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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The graph of $x^{n}+y^{n}=r^{n}$ for sufficiently large $n$

The graph of the function $x^{n}+y^{n}=r^{n}$ for certain large values of $n$ looks suspiciously like a square. See this page from wolframalpha. Have any results been proven regarding this observation? What do we call this figure anyway?
kodyv
  • 1,461
13
votes
3 answers

Why does this function start swinging up and down so weirdly

Please have a look at the function: $$f(x) = \left(x + \frac{1}{x^x}\right)^x - x^x$$ You may see the plot on Wolfram Alpha. Why does it have such a weird behaviour from $x = 13$? It starts swinging up and down so weirdly!
soshial
  • 233
12
votes
3 answers

Does sinc function have any special inverse function defined?

We know that $y=xe^x$ cannot be solved for $x$ using elementary functions. The Lagrange inversion theorem can be used for finding a "new" function that would be the inverse function of the above equation. This special function is named "Lambert W…
curious maths
  • 121
10
votes
2 answers

How to show $\sum_{n=-\infty}^\infty J_n J_{n+m} = \delta(m)$?

The following is an identity concerning the Bessel functions of the first kind $J_n(x)$ for integers $n$ and $m$: $$\sum_{n=-\infty}^\infty J_n(x) J_{n+m}(x) = \delta(m)$$ where $\delta(x)$ is the Kronecker delta function. This can be derived from…
Tobin Fricke
  • 2,381
10
votes
1 answer

Orthogonality of Bessel functions

The orthogonality for Bessel functions is given by $\int_0 ^1 rJ_n(k_1r)J_n(k_2r) dr=0,\ (k_1 \neq k_2)\\ \neq 0, (k_1=k_2,\ J_n(k_1)=J_n(k_2)=0\ \mbox{or}\ J'_n(k_1)=J'_n(k_2)=0)$ This suggests a particular condition at the boundary $r=1$ for…
vijay
  • 327
8
votes
2 answers

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose domain is a subset of $\mathbb{R}^2$ and whose range…
user109107
  • 201
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8
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1 answer

On the Bessel function $J_n(z)$ for high $z$, with respect to $n$

Plotting the Bessel functions of the first kind $J_n(z)$ versus $n$ for some fixed $z\gg1$, it appears that there is a sharp cutoff just before $n=z$. Three questions: What is a reference describing this sharp cutoff? What is an expression for the…
Tobin Fricke
  • 2,381
8
votes
1 answer

Can it be shown that $Y_0(\lambda_n a)J_0(\lambda_n a) - J_0(\lambda_n a)Y_0(\lambda_n a) \ne 0$?

Background I am currently looking into the task of describing a transient temperature field $\theta(r,t)$ across the thickness $a \leq r \leq b$ of an infinitely long and hollow cylinder exposed to a sinusoidal temperature signal applied at the…
Daniel Bremberg
  • 81
6
votes
1 answer

Integral Representation of Bessel Function (K)

There is an integral representation for the modified Bessel function of the second (or third depending on who you talk to) kind (denoted $K_\nu$) that says: $$K_\nu(z) = \dfrac{\sqrt{\pi}(\frac{1}{2}z)^\nu}{\Gamma(\nu+\frac{1}{2})}\int_1^\infty…
Keaton
  • 1,214
6
votes
2 answers

Inverse Function of $(1-e^{-t})t$

I'm interested if there is some hope to obtain the inverse of $$ f(t) = (1-e^{-t})t $$ for $t$ positive. If there is a formula I suspect that the Lambert W function will be involved on it. Clever approximations are also welcomed (not looking for…
Bunder
  • 2,423
6
votes
1 answer

How to prove that only the sine waves keep their shape when they are added together and have the same period?

If $f(t)$ is periodic and $f(t) + C \cdot f(t + t_1)$ has the same shape of $f(t)$ for each value of $C$ and $t_1$, then $f(t)$ has the shape of a sine wave. Is there a simple proof? Is there an intuitive explanation? I mean without using, for…
Andrew
  • 61
6
votes
3 answers

Why can't erf be expressed in terms of elementary functions?

I have seen this claim on Wikipedia and other places. Which branch of mathematics does this result come from?
luqui
  • 713
6
votes
3 answers

if $w(x)=\sum \limits_{n=1}^\infty e^{-n^2\pi x}$ then how to show that $\frac {1+2w(x)} {1+2w(\frac{1}{x})}=\frac{1}{\sqrt{x}}$

$w(x)=\sum \limits_{n=1}^\infty e^{-n^2\pi x}$ $\frac {1+2w(x)} {1+2w(\frac{1}{x})}=\frac{1}{\sqrt{x}}$ I wonder how can be proved such a beautiful relation as shown in wolfram page I need to learn which technics be used to prove such relation…
Mathlover
  • 10,058
5
votes
1 answer

How to prove MeijerG identity

I have an expression which involves Meijer G-functions, to be precise $$ G_{1,4}^{2,1}\left(x \left\vert \begin{array}{c} 1 \\ 1,1,0,\frac{1}{2} \\ \end{array} \right)\right. $$ After messing around a bit with the Mathematica software I found…
Mike Jordan
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5
votes
1 answer

About digamma function

Show that $F(z)$ has the series expansion $\displaystyle F(z)=-\gamma + \sum_{n=2}^{\infty }(-1)^n\zeta (n)z^{n-1}$ where $\displaystyle \zeta (n)$ is the Riemann zeta function.
tweelly
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