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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About The Selected Identity $ J_\nu(z)=\frac z {2 \nu} (J_{\nu-1}(z)+J_{\nu+1}(z));$

I found this $$ J_\nu(z)=\frac z {2 \nu} (J_{\nu-1}(z)+J_{\nu+1}(z));$$ at the Selected Identities at Wikipedia's site on the Bessel function. How to prove this? The rest of the site doesn't give me anything to start with...
draks ...
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product of Meijer-G function and exponential

I have a product of the form $e^{-tx}\times G_{p,q}^{m,n}\left(x\left|\begin{smallmatrix}\mathbf{a}_n, \mathbf{a}_{p-n}\\ \mathbf{b}_m, \mathbf{b}_{q-m}\end{smallmatrix}\right.\right)$. I need to write the product as Meijer's-G function. I know the…
K.K.McDonald
  • 3,127
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Expression has sinc-like properties

I have observed that the expression $$\frac{\sin(\pi f_n t)}{f_n \sin(\pi t)}.\Pi(t)\approx\text{sinc}(\pi f_n t).\Pi(t)$$ as $f_n$ becomes large and $\Pi(t)$ is the rect($t$) function. Without the $\Pi(t)$, the left hand term is periodic so the…
HVW
  • 87
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Explicit form of Meijer G-function $G_{1,3}^{3,0}$

Please, can the following Meijer G-function $$G_{1,3}^{3,0}\left(a^2\Bigg| \begin{array}{c} n+2 \\ -\dfrac{1}{2},1,\dfrac{3}{2} \\ \end{array} \right)$$ MeijerG[{{},{2+n}},{{-(1/2),1,3/2},{}},a^2] be expressed more explicitly in…
Gallagher
  • 217
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How to find $xy=F(t)$ from $- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$?

I have the integral equation $$- \operatorname{Ei}{\left(x y e^{i \pi} \right)} = A t^{2} + B t + C$$ where $\operatorname{Ei}$ is the exponential integral and $A$, $B$, and $C$ are (finite) arbitrary constants. The approach I have thought of so far…
Galen
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Question regarding Langevin function and modified Bessel functions

Upon noticing that the plots of the functions $f(x) = \frac{I_1(x)-I_1(-x)}{I_0(x)+I_0(-x)}$ and $L(x) = \coth(x)-\frac1{x}$ are nearly the same I wanted to ask whether the functions are just generally equal or at least approximately the same for…
goalgetter666
  • 75
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Another way to write $ \frac{\Gamma(\frac{1}{2}-x)}{\Gamma(x)} $

How can I evaluate the value of $$ \frac{\Gamma(\frac{1}{2}-x)}{\Gamma(x)} $$ Is there a simple way to write this term, (Simple in the sense that there is no denominator term)? I have tried to use the duplication and the reflection formula of the…
Kashif
  • 710
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Using the beta function

Show that $\displaystyle \int_{0}^{\frac{\pi }{2}}\cos^{n} \theta d\theta=\int_{0}^{\frac{\pi }{2}}\sin^{n} \theta d\theta=\frac{\sqrt{\pi}[\frac{(n-1)}{2}]!}{2(\frac{n}{2})!}$
tweelly
  • 117
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Simplify square of sinc functions

I need to simplify if possible the following: $$\left(i^n\cdot \operatorname{sinc}\big(\pi(x-\tfrac{n}{2})\big)+(-i)^n\cdot \operatorname{sinc}\big(\pi(x+\tfrac{n}{2})\big)\right)^2$$ with $n \in \mathbf{N}$ and…
JFNJr
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How can I solve an integral by using hypergeometric function of the second kind?

This is an article that I've been studing for a while. I came across this integral equation: $$\mathrm dm=\frac{\mathrm dr}{{r}^{z+1}\left[1-{\left(\dfrac{r_h}{r}\right)}^{n+z-1}\right]}$$ in which $n$, $z$ and $r_h$ are parameters and $r$ and $m$…
Q Tara
  • 21
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Proving the identity of two functions

F1[b_, r_] := 1/(2 π) Integrate[ x*BesselK[0, Sqrt[(r - x*Cos[y])^2 + x^2*Sin[y]^2]], {x, 0, b}, {y, 0, 2 π}]; F2[b_, r_] := 1 - b*BesselK[1, b]*BesselI[0, r]; Using Mathematica, I can find that F1 and F2 give the same…
bendm
  • 51
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Existence of a continuos function in $[0, 1]$

Can exist a continuous function $f:[0, 1]\rightarrow \mathbb{R}$ so that $f(x)\in\mathbb{Q}$ if $x\in\mathbb{I\cap [0, 1]}$ and $f(x)\in\mathbb{I}$ if $x\in\mathbb{Q\cap [0, 1]}$? Why yes? why not?
Roiner Segura Cubero
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Gamma function minimum and maximum possible Value

Hello everyone I tried to find out the range of $\Gamma$ function but, I didn't .Can anyone please explain the range of $\Gamma$ i.e. the minimum and maximum value of $\Gamma(x)$ of any number $x$.
Harikesh
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Special function generating function

Can this be represented as linear combinations of complex exponential: $\frac{1}{\sqrt{(1-(e^{(i\theta)}+e^{-i\theta})t+t^2)}}$. This is the generating function of Legendre polynomials $P_l(cos(\theta)$.
User1825
  • 5
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solutions of exponential functional equation

Let $G$ be a group and $m_1, m_2: G\to \Bbb C$ are unbounded functions satisfying the exponential functional equation $$ m_j(x+y)=m_j(x)m_j(y),\,\, j=1, 2 $$ for all $x, y\in G$. If $|m_1(x)-m_2(x)|$ is a bounded function, then is it true…
Chung. J
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