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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Properties of greatest integer function

I am curious to know some properties of the floor functions, for instance, $\lfloor a \cdot x \rfloor$, $\lfloor a1\cdot x1+a2\cdot x2 \rfloor$, etc. Is there any book that contains such properties ?
Naveen Crasta
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How to learn Digamma Function and how to take derivative of Gamma function?

How can I learn polygamma function?(More precisely digamma function)As I was learning Bessel Function of Second kind expressed in terms of power series digamma function is used.I have firm-grasp in Calculus(Single and Multivariable).But no exposure…
Deepnarayan Neogi
  • 5
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1 answer

Legendre's Chi- Function

I want to get the numerical value(twenty at thirty decimals) of $$\operatorname\chi_{2}(\frac{1}{\sqrt{2}})$$ Thanks you very much.
user178256
  • 5,467
-1
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1 answer

Gamma function expansion

I am trying to find the expansion of the $\Gamma(1+\epsilon)$ up to the quadratic order. The simplest way is (Can be found some other answers for almost the same question). $$ \Gamma(1+\epsilon)=\Gamma(1)+\epsilon \Gamma…
Tanmoy Pati
  • 169
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4 answers

Special Products Question:$ (p+\frac1p)=5, \ (p-(\frac1p))^2=?$

This is a special productions Question. $ (p+\frac1p)=5, \ (p-(\frac1p))^2=?$ Anyone has solution? Please answer fast if you know. I have exam tomorrow. I tried to square both sides and use formula $a^2\ + 2ab+ b^2$ but could do only half.
user545481
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2 answers

Prove bounded function

if f+g is bounded, then f and g are bounded Counter-example: if $f(x) = (x-2)$ and $g(x) = (-x+3) |f(x) + g(x)| = 1$. How, do i prove that f and g are bounded or not? Also |f(x) +g(x)| bounded?
Bob
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If $f,g>0$, f increasing and g decreasing on interval $(0,1)$ then, what conditions imposed for that $fg$ is decreasing in $(0,1)$?

The question is about of the conditions over $f$ and $g$ for the next: If $f,g>0$, $f$ increasing, $g$ decreasing then what conditions imposed for that $fg$ is decreasing in $(0,1)$? Where $f,g,fg: (0,1)\rightarrow \mathbb{R}$
juank2016
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