# Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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### Many contradictory results for a single integral

I am interested in solving the integral $$\int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d}$$ for $a,b,c \in \mathbb{R}$ and $d \in \mathbb{N}$. As an early step, I have asked ...
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### How to port Matlab/Python's multivariate FoxH implementation in Mathematica?

Ref code MATLAB Python My implementation Are there some mistakes in my Mathematica code? Any help would be greatly appreciated. Attempt Version 1 (❌) ...
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1 vote
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### How to verify a FactorialPower identity?

How to verify that FactorialPower[x, m*n, k] is always the same as Product[FactorialPower[x - i*k, n, m*k], {i, 0, m - 1}] ...
62 views

### Clebsch–Gordan coefficient calculation for L-S basis in system of three particles

I am trying to calculate eq (A3) ...
1 vote
153 views

### Confused about the output of CosIntegral

I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ...
• 2,335
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### Unexpected problem while integrating the Weierstrass $\wp$ function

I would like to calculate some integrals in terms of the Weierstrass $\wp$ function. My code is as follows. ...
63 views

### How does the Prime function work?

This is somewhat interesting. I was trying to do a demo to abort a hopeless computation, so I decided to ask Mathematica for the ten trillionth prime. To be honest, I just typed ...
254 views

### Integration involving Piecewise function and DiracDelta function

I want to calculate an integration, which reads where $\delta\left(q_{23}^{01}\right)=\delta\left(1+q_1-q_2-q_3\right)$ and $\mathrm{min}(1,q_1,q_2,q_3)$ means the minimum of $(1,q_1,q_2,q_3)$. What ...
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### Issue in HypergeometricPFQ function:

I have a solution from integral: A = Integrate[x^n*(1 + x)^n*Exp[-n*x^2] /. n -> 1, {x, 0, Infinity}] //Expand (*1/2 + Sqrt[π]/4*) %//N (*0.943113*) Then I ...
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1 vote
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### Mathematica can't handle expressions with Bessel functions in the limit of a large argument [duplicate]

Consider the following function: ...
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### Question about SystemMeijerGDump* [closed]

Steps Start a fresh kernel, then copy the below sentence, run it. ...
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### How to transform this combination of $_2F_1$?

Following my previous question How to transform this MeijerG $G_{3,3}^{2,3}$ into a hypergeometric function $_2F_1$, I transformed a MeijerG function $G_{3,3}^{2,3}$ below into a combination of ...
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1 vote
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### Calculation time too long for FoxH

Clear["Global*"]; FoxH[{{}, {}}, {{{0, 0.5}, {-3, 1}}, {}}, 0.2] Related: No result from FoxH
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1 vote
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### Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order

When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following: ...
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### Abnormally long computation time using AppellF1 function

I am trying to use the AppellF1 function in Mathematica 13.3.1 on an Ubuntu machine with an i7 13700. The inbuilt function seems to be much slower in some cases ...
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### What is the exact formula Mathematica uses for Riemann zeta function? [closed]

I'm trying to find the formula of the zeta function used to plot this graph: This is taken from Wikipedia Riemann Zeta Function. It was created in Mathematica by ...
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### Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions

The integral of the product of Legendre polynomials and power functions: $I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$ The calculation result from the textbook is:  \begin{aligned} I & =0 \...
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1 vote
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### How to integrate Legendre polynomials with parameters?

The orthogonality of Legendre polynomials: $\int_{-1}^1 \mathrm{P}_l(x) \mathrm{P}_k(x) \mathrm{d} x=0, \quad k \neq l$ But ...
• 2,183
1 vote