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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Proof to $\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z)$

I tried to check the source of the proof to the equation $$\int_{0}^{\infty}\sin(t)t^{z-1}\,\mathrm{d}t= \sin\left ( \frac{\pi z}{2} \right )\Gamma(z),\qquad -1<\Re z < 1$$ but it only has a sketch of proof, since it was meant to be an exercise. In…
Hopeless
  • 906
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2 answers

Identifying a function

I am reading a piece of a physic paper where a function is mentioned without being given a name or reference - I guess it is a canonical one and that I should be familiar with. The expression goes as: $$|_2F_1(a,b;c;z)|$$ with $a,b,c \in {\mathbb…
Learning is a mess
  • 889
2
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1 answer

Hilbert Curve and Spatial properties

I'm trying to understand the following proposition about the Hilbert Curves: If (x,y) are the coordinates of a point within the unit square, and d is the distance along the curve when it reaches that point, then points that have nearby d…
Tarantula
  • 196
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1 answer

Special case of Meijer G function

I have an instance of the Meijer G function (using the definition from http://en.wikipedia.org/wiki/Meijer_G-Function, first equation there) that seems like, given its simplicity, it should be expressible in terms of another special function. The…
MarkWayne
  • 245
2
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3 answers

Function to express a time interval with results between 0 and 1

Its my first time on here and my maths is poor so please be kind. I am working on a Masters dissertation focused on document clustering methods in which I would like to apply a weight based on the time interval between two documents. I am looking…
Claire McMahon
  • 21
2
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0 answers

Proof for the Rodrigues formula for Neumann's Spherical functions.

I've been trying to prove the Rodrigues formula for Neumann's Spherical functions. The Neumann's Spherical functions are: $$N_n(x))=-(-x)^n\left[\frac{1}{x}\frac{d}{dx}\right]^n\frac{\cos{(x)}}{x}$$ As a guide, I followed the derivation of the…
Arturo Miranda
  • 21
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1 answer

inequality based on Hermite polynomial

How to prove that $$|H_n(x)| \leq |H_n(ix)|?$$ I have tried with the explicit representation of the Hermite polynomial, but can't reach the target. Any clue please.
user3575652
  • 347
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1 answer

Evaluation of definite integral in terms of Bessel function

Can I express the integral $\int_0^1[\cos (xt)/(1-t^2)]dt$ in terms of Bessel Polynomial? I tried by putting $t=\sin \theta$ and used the integral representation of Bessel's polynomial $J_n(x)=(1/\pi)\int_0^\pi \cos(n\theta-x\sin \theta)d\theta$. I…
Purushothaman
  • 497
1
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1 answer

Is this a special function?

Suppose $$ f(z;a) = \int_0^z t^{-a-1}\,(1+t)^{a}\,dt, $$ where $a>1$. Is this function known as a special function? It appears to be close to the following representation of the beta…
Jason
  • 765
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0 answers

Manipulating constants inside the Exponential Integral function

In the following form of the Exponential Integral function; $$ E_{n}(x+c) $$ where $E_{n}(x+c)$ is the exponential integral function, $x\in\mathbb{R}^+$ , $n\in\mathbb{N}$ , $c$ is a constant Is it possible to take out the constant part $c$ from the…
sky-light
  • 468
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0 answers

Identity involving the hypergeometric function

Let $n$ be an integers greater than one and $p,q$ be real numbers.How do I prove the following identity: \begin{equation} F_{2,1}\left[ \begin{array}{cc} \frac{3}{2} - n && 2-n \\ & \frac{5}{2} \end{array}; \left(\frac{q}{p}\right)^2 \right] = 3…
Przemo
  • 11,331
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1 answer

BesselJ function on negative real numbers

I have evaluated Bessel$J_v(x)$ with some real $v$ and negative real $x$ in MATHEMATICA. I cannot understand how the result is complex (non-real). I look at the series definition of BesselJ and I cannot see where an imaginary number comes in. What…
Fred
  • 75
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1 answer

Integral Bessel recurrence relation

I want to show that $\int x^vJ_{v-1}(x)dx = x^vJ_v(x) + C$. Now I know the recurrence relations of the Bessel equation/function and the one I need to use is $x^vJ_v(x) = x^vJ_{v-1}(x)$ I'm just thinking to set v as a constant and just integrate with…
Kennan
  • 351
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1 answer

What is the value of bessel functions at 0?

I would like to know what the value of the bessel functions of the first kind and the modified bessel of the first kind is at 0. I think for order 0 they are 1 and for orders greater than 0 they are 0. ie $J_0 (0) = 1$ and $J_v(0) = 0$ for $v$ >…
evan54
  • 261
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0 answers

How to solve this infinte series related to bessel integral?

$$\int_0^z t^{\rho} J_{\mu}(at) J_{\nu}(bt) dt = \frac{(\frac 1 2 a z)^{\mu} (\frac 1 2 bz)^{\nu} z^{\rho + 1}}{\Gamma(\mu + 1)\Gamma(\nu + 1)} \times \sum_{k = 0}^{\infty} \frac{(-1)^k (\frac 1 2 a z)^{2k} {}_2 F_1(-k, -\mu - k; \nu + 1;…
Pali majumder
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