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
2
votes
1 answer

"Comfort" function with tunable parameter

I'm trying to create a "comfort" function with the following characteristics: its domain is $(-\infty, +\infty)$; its range is $[0,1]$; it is at or near its maximum value ($1$) in some interval $[x_{c}-\delta, x_{c}+\delta]$; it is at or near its…
Orestis Tsinalis
  • 153
2
votes
1 answer

What is the relationship between hypergeometric function and Legendre polynomials?

I have an equation $$ -\frac{1}{2}y''(x)-b~\text{sech}^2(ax)~y=-\frac{k^2}{2}y $$ and I know that it has solution in terms of Legendre polynomials: $$ y_1(x)=P_s^\epsilon(\zeta) $$ $$ y_2(x)=Q_s^\epsilon(\zeta) $$ where $$ s=\frac{1}{2}…
xslittlegrass
  • 471
2
votes
2 answers

Simplify or FullSimplify doesn't work on Gamma function

I have this expression: Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I] which returns real values for all $n$. However, neither Simplify nor FullSimplify can reduce it to a form without involving $i$. Is there a way to force these commands to…
user91875
2
votes
0 answers

Generalizing infinite q - Pochammer symbols to more q symbols

The (infinite) q-Pochhammer symbols are defined as $$(a;q)_{\infty} := \prod_{k=0}^{\infty}(1-aq^{k})$$ I am interested a natural generalization of this symbol to multiple q's. In specific the 2 D and 3D cases which are defined as, For 2 symbols…
Prathyush
  • 341
2
votes
1 answer

Modified Bessel functions identity

Show that $\displaystyle e^x=I_0(x)+2\sum_{n=1}^\infty I_n(x)$ Using the following expansion $\displaystyle e^{i\rho\cos \varphi}=\sum_{m=-\infty}^{\infty}J_m(\rho)[ie^{i\varphi}]^m$ with $\varphi=0 , \rho = -ix$
tweelly
  • 117
2
votes
1 answer

trigamma fuction

Explain me please why the equality $\ \displaystyle\psi_1(z)=\sum_{n=0}^\infty \frac 1{(z+n)^2}\ $ is right if we define trigamma function like this : $\ \displaystyle\psi_1(z)=\frac{d^2}{dz^2}\ln\Gamma(z)$.
Alex
  • 21
2
votes
0 answers

Why is the Airy function defined with the pre-factor $(2\pi i)^{-1}$? For normalisation?

The Airy function is a solution to the differential equation \begin{align}y''(x)+x\, y(x)=0.\end{align} Using methods from complex analysis to find solutions to ODEs in terms of contour integrals, it is relatively easy to show that one solution to…
OldTomMorris
  • 121
2
votes
0 answers

Certain ratio/product of Jacobi theta functions appears periodic, simplification?

I have noticed through tinkering in Mathematica that this particular combination of theta functions of nome $e^{-\pi}$ is periodic and looks alot like cosine. Is there any identity I can use to realize a simpler form? $\frac{(\theta_4(ix/2))^2…
Dominic Reiss
  • 374
2
votes
0 answers

Relation between two Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
Dana
2
votes
0 answers

Sine integral and hyperbolic tangent.

Why does $\operatorname{Si}(x)$ resemble $\tanh(x)(1+\frac{\sin x}{x})$? How are both functions related? What is the simplest function that produces such a shape? What are the uses of such a function, and where does it arise naturally? Can such a…
user575696
  • 93
2
votes
1 answer

Special functions related to $\int_0^{2\pi} e^{(\cos{t})^2 + k \cos{t}} dt$

I'm interested in whether there are any closed-form representations of $$ \int_0^{2\pi} e^{(\cos{t})^2 + k \cos{t}} dt \quad \text{ or } \quad \int_0^{2\pi} e^{(q+\cos{t})^2} dt $$ in terms of other special functions, where $k$ and $q$ are real…
sourisse
  • 951
2
votes
1 answer

Identity concerning $e^{ia\sin{x}}$ as a series of bessel functions

Prove the following identity: \begin{equation} e^{ia\sin{x}}=\sum_{-\infty}^{+\infty}J_k(a) e^{ikx}, \end{equation} where $a$ is a real constant and $J_k$ is the Bessel function of the first type of order $k$.
quark1245
  • 1,001
2
votes
2 answers

Scale modified Bessel functions to then unscale later

So I have some variables $\,x_{1},\, x_{2},\, \nu\, =\, 12.654,\, 13.487,\, 0\,$ and the following function: $\dfrac{(x_{1}\cdot(-BesselK(\nu,x_{1}\cdot125))\cdot BesselI(\nu,x_{2}\cdot125))-(x_{2}\cdot BesselK(\nu+1,x_{1}\cdot 125)\cdot…
Rapid
  • 171
2
votes
1 answer

Question on legendre equation - part 2

I would like to know if is possible to have regular solutions of Legendre equation when the constant $l$ in the Legendre equation $(1-x^2)u''-2xu''+l(l+1)u=0$ is a non integer number? I am interested in polynomial solutions for non integer. Thanks…
TheStudent
  • 107
2
votes
1 answer

Question on Legendre equation

I have a doubt. If Legendre equation has a polynomial solution, is the constant $l$ in $l(l+1)$ necessarily a integer number? Asked in another way, is possible $l(l+1)$ be a integer if $l$ is not an integer? Thanks in advance.
TheStudent
  • 107
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