# Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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I have: ...
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### Strange result simplifying higher order BesselJ

Consider the following integral: $$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$. The solution of this should contain a single Bessel function: $$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
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### Expanding Pochammer symbols/Gamma function for simplifying expressions

TLDR: How to expand gamma functions or Pochammer symbols in an arbitrarily long product? Some context I am trying to find out a closed-form expression for $\langle r^\alpha\rangle$ for the non-...
1 vote
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1 vote
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### How to get the definite product of Sin squared? [closed]

I want to solve this for different values of D using mathematica. But I don't know how to write the program for this calculation. Please help. Edit: I tried to use the "Definite product" ...
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1 vote
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### Complicated expressions involving hypergeometric functions

This is in relation to my last post which was unanswered but I think I may have found somewhat of a workaround. I have the following preliminary definitions ...
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### Simplification of integration of product of Bessel functions

I have been trying to evaluate the following integral involving modified Bessel functions: \begin{align} \int r I_1 (r) K_1(r) dr \end{align} This integral has an explicit expression given in the ...
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### Manipulating Dirichlet characters and L functions

I read some basic knowledge about characters and L functions, and would like to play around with them in MMA. I tried to do the following things, but ending in minor success. (MMA notation used) ...
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### Can you give a faster implementation with Mathematica for these q-analog functions?

Now(2023), Mathematica has some QFunctions for q-anlaog. See Official Documentation. e.g.: QPochhammer QFactorial ...
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1 vote
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### A request on the kernel of the function ”SpheroidalEigenvalue“ in Mathemitica 11

I want to understand the algorithm of the function "SpheroidalEigenvlaue" in Mathematica 11, especially the code of the following command. Then I will try to use this algorithm to reproduce ...
120 views

### Complex integral with branch cuts

I am struggling with a complex double integral with multiple branch cuts. Even the single variable complex integral I find quite complicated due to the branch cut and the special functions involved in....
1 vote
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### Calculate $\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4$

I need to calculate the limit $$\lim_{n\to\infty} \left(\frac{e}{3}\right)^{3n} a_n^4$$ where $a_n=\sum_{r=0}^{n}\left(\binom{n}{r}\binom{n+r}{r}\right)^2$ and $e$ is the natural base of logarithm. ...
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### Is EllipticK defined correctly in Mathematica? [duplicate]

I needed to calculate F[a]=Integrate[1/(Sqrt[x^2+1]Sqrt[x^2+a^2]),{x,0,Infinity}] so I fed it to Mathematica. The result I got was: ...
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### Mathematica 13.3 forgot integral defining Bessel functions [duplicate]

I installed the newest version of Mathematica 13.3.0.0 on Mac. It looks like it forgot how to compute a simple integral Integrate[Cos[n t - z Sin[t]], {t, 0, Pi}] ...
1 vote
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### Mathematica function B

I came across this page on the Wolfram functions site: The conformal mapping from the triangle to the half plane. It describes a conformal map from an equilateral triangle to the upper half-plane ...
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### How is the analytical continuation for the HurwitzZeta function implemented?

Following up on this question, I am trying to understand the implementation details of the HurwitzZeta[x,y] function in Mathematica, particularly when the first ...
1 vote
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### Discrepancy with Hurwitz Zeta function

I've come across an issue while using Wolfram Mathematica that I don't quite understand. Consider the following symbolic computation: ...
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### Simplifying expression with BesselJ and BesselY and Gamma

I solved this ode by hand and got much simpler solution than Mathematica's. Both are correct. But I could not find a way to simplify Mathematica's solution to the simpler one. Could someone find a ...