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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Derivated function $f$ so that $f(x+y)=f(x)f(y)$ and $f'(x)f(y)=f(x)f'(y)$ for all $x,y \in \Bbb R$

Let $f:\Bbb R\to \Bbb R$ a derivated function in all $\Bbb R$ that satisfies the condition $$f(x+y)=f(x)f(y),\;\,\,\text{for all $x,y \in \Bbb R$}$$ I already tried that $f'(x)f(y)=f(x)f'(y)$ for all $x,y \in \Bbb R$ and that exists $c\in \Bbb R$…
Roiner Segura Cubero
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Integral involving the Legendre Polynomials

I'm trying to compute the following integral: $\int_a^{+\infty} \frac{dt}{(P_\lambda(\tanh{t}))^2}$ and to be honest I have no idea on how I should attack this problem. Is there any reference that I could look at to get on track (or where I'll find…
Nicolas
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Recognize as a special function?

Is the following function a special function of some kind $$ f(x) = \int_0^x (1+e^{-t})^{b}\,dt, $$ where $b>1$?
Jason
  • 765
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Asymptotic Expansions of Exponential Integral function

In NIST equation 8.20.2 what is meant by $(p)_{k}$ $$\mathop{E_{p}}\nolimits\!\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{% \infty}(-1)^{k}\frac{\left(p\right)_{k}}{z^{k}},$$
sky-light
  • 468
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Looking for a magnifyer function

I am looking for a function $f$ having the following characteristics: $f$ defined on $[0,1]$ $f(0)=0$ $f(1)=1$ $ \forall x \in ]0,1[, x 0$ $f'(1)=1$ $\lim\limits_{x\to0} f'(x)=+\infty$ Finally, I will…
julien
  • 65
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spherical Bessel function of the first kind

I'm trying to find the first few terms in the spherical Bessel functions of the $1^{st}$ kind and am not getting the third term correct. $j_l(kr)=\left(\frac{-r}{k}\right)^l\left(\frac{1}{r}\frac{d}{dr}\right)^lj_{0}(kr)$, and…
CuriousGeorge119
  • 1,207
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Mathieu function rescale problem

The Mathieu functions are the solutions for the equation $$ y''+(a-2q\cos(2z))y=0 $$ If we require the solution has the form $$ y(z) = e^{i r z}f(z) $$ where $f(z)$ is a periodic function with period of $2\pi$, then the parameter $a$ should…
xslittlegrass
  • 471
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multiderivative of incomplete gamma function

I spend 4 hours trying to solve it. I believe that I am either stuck or I am approaching the problem at the wrong angle. Here is my challenge: $$\frac{\partial^m }{\partial x^m}\left [ x^{-a}\gamma \left ( a,x \right ) \right ]=\left ( -1 \right…
Adam S
  • 157
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Solutions in terms of Bessel functions

I came across this question that asked to express solutions to: $$x^2y'' + xy' + (4x^2 - v^2)y = 0,\quad 0\le x<\infty$$ in terms of Bessel functions subject to the boundary conditions y(x) is bounded in the interval $0\le x< \infty$ and $y(0) =…
Kennan
  • 351
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What will the integration be ?

Using the orthogonality property of bessel function we have Now what will be the formula for the following
ARIJIT
  • 25
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Determine whether the functions are odd, even, or neither without using graphs

Determine whether the following functions are odd, even, or neither: $y=10(e^x+e^{-x})$ $y=e^{-x} \cos(2x)$ $y=x^8\sin (2x)$
bobby
  • 63
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Legendre associated functions

From this $$P^m_n(x)=\displaystyle\frac{1}{2^nn!}(1-x^2)^{m/2}\displaystyle\frac{d^{n+m}}{dx^{n+m}}(x^2-1)^n$$ I should derive that $$P^{-m}_n(x)=(-1)^n\displaystyle\frac{(n-m)!}{(n+m)!}P_m^n(x)$$ Using the Leibniz's formula to…
EQJ
  • 4,369
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1 answer

Show that these identities are true.

$\dfrac{\text{d}}{\text{d}x}[x^{-n}J_n(x)]=-x^{-n}J_{n+1}(x)$ $\dfrac{\text{d}}{\text{d}x}[x^{n}J_n(x)]=x^{n}J_{n-1}(x)$ $xJ_n'(x)=nJ_n(x)-xJ_{n+1}(x)$ Where…
Chuck Franklin
  • 550
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2 answers

Proving that a function is grounded

I need to prove that a function $f:[0,1]\times[0,1]\rightarrow [0,1]$, which is nondecreasing in each variable, is grounded (i.e. that $f(0,y)=0=f(x,0)$ for all $(x,y)$ in $[0,1]\times[0,1]$). Additionally, it is known that $f(t,1)=t=f(1,t)$ for all…
user109107
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Evaluating integral with Legendre polynomials

I want to do the following integral: $$ I=\int_{-1}^{1} x^n P_{n}(x) \rm{d}x $$ WITHOUT using Rodrigues' formula. I'm required to use $$ P_{n}(x) = \sum_{r=0}^{[n/2]} \frac{(-1)^r (2n-2r)!}{2^n r! (n-r)! (n-2 r)!} x^{n-2 r}. $$ Substituing $…
milmal
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