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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Lambert Function as a solution

The solution to the equation $$Xe^X=K$$ is given by $$X=W(K)$$ where $W$ is the Lambert function. Is it possible to adapt this such that we can find a solution for $$\frac{1-e^X}{X}=K?$$
David Simmons
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Bessel J function problem

Let $\xi_{0k}$ be the k-th positive zero of $J_{0}$ Bessel function. Determine the coefficients $c_k$, so that $1 = \sum^{\infty}_{k=1} c_kJ_0(\frac{x \xi_{0k}}{2})$. I don't see what to do, is this solvable?
James
  • 31
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Integral involving Marcum Q-function

I'm struggling to find an approximation for the following integral: $$\int\limits_0^\infty {{{\left[ {1 - Q\left( {a\sqrt t ,b} \right)} \right]}^n}{e^{ - t}}dt} $$ where ${Q\left( {a\sqrt t ,b} \right)}$ is the first order Marcum Q-function. Do…
BinhDDT
  • 301
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0 answers

Identity involving Bessel functions.

Mathematica gives a following identity: \begin{equation} (1) J_{1/3}(x) J_{2/3}(x) + J_{-1/3}(x)J_{-2/3}(x) = \frac{\sqrt{3}}{\pi x} \end{equation} How do I prove this identity? I was trying to use the recurrence…
Przemo
  • 11,331
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1 answer

Properties of Inverse Lambert W function

Starting with the two-branched Lambert W function (from Wikipedia): Suppose we just flip it like this: Is there a single power series for this $y=W^{-1}(x)$?
graveolensa
  • 5,678
3
votes
1 answer

Is this function known as a special function?

Let the function (of $z$) be defined by the formula $$ \int \limits_1^\infty\frac{dt}{t^{\large z}\sqrt{t^2-1}}. $$ Is it known as a special function?
limanac
  • 619
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3 answers

Airy function and modified Bessel function

I have got a question concerning the Airy functions in relation to the Bessel function. From Wiki, it is possible to see how $$ Ai(x)=\frac{1}{\pi}\sqrt{\frac{x}{3}}K_{1/3}\left(\frac{2}{3}x^{\frac{3}{2}}\right) $$ The question is: how can the Airy…
fpe
  • 247
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0 answers

Is there a Mathematics Database to find specific formula's meaning in real world?

I am doing a Science outreach project on Mathematics. I want make mathematics more interesting for students, since only when they realize its' beauty and useful in real world applications, they will learn mathematics in a more active manner. But one…
Dexter
  • 39
3
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1 answer

How to show $ \frac{Q_{k}(1-z^2)-zP_{k}(1-z^2)}{Q_{k}(1-z^2)+zP_{k}(1-z^2)}=\left(\frac{1-z}{1+z}\right)^{2k+1} $ analytically?

Let \begin{eqnarray*} P_{k}(z)={_2F_1}(-k,\frac{1}{2}-k; -2k; z), \ \label{e0} Q_{k}(z)={_2F_1}(-k,-\frac{1}{2}-k; -2k; z) , \end{eqnarray*} where $k\ge 1$ is an integer. How to show \begin{eqnarray*}\label{e1} …
Sunni
  • 4,536
3
votes
1 answer

Iterated exponential function

Is it possible to infinitely differentiably extend the function defined as $f(x+1,a)=e^{f(x,a)}$, $f(0,a)=a$ to non-integers? What I’m trying to do is derive a sort of «half logarithm», a function that if applied twice gives the natural logarithm.
Fernando Martínez
  • 75
  • 5
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0 answers

Proof of converting the product of Bessel function of the second kind and the sine function into Meijer $G$-function.

How can I derive the formula which converts the product of Bessel function of the second kind and the sine function into Meijer $G$-function, $$ \sin(\sqrt{z})Y_v(\sqrt{z})=\frac{1}{\sqrt{2}}G_{3,5}^{2,2} \left( z\left|…
mywyk5522
  • 31
3
votes
2 answers

Extension of trigonometric functions (like Bessel functions)

I'm physics student learning mathematical physics, now especially Bessel functions. I learned that the Bessel functions and that family are derived from the ODE $$ y''+y'/x+(1-\nu^2/x^2)y=0, $$ and $J_\nu(x)$, $Y_\nu(x)$, $H_\nu^{(1)}(x)$,…
Septacle
  • 461
3
votes
2 answers

error function (erf) with better precision

Currently I'm using this C++ routine to approximate the error function inline double erf(double x) { ASSERT(x == x); // check for invalid number const double a1 = 0.254829592; const double a2 = -0.284496736; const double a3 = …
demorge
  • 293
3
votes
1 answer

Question about the modified Bessel functions

I need to show that the following expression is always increasing. $$\frac{I_1(x)}{x I_0(x)} + \left(\frac{I_1(x)}{I_0(x)}\right)^2 \quad x>0,$$ where the $I_n$ represent the modified Bessel functions of the first kind. Wolfram Alpha tells me this…
almosteverywhere
  • 2,080
3
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0 answers

Solutions of legendre equation for $\vert x\vert \leq 1$

Why books say that is necessary in Legendre equation to have $l$ integer if you want regular solutions in $\vert x\vert \leq 1$. It seems not necessary. Thanks in advance.
TheStudent
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