Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [special-functions]

This tag is for questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

Higher transcendental functions are frequently termed special functions. These functions were studied extensively in the eighteenth and nineteenth centuries-by Gauss, Euler, Abel, Jacobi, Weierstrass, Riemann, Hermite, Poincare, and other leading mathematicians of the day. Although many of the functions that they treated were quite recondite and are no longer of much interest today, others (such as the Riemann zeta function, the gamma function, and elliptic functions) are still intensively studied.

Before asking, please make sure that you define your notation very precisely, as
1. not everybody is familiar with the notation for special functions; and
2. a lot of special functions have different notational conventions, depending on the paper/book.

You might want to first check if the special function you are considering is discussed in Abramowitz and Stegun, the Digital Library of Mathematical Functions, or the Wolfram Functions site.

References:

https://en.wikipedia.org/wiki/Special_functions

https://en.wikipedia.org/wiki/List_of_special_functions_and_eponyms

"Special Functions and Their Applications" by R. Silverman

4626 questions
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Natural logarithm 2F1,hypergeometric

I find out that $\ln (1+z) = z _2F_1(1,1,2,-z)$ and that $\ln (1-z) = -z _2F_1(1,1,2,z)$, but what is $\ln\Big( \dfrac{1+z}{1-z} \Big)$? Is there a possibility to add two $_2F_1$? I mean what can I do with this $\ln\Big( \dfrac{1+z}{1-z} \Big) =…
aglaia
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3 answers

Derive the indefinite integral of a Heaviside Function

How can I prove that $$\int H(x-a)dx=(x-a)H(x-a)+constant$$ where $H(x-a)$ represents a jumped Heaviside function Thanks in advanced.
Celsk
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How to convert the following Meijer G function to hypergeometric functions

$$ G_{1,4}^{2,0}\left(z\left|\begin{smallmatrix}1\\ 0,0,\frac{1}{2},\frac{1}{2}\end{smallmatrix}\right.\right) $$
Dilruk Gallage
  • 95
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Question concerning the domain of a function

In the function $f:\mathbb{N}\rightarrow\mathbb{N}$ defined by $f(x)= 5x$ why is range not equal to co domain? I don't understand why this is not a surjective function while the same function is surjective if $f:\mathbb{R}\rightarrow\mathbb{R}$.
Papa Jones
  • 19
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Double S-Curve Function

I am looking for a function for a double s-curve. I am using hyperbolic tan function for an S-curve, however, I would like to have a function that has two ranges of higher slope and three ranges of lower slope. To have better classification in three…
profaisal
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Transformation formula of hypergeometric function

I think it's easier to ask your help about my secial case. So, can you please help me to find a relationship wetween $ _2F_1 (1,1,2-2/a; \frac{1}{1+\frac{R^a}{u.s.P}}) \;$ and $\; _2F_1 (1,1,2-2/a; \frac{1}{1+R^{a/2}}) $. Many thanks in advance.
adil
  • 73
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Integral involving Bessel functions

I am interested in evaluating the integral: $$\small\iint\limits_{[0,2\pi) \times [0,2\pi)} (x^2 + y^2)^{-1/2}((b-x)^2 + (c-y)^2)^{-1/2} J_{1}\left(\rho \sqrt{x^2 + y^2}\right)J_{1}\left(\rho \sqrt{(b-x)^2 + (c -…
user363087
  • 1,135
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0 answers

Deducing Laguerre Polynomials

Studying for a final and came accross this problem in the textbook. Considering I have no idea how to even start im a bit scared :). Any explanation would be greatly appreciated. problem: If f(x)is a polynomial of degree m, show that f(x) may be…
Aksel'sRose
  • 373
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2 answers

Beta function problem

Write the following integral in the form of Beta function $$\int_{0}^{\pi/4} \tan(2x)\, \mathrm{d}x$$ I know that I can use this $$B(p,q)=2 \int_{0}^{\pi/2} \sin^{2p-1}(x) \cos^{2q-1}(x)\, \mathrm{d}x$$
Br0kenb0W
  • 43
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Meijer's G-function differentiation

I am trying to calculate the derivative of the Meijer's G function, Based on wolfram function identities I have found in (07.34.20.0003.01) that the derivative is expressed…
mehyeddine
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Building a good penalizing function

Currently I'm working with the following penalizing function: $$ \psi(x) = \left\{ \begin{array}{lr} 0 & : x < 0 \\ \frac{1}{1+e^{\frac{1}{x-1}+\frac{1}{x}}} = \frac{g(x)}{g(x)+g(1-x)} & : x \in [0,1]\\ 1 & : x >1 …
mavillan
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1 answer

Simple form of LegendreQ function

for any n is positive integer LegendreP function can be expressed as $\displaystyle P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}\left[(x^2-1)^n\right]$. Let $\displaystyle q_n(x)=Q_n(x)-P_n(x)\log\left(\frac{1+x}{1-x}\right)$, then…
Liding Yao
  • 1,457
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1 answer

Legendre functions - Derivation of the recursion relation

From the following: $$\sum_0^\infty [ n(n-1)a_nx^{n-2} - n(n-1)a_n x^n -2na_nx^n + l(l+1)a_n x^n ] = 0$$ (a) I'm trying to get to: $$\sum_0^\infty [ (n+2)(n+1)a_{n+2} - [n(n+1) + l(l+1)]a_n]x^n = 0$$ (b) Unfortunately, I'm not being very…
iamatrain
  • 125
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How to prove the following function is independent of z?

I series expanded the following expression in Mathematica and the result is independent of z: $$(1-z)^{-a }\left(\, _2F_1(1,-a ;1-a ;1-z)+\, _2F_1\left(1,a ;a+1;\frac{1}{1-z}\right)-1\right)-(1-z)^{a } \left(\, _2F_1\left(1,-a ;1-a…
Nahc
  • 121
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1 answer

Derivation of an identity with beta function.

Beta function is defined as: $$B(x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for $\Re(x) , \Re(y)>0$, I want to show that:$\frac{B(x,y)}{c^y}=\int_0^\infty \frac{t^{x-1}dt}{(c+t)^{x+y}}$. I thought of changing variables to $s=\frac{c}{t}-c$, but only for…
MathematicalPhysicist
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