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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Why does $\frac{n!n^x}{(x+1)_n}=\left(\frac{n}{n+1}\right)^x\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x$

Why does $$\frac{n!n^x}{(x+1)_n}=\left(\frac{n}{n+1}\right)^x\prod_{j=1}^{n}\left(1+\frac{x}{j}\right)^{-1}\left(1+\frac{1}{j}\right)^x$$ where the subscript n is the rising factorial in the left denominator my attempt: the index n in the product…
user5389726598465
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Prove that $|Si(1/x) - \pi/2| \leq 2x$ for $x>0$

Prove that $|Si(1/x) - \frac \pi 2| \leq 2x$ for $x>0$, where $Si(x)=\int_0^x \frac{\sin t}t \, dt$.
wlad
  • 8,185
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Asymptotic approximate solution of the parabolic cylinder differential equation

In chapter 3 (example 4) of the book "Advanced Mathematical Methods for Scientists and Engineers", by Bender and Orszag, I want to get the approximate solution for $+\infty$ for the parabolic cylinder differential equation: $$y'' + (\nu + 1/2…
Edison Santos
  • 133
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Special Functions.

How do I show that $\Gamma(v,x)=(v-1)\Gamma(v-1,x) + x^{v-1} e^{-v}$. I know I should use Integration by parts on the formula for upper incomplete gamma function. But the result is not coming out. A hint would be helpful, Thanks.
Prince
  • 57
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Is there a general form in terms of Euler constant for the integral: $\int_{0}^{\infty} (\ln x)^i e^{-x} dx$

The Euler constant arises in the integrals $$ \int_{0}^{\infty} \ln x ~e^{-x} dx=-\gamma $$ $$ \int_{0}^{\infty} (\ln x)^{2} e^{-x} dx=\gamma^2+\frac{\pi^2}{6}$$ $$ \int_{0}^{\infty} (\ln x)^{3} e^{-x} dx=-\gamma^3-\frac{\gamma…
ELAWADY
  • 101
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Partial derivatives of m=l spherical harmonics

The spherical harmonics, $Y^m_n(\theta, \varphi)$, are only defined when $m$ $\in$ $[n,n-1,...,-n+1,-n]$. However, the derivative relation with respect to $\theta$ requires $Y^{m+1}_n(\theta, \varphi)$ (see…
D. Price
  • 53
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Derivative of Hankel functions and Bessel functions

Dose anyone know about the formulations of derivative of Bessel and Hankel function as below, because when I just used the derivative of Bessel function and Hankel function as in the following…
gemmy9492
  • 99
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Special values of Meijer G-Function

We define a function $$ f(k)= G_{3,3}^{3,2}\left(1\left| \begin{array}{c} -1,\frac{2}{k}-2,\frac{1}{k}-1 \\ 0,\frac{1}{k}-2,\frac{2}{k}-2 \\ \end{array} \right.\right) $$ where $G(\cdot)$ denote Meijer G-Function. I noticed that for $k=2,3,4,6$,…
NonalcoholicBeer
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Why is the unit step function in the end is written as $u(t-t_1-t_0)$ instead of just $u(t-t_0)$?

So this is basically about signals and systems and how to compute the output of the system using only the impulse response and the given input signal- where x(t) is the input signal and h(t) is the impulse response. the output y(t) is simply the…
user507170
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Can a logarithm function with two variable be expressed as a Meijer-G function?

Let {x},{y}>0 and {a},{b}>0. can the function log(1+ax+by) be expressed as a Meijer-G function?
fatimaaa
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Need a function to fit the following data

I need to fit a function to this particular set of data points, but I can't find anything with this general form. I have found a great fit for the inverse of this graph using the form $$x(y) = \frac{a}{b\exp{(cy-y_c)}+d}+e$$ with the values…
Matheus Leão
  • 121
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Summing equally spaced samples of a periodic function

I'm a little stuck at the moment and wondered if someone could point me in the direction of the theory I need to read. I have a $2\pi$-periodic function, $f:\mathbb{R}\rightarrow\mathbb{R}$ which I have sampled at $N\in\mathbb{N}$ and $3N$ regularly…
Alanaj5
  • 49
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Extension of trigonometric functions (like Bessel functions)

This may be more related to math SE, but I got no answer from there, and maybe physics SE can tell me more about practicality of my question. I learned that the Bessel functions and that family are derived from the…
Septacle
  • 461
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W(a,b,x)=x＋(a－b)x^2/2!＋(a－b)(a－2b)x^3/3!＋...

Given $W(a,b,x)=x＋(a－b)x^2/2!＋(a－b)(a－2b)x^3/3!＋...$ and $Abs(x)＜1/(Abs(b))$ Prove: if $x=W(a,b,y)$ then $y=W(a,b,x)$
Gallo
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Contour integral of hypergeometric function

I need to numerically compute the hypergeometric function $$ _2F_1(k,1,c,z) $$ where $k$ is an integer, $c>2$ is a real number and $|z|<1$, using the integral representation $$ {}_2F_1[a,b;c;z] = {i \, \Gamma(c) \, e^{i\pi (b-c)} \over \Gamma(b)…
ABarr
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