)},$

the integral representations

$\displaystyle J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin(\tau)) \,d\tau = \frac{1}{2 \pi} \int_{-\pi}^\pi e^{i(n \tau - x \sin(\tau))} \,d\tau$

and satisfies the orthogonality relation

$\displaystyle \int_0^1 x J_\alpha(x u_{\alpha,m}) J_\alpha(x u_{\alpha,n}) \,dx = \frac{\delta_{m,n}}{2} [J_{\alpha+1}(u_{\alpha,m})]^2 = \frac{\delta_{m,n}}{2} [J_{\alpha}'(u_{\alpha,m})]^2,$

where $\alpha > -1$ , $\delta_{m,n}$ Kronecker delta, and $u_{\alpha, m}$ is the m-th zero of $J_{\alpha}(x)$ .

Modified Bessel functions of the first ( $I_{\alpha}(x)$ ) and second ( $K_{\alpha}(x)$ ) kind of order $\alpha$ satisfy:

$\displaystyle x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} - (x^2 + \alpha^2)y = 0$

(replacing $x$ with $ix$ in the previous equation).

The four functions may be expressed as follows.

$\displaystyle J_{\alpha}(x) = \sum_{m=0}^\infty \frac{(-1)^m}{m! \, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m+\alpha}$

$\displaystyle I_\alpha(x) = \sum_{m=0}^\infty \frac{1}{m! \, \Gamma(m+\alpha+1)} {\left(\frac{x}{2}\right)}^{2m+\alpha}$

$\displaystyle Y_\alpha(x) = \frac{J_\alpha(x) \cos(\alpha\pi) - J_{-\alpha}(x)}{\sin(\alpha\pi)}$

$\displaystyle K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)}$

(In the last formula we need to take a limit when $\alpha$ is an integer.)

Note that $K$ and $Y$ are singular at zero.

The Hankel functions $H_\alpha^{(1)}(x) = J_\alpha(x)+iY_\alpha(x)$ and $H_\alpha^{(2)}(x) = J_\alpha(x)-iY_\alpha(x)$ are also known as Bessel functions of the third kind.

The functions $J_\alpha$ , $Y_\alpha$ , $H_\alpha^{(1)}$ , and $H_\alpha^{(2)}$ all satisfy the recurrence relations (using $Z$ in place of any of these four functions)

$\displaystyle \frac{2\alpha}{x} Z_\alpha(x) = Z_{\alpha-1}(x) + Z_{\alpha+1}(x),$
$\displaystyle 2\frac{dZ_\alpha}{dx} = Z_{\alpha-1}(x) - Z_{\alpha+1}(x).$

Bessel functions of higher orders/derivatives can be calculated from lower ones via:

$\displaystyle \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ x^\alpha Z_{\alpha} (x) \right] = x^{\alpha - m} Z_{\alpha - m} (x),$
$\displaystyle \left( \frac{1}{x} \frac{d}{dx} \right)^m \left[ \frac{Z_\alpha (x)}{x^\alpha} \right] = (-1)^m \frac{Z_{\alpha + m} (x)}{x^{\alpha + m}}.$

In particular, note that $-J_1(x)$ is the derivative of $J_0(x)$ .

The Airy functions of the first ( $Ai(x)$ ) and second ( $Bi(x)$ ) kind satisfy

$\displaystyle \frac{d^2y}{dx^2} - xy = 0$ .

This arises as a solution to Schrödinger’s equation for a particle in a triangular potential well and also describes interference and refraction patterns.

2. Orthogonal polynomials

Hermite polynomials (the probabilists’ defintion) can be defined by:

$\displaystyle \mathit{He}_n(x)=(-1)^n e^{\frac{x^2}{2}}\frac{d^n}{dx^n}e^{-\frac{x^2}{2}}=\left (x-\frac{d}{dx} \right )^n \cdot 1$ ,

and are orthogonal with respect to weighting function $w(x) = e^{-x^2}$ on $(-\infty, \infty)$ .

They satisfy the differential equation

$\displaystyle \left(e^{-\frac{x^2}{2}}u'\right)' + \lambda e^{-\frac{1}{2}x^2}u = 0$

(where $\lambda$ is forced to be an integer if we insist $u$ be polynomially bounded at $\infty$ )

and the recurrence relation

$\displaystyle {\mathit{He}}_{n+1}(x)=x{\mathit{He}}_n(x)-{\mathit{He}}_n'(x)$ .

The first few such polynomials are $1, x, x^2-1, x^3-3x, \ldots$ . The Physicists’ Hermite polynomials $H_n(x)$ are related by $H_n(x)=2^{\tfrac{n}{2}}{\mathit{He}}_n(\sqrt{2} \,x)$ and arise for example as the eigenstates of the quantum harmonic oscillator.

Laguerre polynomials are defined by

$\displaystyle L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right) =\frac{1}{n!} \left( \frac{d}{dx} -1 \right) ^n x^n = \sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k$ ,

and are orthogonal with respect to $e^{-x}$ on $(0,\infty)$ .

