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What is a Special Function? 📂Functions

What is a Special Function?

Description

Special functions, as called in mathematics, are usually the solutions to certain differential equations, defined by complex integrals, can’t be expressed by elementary functions, or have mathematically interesting properties. They often bear names of people, letters of the alphabet, or Greek letters, and it can be said that almost all named functions, except polynomial, trigonometric, exponential, and logarithmic functions, are referred to as special functions. As you can see from the explanation, whether a function is considered special or not is not conceptually and strictly distinguished.

Examples

Gamma Function

The Gamma Function is a special function defined by the following improper integral.

Γ(x):=0tx1etdt \Gamma (x) := \int\limits_{0}^{\infty} t^{x-1} e^{-t} dt

It appears in many fields including mathematics and physics, and can be understood as an extension of the factorial’s domain from natural numbers to real (complex) numbers. Indeed, for natural numbers nNn \in \mathbb{N}, the following holds.

Γ(n)=(n1)! \Gamma (n) = (n-1)!

List

Defined by Integrals

  • Gamma Function:

    Γ(x):=0tx1etdt \Gamma (x) := \int_{0}^{\infty} t^{x-1} e^{-t} dt

  • Beta Function:

    B(p,q):=01tp1(1t)q1dt B(p,q) := \int_{0}^{1}t^{p-1}(1-t)^{q-1}dt

  • Elliptic Integral of the Second Kind:

    E(ϕ,k)=0ϕ1k2sin2θdθ E(\phi, k)=\int_{0}^{\phi}\sqrt{1-k^{2} \sin ^{2} \theta}d\theta

Defined by Series

  • Bessel Function:

    Jν(x)=n=0(1)nΓ(n+1)Γ(n+ν+1)(x2)2n+ν J_{\nu}(x)=\sum \limits_{n=0}^{\infty} \frac{(-1)^{n} }{\Gamma (n+1) \Gamma (n+\nu+1)} \left(\frac{x}{2} \right)^{2n+\nu}

  • Bessel Function of the Second Kind:

    Nν(x)=Yν(x):=cos(νπ)Jν(x)Jν(x)sin(νπ) N_{\nu}(x) = Y_{\nu}(x) := \frac{\cos (\nu \pi)J_{\nu}(x)-J_{-\nu}(x)}{\sin (\nu\pi)}

  • Bessel Function of the Third Kind:

    Hp(1)(x):=Jp(x)+iNp(x)Hp(2)(x):=Jp(x)iNp(x) H_{p}^{(1)}(x) := J_{p}(x)+iN_{p}(x) \\ H_{p}^{(2)}(x) := J_{p}(x)-iN_{p}(x)

  • Modified Bessel Function:

    Iν(x)=iνJν(ix)Kν(x)=π2iν+1[Jν(ix)+iNν(ix)]=π2iν+1Hp(1)(ix)=π2Iν(x)Iν(x)sin(νπ) \begin{align*} I_{\nu}(x) & =i^{-\nu}J_{\nu}(ix) \\ K_{\nu}(x) &= \frac{\pi}{2}i^{\nu+1}\left[ J_{\nu}(ix)+iN_{\nu}(ix) \right] \\ &= \frac{\pi}{2}i^{\nu+1}H_{p}^{(1)}(ix) \\ &=\frac{\pi}{2}\frac{I_{-\nu}(x)-I_{\nu}(x)}{\sin (\nu\pi )} \end{align*}

  • Airy Function:

    Ai(x)=1πx3K1/3(23x2/3)Bi(x)=x3[I1/3(23x3/2)+I1/3(23x2/3)] \begin{align*} \operatorname{Ai}(x) &= \frac{1}{\pi}\sqrt{\frac{x}{3}}K_{1/3}\left( \frac{2}{3}x^{2/3} \right) \\ \operatorname{Bi}(x) &= \sqrt{\frac{x}{3}}\left[ I_{-1/3}\left( \frac{2}{3}x^{3/2} \right) + I_{1/3} \left( \frac{2}{3}x^{2/3} \right) \right] \end{align*}

  • Riemann Zeta Function:

    ζ(s):=nNns=p:prime(1ps)1 \zeta (s) := \sum_{n \in \mathbb{N}} n^{-s} = \prod_{p : \text{prime}} \left( 1- {p^{-s}} \right)^{-1}

Defined by Polynomials

  • Legendre Polynomials:

    Pl(x)=12ll!dldxl(x21)l P_{l}(x)=\dfrac{1}{2^{l} l!} \dfrac{d^{l}}{dx^{l}}(x^{2}-1)^{l}

  • Associated Legendre Polynomials:

    Plm(x)=(1x2)m2dmdxmPl(x)=(1x2)m2dmdxm[12ll!dldxl(x21)l] \begin{align*} P_{l}^{m}(x) &= (1-x ^{2})^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} } P_{l}(x) \\ &=(1-x ^{2})^{\frac{|m|}{2}} \frac{ d^{|m|} }{ dx^{|m|} }\left[ \dfrac{1}{2^l l!} \dfrac{d^l}{dx^l}(x^2-1)^l\right] \end{align*}

  • Hermite Polynomials:

    Hn=(1)nex2dndxnex2 H_{n} = (-1)^{n} e^{x^2} {{d^{n}} \over {dx^{n}}} e^{-x^2}

  • Laguerre Polynomials:

    Ln(x)=1n!exdndxn(xnex) L_{n}(x) = \frac{1}{n!}e^{x}\frac{ d ^{n}}{ dx^{n} }(x^{n}e^{-x})