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
1
vote
1 answer

Bessel function radius of convergence

Assuming the series representation of the Bessel function $J_v(x)$ I was given that the radius of convergence was ∞. I've tried using the ratio test but I dunno how the gamma function would disappear...Would someone be able to explain it to me?
Kennan
  • 351
1
vote
1 answer

product of different order Bessel function integral

$\displaystyle w = \int_0^\infty r\; J_\mu(ar)\;J_\theta(br)\; \text{d}r $ I'd like to solve this integral ,where a and b are real and positive constant. any information regarding this integral help me alot.
pali
  • 11
1
vote
1 answer

Taylor series of the upper incomplete gamma function

What is the taylor series of the upper incomplete gamma function? I need it to approximate a difficult integration.
kazekage
  • 657
1
vote
0 answers

integral involving a bessel function

Try to evaluate the integral: $$ \int e^{-x^2}\dfrac{\Gamma(-1/2+ix)\Gamma(-1/2-ix)}{\Gamma(ix)\Gamma(-ix)}\Gamma(-1/4+ix)\Gamma(-1/4-ix)K_{2ix}(z)x, $$ either over $\mathbb{R}$ or over $\mathbb{R}^+$ (there is some symmetry here). Trying to use the…
Jason
  • 765
1
vote
1 answer

Define Beta function in terms of recursive relation?

The Beta function has recursive relation: $$ Beta(x,y) = Beta(x, y+1) + Beta(x+1, y) \! $$ Is there some least redundant condition, if added, will make the recursive relation an equivalent definition of the Beta function? The purpose of my…
Tim
  • 47,382
1
vote
2 answers

How prove this $|P_{n}(x)|\le 1$

let $0
math110
  • 93,304
1
vote
1 answer

parabolic cylinder function with negative argument

Say I wish to calculate $U(0,-1)$ which is $D_{-1/2}(-1)$ using the whitaker notation. According to http://dlmf.nist.gov/12.7.10, the identity is $$U(0,z)=\sqrt{\frac{z}{2\pi}}\mathcal{K}_{1/4}\left(\frac{1}{4}z^2\right)$$ However, the fact that z…
sachinruk
  • 931
  • 2
  • 9
  • 21
1
vote
1 answer

About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by $$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} \ln(\frac{u}{\Lambda^2})$$ According to Francis…
Craig Thone
  • 505
  • 2
  • 9
1
vote
0 answers

How to find the non singular condition for this equation

I'm trying to solve this one dimensional time-independent Schrodinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2}+V(x)\psi=\frac{\hbar^2k^2}{2m}\psi $$ where $$ V(x)=-\frac{\hbar^2b}{m}sech^2(a x) $$ ,$k$ is a parameter, $a$ and $b$ are…
xslittlegrass
  • 471
1
vote
1 answer

A function that creates a partition of values such that the sum is 1

I'm trying to find a function $g:\mathbb N\cup\left\{0\right\} \rightarrow\left(0,1\right)$, such that, given a (real) value $k \in \left(0,1\right)$ and an integer $i>1$, allows me to calculate a partition of the interval $(0,1)$ that fulfills the…
Barranka
  • 239
1
vote
2 answers

showing $\frac{H_{2n}(0)}{(2n)!} = \frac{(-1)^n}{n!}$

How do I show that, given the Hermite polynomials $H_n(x)\equiv (-1)^n e^{x^2}\frac{d^n}{dx^n}(e^{-x^2})$, $$\frac{H_{2n}(0)}{(2n)!} = \frac{(-1)^n}{n!}?$$
hasExams
  • 2,285
1
vote
0 answers

Recurrence relation for hypergeometric function

Are there known recurrence relations of the hypergeometric function with the structure: $$x{}_2F_1(a,b,c,x)=\sum_{n=1}^m c_n {}_2F_1(a+n,b-n,c,x)?$$ I have been looking around for a while, but I couldn't find anything
mattiav27
  • 413
1
vote
1 answer

Express definite integral in terms of special functions?

The integral is $$\int_0^{2\pi} \sqrt{1+a \sin x}\, dx$$ where $a$ is a parameter with $|a|<1$. I don't believe this can be expressed in terms of elementary functions, but surely it is related to some kind of special function? My main concern is a…
Cass Sackett
  • 11
1
vote
1 answer

Nonhomogeneous equation involving logarithm

This is probably a lame question, but what is the general approach to solving $\log z +z \sigma +1=0$ for $z$? Wolfram Alpha obtains a Lambert W-function, but I don't quite see how.
sigma.z.1980
  • 1,717
1
vote
0 answers

How to simplify the multiplication of Bessel and modified Bessel function of the first kind

I was wondering if anyone can help me with the procedure of simplifying the following formula: $$ e^{ - \beta x} \mathcal{T}_b \sum\limits_m {i^m J_m (kr)e^{im(\theta - \alpha )} } $$ using the following identity: $$ e^{ - \beta x} = e^{ - \beta…
math student
  • 33
  • 3
1 2 3
…
14 15