They satisfy the differential equation

$\displaystyle xy'' + (1 - x)y' + ny = 0$ ,

recurrence relation

$\displaystyle L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}$ ,

and have generating function

$\displaystyle \sum_n^\infty t^n L_n(x)= \frac{1}{1-t} e^{-\frac{tx}{1-t}}.$

The first few values are $1, 1-x, (x^2-4x+2)/2$ . Note also that $L_{-n}(x)=e^xL_{n-1}(-x)$ .

The functions come up as the radial part of solution to Schrödinger’s equation for a one-electron atom.

Legendre polynomials can be defined by

$\displaystyle P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right]$

and are orthogonal with respect to the $L^2$ norm on $(-1,1)$ .

They satisfy the differential equation

$\displaystyle {d \over dx} \left[ (1-x^2) {d \over dx} P_n(x) \right] + n(n+1)P_n(x) = 0$ ,

recurrence relation

and have generating function

$\sum_{n=0}^\infty P_n(x) t^n = \displaystyle \frac{1}{\sqrt{1-2xt+t^2}}$ .

The first few values are $1, x, (3x^2-1)/2, (5x^3-3x)/2$ .

They arise in the expansion of the Newtonian potential $1/|x-x'|$ (multipole expansions) and Laplace’s equation where there is axial symmetry (spherical harmonics are expressed in terms of these).

Chebyshev polynomials of the 1st kind $T_n(x)$ can be defined by

$T_n(x) =\begin{cases} \cos(n\arccos(x)) & \ |x| \le 1 \\ \frac12 \left[ \left (x-\sqrt{x^2-1} \right )^n + \left (x+\sqrt{x^2-1} \right )^n \right] & \ |x| \ge 1 \\ \end{cases}$

and are orthogonal with respect to weighting function $w(x) = 1/\sqrt{1-x^2}$ in $(-1,1)$ .

They satisfy the differential equation

$\displaystyle (1-x^2)\,y'' - x\,y' + n^2\,y = 0$ ,

the relations

$\displaystyle T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$

$\displaystyle (1 - x^2)T_n'(x) = -nx T_n(x) + n T_{n-1}(x)$

and have generating function

$\displaystyle \sum_{n=0}^{\infty}T_n(x) t^n = \frac{1-tx}{1-2tx+t^2}.$

The first few values are $1, x, 2x^2-1, 4x^3-3x, \ldots$ . These polynomials arise in approximation theory, namely their roots are used as nodes in piecewise polynomial interpolation. The function $f(x) = \frac1{2^{n-1}}T_n(x)$ is the polynomial of leading coefficient 1 and degree n where the maximal absolute value on (-1,1) is minimal.

Chebyshev polynomials of the 2nd kind $U_n(x)$ are defined by

$\displaystyle U_n(x) = \frac{\left (x+\sqrt{x^2-1} \right )^{n+1} - \left (x-\sqrt{x^2-1} \right )^{n+1}}{2\sqrt{x^2-1}}$

and are orthogonal with respect to weighting function $w(x) = \sqrt{1-x^2}$ in $(-1,1)$ .

They satisfy the differential equation

$\displaystyle (1-x^2)\,y'' - 3x\,y' + n(n+2)\,y = 0$ ,

the recurrence relation

$\displaystyle U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$

and have generating function

$\displaystyle \sum_{n=0}^{\infty}U_n(x) t^n = \frac{1}{1-2 t x+t^2}.$

The first few values are $1, 2x, 4x^2-1, 8x^3-4x, \ldots$ . (There are also less well known Chebyshev polynomials of the third and fourth kind.)

Bessel polynomials $y_n(x)$ may be defined from Bessel functions via

$\displaystyle y_n(x)=\sqrt{\frac{2}{\pi x}}\,e^{1/x}K_{n+\frac 1 2}(1/x) = \sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\left(\frac{x}{2}\right)^k$ .

They satisfies the differential equation

$\displaystyle x^2\frac{d^2y_n(x)}{dx^2}+2(x\!+\!1)\frac{dy_n(x)}{dx}-n(n+1)y_n(x)=0$ .

The first few values are $1, x+1, 3x^2+3x+1,\ldots$ .

3. Integrals

The error function has the form

$\displaystyle \rm{erf}(x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt$ .

This can be interpreted as the probability a normally distributed random variable with zero mean and variance 1/2 is in the interval $(-x,x)$ .

The cdf of the normal distribution $\Phi(x)$ is related to this via $\Phi(x) = (1 + {\rm erf}(x/\sqrt{2})/2$ . Hence the tail probability of the standard normal distribution $Q(x)$ is $Q(x) = (1 - {\rm erf}(x/\sqrt{2}))/2$ .

