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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Iterating $x^x$

If we set $f(x):=x^x$, then the iterations of $f(x)$ follow a steadily increasing in the complexity of the function as shown by https://www.desmos.com/calculator/wpgpilljhg. The iterations also show a very nice order to them, with their steadily…
Jacob Claassen
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Variation on Hermite Generating Function

I am having trouble using the Hermite generating function to determine $e^{t^2}\cos(2xt)$. I know the generating function is $e^{2tx-t^2}=\sum_{n=0}^\infty (-1)^n \frac{t^n}{n!}H_n(x)$ but can't seem to get anywhere. Can anyone help direct me on…
Aksel'sRose
  • 373
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How can I integrate a Bessel function divided by a "shifted" value?

Sorry to ask yet another "how do I do this integral" question! But I've really been having a hard time with this one. $$\int_0^\infty \frac{J_0(x)}{|C^2 - x^2|}\mathrm{d}x$$ I've been through a lot of identities giving results for expressions of the…
David Z
  • 3,539
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Hypergeometric series

If found that : "Assume further that this equation has e series solution $\sum a_ix^i$ whose coefficients are connected by two term recurrence formula. Then, such a series can be expressed in terms of hypergeometric series." [Bragg, 1969] how can…
MAK
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Expressing a series involving the Riemann zeta function in terms of known functions

We have the series: $$\sum_{n=1}^{\infty}\sin\left(\frac{\pi n}{2}\right)\frac{\zeta(n+1)}{(2\pi)^{n+1}}\frac{\Gamma(z)}{\Gamma(z-n)}\left[\psi^{0}(z-n)-\psi^{0}(z)\right]$$ Where $\psi^{0}(\cdot)$ is the digamma function, and $z$ is a complex…
Mohammad Al Jamal
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What is the closed form of certain sum in Mathematical Epidemiology?

The following sum appears in Mathematical Epidemiology in the context of the schistosomiasis: $$\sum _{p=0}^{\infty } \left( \sum _{q=0}^{\infty }{\frac {\min \{ p,q \} {\lambda}^{p+2+q}}{ \left( p+2 \right) !\,p!\, \left( q+2 \right) !\,q!}}…
Juan Ospina
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The integral $\int_{0}^\pi\frac{d\omega}{(2\sin(\omega/2))^{2\alpha}+2\lambda(2\sin(\omega/2))^\alpha\sin(\alpha\omega/2)+\lambda^2} $?

Can the integral \begin{equation} \int_{0}^\pi\frac{d\omega}{(2\sin(\omega/2))^{2\alpha}+2\lambda(2\sin(\omega/2))^\alpha\sin(\alpha\omega/2)+\lambda^2},\quad 0<\alpha<2,\quad \lambda>0 \end{equation} be evaluated? At first thought, I tried to write…
ecook
  • 399
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Uniqueness proof for a certain functional equation

Consider the following functional equation: \begin{equation}f(x)=kf(mx)\end{equation} where $x \in [0,1]$; $k>0$; $0
malenkiy_scot
  • 517
2
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Kronecker delta function notation

Can someone please help me, what does $\delta_{i-j-1}$ stand for? I have a matrix with elements $z_{ij}=\delta_{i-j-1}$ where $\delta_k$ is the Kronecker delta function (that's how it's written in the text). I'm confused with minus sign in the…
djogani
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Associated Legendre functions special values

I should prove that $$P_n^n(\cos \theta)=(2n-1)!! \sin^n\theta$$ $$P_n^m(0)=\begin{Bmatrix} (-1)^{(m+n)/2}\displaystyle\frac{(n+m-1)!!}{(n-m)!!} & \mbox{ if }& n+m \text{ even}\\ 0 & \mbox{if}& n+m \text{ odd}\end{Bmatrix}$$ $P_n^m(x)$ is a…
EQJ
  • 4,369
2
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1 answer

asymptotics of tricomi function

What's the asymptotic behavior of the Tricomi confluent hypergeometric function $U(a,b,z)$ when $|z|\to0$ and $b$ is complex but with $Re(b)=1$. The Abramowitz and Stegun handbook does not seem to include this case; They have the cases when…
Jason
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Solve a function and save the curve

I have a function BesselJ[1, X] = BesselJ[0, Y] and I need to solve Y for a range of X and then use Y in a different equation. Z(X) = Y + const*X I was only able to use ContourPlot to plot the first function but I want to create [X,Y] pair and then…
user1078719
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Do Bessel functions have any "nice" input/output pairs?

By analogy with the sine function, we have the following: $$\begin{align}\sin(0) &= 0, \\ \sin\left(\frac{\pi}{2}\right) &= 1, \\ \sin(\pi) &= 0,\end{align}$$ etc. I'm wondering if Bessel functions have anything similar.
William Jockusch
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What is a name for given below polynomials?

Physicists has a lot work with special polynomials, so I want to ask: Is there anybody who knows what are called this polynomials f(q) ( q as…
kakaz
  • 174
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Lipschitz bump function

Does anyone know of an example of a Lipschitz or Holder continuous bump function on $\mathbb{R}^n$? Any help is appreciated. Thank you.
canis89
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