Fresnel integrals are defined by

$\displaystyle S(x) =\int_0^x \sin(t^2)\,\mathrm{d}t=\sum_{n=0}^{\infty}(-1)^n\frac{x^{4n+3}}{(2n+1)!(4n+3)}$
$\displaystyle C(x) =\int_0^x \cos(t^2)\,\mathrm{d}t=\sum_{n=0}^{\infty}(-1)^n\frac{x^{4n+1}}{(2n)!(4n+1)}$

They have applications in optics.

The exponential integral ${\rm Ei}(x)$ (used in heat transfer applications) is defined by

$\displaystyle {\rm Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}t\,dt$ .

It is related to the logarithmic integral

$\displaystyle {\rm li} (x) = \int_0^x \frac{dt}{\ln t}$

by $\mathrm{li}(x) = \mathrm{Ei}(\ln x)$ (for real $x$ ).

The incomplete elliptic integral of the first, second and third kinds are defined by

$\displaystyle F(\varphi,k) = \int_0^\varphi \frac {d\theta}{\sqrt{1 - k^2 \sin^2 \theta}}$

$\displaystyle E(\varphi,k) = \int_0^\varphi \sqrt{1-k^2 \sin^2\theta}\, d\theta$

$\displaystyle \Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac{1}{1-n\sin^2 \theta} \frac {d\theta}{\sqrt{1-(\sin\theta\sin \alpha)^2}}$

Setting $\varphi = \pi/2$ gives the complete elliptic integrals.

Any integral of the form $\int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt$ , where $c$ is a constant, $R$ is a rational function of its arguments and $P(t)$ is a polynomial of 3rd or 4th degree with no repeated roots, may be expressed in terms of the elliptic integrals. The circumference of an ellipse of semi-major axis $a$ , semi-minor axis $b$ and eccentricity $e = \sqrt{1-b^2/a^2}$ is given by $4aE(e)$ , where $E(k)$ is the complete integral of the second kind.

(Some elliptic functions are related to inverse elliptic integral, hence their name.)

The (upper) incomplete Gamma function is defined by

$\displaystyle \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t$ .

It satisfies the recurrence relation $\Gamma(s+1,x)= s\Gamma(s,x) + x^{s} e^{-x}$ . Setting $s= 0$ gives the Gamma function which interpolates the factorial function.

The digamma function is the logarithmic derivative of the gamma function:

$\displaystyle \psi(x)=\frac{d}{dx}\ln\Big(\Gamma(x)\Big)=\frac{\Gamma'(x)}{\Gamma(x)}$ .

Due the relation $\psi(x+1) = \psi(x) + 1/x$ , this function appears in the regularisation of divergent integrals, e.g.

$\sum_{n=0}^{\infty} \frac{1}{n+a}= - \psi (a)$ .

The incomplete Beta function is defined by

$\displaystyle B(x;\,a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,\mathrm{d}t$ .

When setting $x=1$ this becomes the Beta function which is related to the gamma function via

$\displaystyle B(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ .

This can be extended to the multivariate Beta function, used in defining the Dirichlet function.

$\displaystyle B(\alpha_1,\ldots,\alpha_K) = \frac{\Gamma(\alpha_1) \cdots \Gamma(\alpha_K)}{\Gamma(\alpha_1 + \ldots + \alpha_K)}$ .

The polylogarithm, appearing as integrals of the Fermi–Dirac and Bose–Einstein distributions, is defined by

$\displaystyle {\rm Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s} = z + \frac{z^2}{2^s} + \frac{z^3}{3^s} + \cdots$

Note the special case ${\rm Li}_1(z) = -\ln (1-z)$ and the case $s=2$ is known as the dilogarithm. We also have the recursive formula

$\displaystyle {\rm Li}_{s+1}(z) = \int_0^z \frac {{\rm Li}_s(t)}{t}\,\mathrm{d}t$ .

4. Generalised Hypergeometric functions

All the above functions can be written in terms of generalised hypergeometric functions.

$\displaystyle {}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n} \, \frac {z^n} {n!}$

where $(a)_n = \Gamma(a+n)/\Gamma(a) = a(a+1)(a+2)...(a+n-1)$ for $n > 0$ or $(a)_0 = 1$ .

The special case $p=q=1$ is called a confluent hypergeometric function of the first kind, also written $M(a;b;z)$ .

This satisfies the differential equation (Kummer’s equation)

$\displaystyle \left (z\frac{d}{dz}+a \right )w = \left (z\frac{d}{dz}+b \right )\frac{dw}{dz}$ .

The Bessel, Hankel, Airy, Laguerre, error, exponential and logarithmic integral functions can be expressed in terms of this.

The case $p=2, q=1$ is sometimes called Gauss’s hypergeometric functions, or simply hypergeometric functions. This satisfies the differential equation

$\displaystyle \left (z\frac{d}{dz}+a \right ) \left (z\frac{d}{dz}+b \right )w =\left (z\frac{d}{dz}+c \right )\frac{dw}{dz}$ .

The Legendre, Hermite and Chebyshev, Beta, Gamma functions can be expressed in terms of this.

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