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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

See Special Functions for a listing of the named mathematical functions currently supported by Mathematica, and The Wolfram Functions Site for an overview of the properties and identities satisfied by these functions. For additional information on special functions, you might be interested in the NIST Digital Library of Mathematical Functions.

Questions on how to use functions as a programming device (i.e. not about mathematical functions) should use the tag.

1420 questions
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How to define Mc and Ms Mathieu functions in terms of MathieuC and MathieuS?

Reading multiple books about Mathieu functions, I always come across notation like $\operatorname{ce}_r(z,q)$, $\operatorname{se}_r(z,q)$ for angular functions and $\operatorname{Ce}_r(z,q)$, $\operatorname{Se}_r(z,q)$ as well as…
Ruslan
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Multi-Factorial and Series with Triple-factorial terms

Let $n!^{(k)}$ denote a multi-factorial which is defined by $$ n!^{(k)} = \begin{cases} 1 & n \leqslant 0, \\ n, & 0 < n < k,\\ n\times(n-k)!^{(k)}, & n \geqslant k. \end{cases} …
Frenzy Li
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Lower branch of Lambert W function in mathematica

I am interested in values of the Lambert W function, which is defined as the solution to the equation $ z = W(z) e^{W(z)} . $ The solution is not, however, single-valued, but branches into two solutions for $z < 0$. See for example the first figure…
Echows
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why do these two Meijer G functions not cancel each other?

I encountered such expressions in Mathematica MeijerG[{{ }, {1, c + 1/2}}, {{0, c, c, c}, { }}, 1] + MeijerG[{{1}, {c + 1/2}}, {{c, c, c}, {0}}, 1] which in the notation of Wiki is given by the sum of the following Mellin–Barnes…
Jing
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CatalanNumber[-1] formula on functions.wolfram.com unequals result from Mathematica

The formula for CatalanNumber[-1] on this functions.wolfram.com page gives: -(1/2) whereas CatalanNumber[-1] directly evaluated gives? -1
axelclk
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7
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Reproducing the Integral Definition of the Modified Bessel function

I need to simplify some integral expressions in terms of special functions, such as the modified Bessel function of the first kind. See for example Eq. (5) on this page. Notice that the real expressions I am working with are somewhat more complex,…
TriSSSe
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AppellF1 calculation hangs indefinitely

The built-in AppellF1 function seems generally useless. For example, AppellF1[3/4, 1/2, 1/2, 7/4, (7 + 4 Sqrt[3]), (7 - 4 Sqrt[3])] hangs indefinitely on my system. For the curious, this particular instance can be naturally encountered in the…
Nickolas
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Legendre's relation for Elliptic Functions not satisfied in Mathematica?

I've been playing around with elliptic integrals lately and I came across Legendre's relation, relating the complete integrals of the first and second kind, $K(k)$ and $E(k)$…
ellipticmachinekk
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Is there a function for falling factorial in Mathematica

If I had to construct a function for falling factorial in mathematica I'd do something like that (hope I'm not mistaken): fallfact[x_,k_]:=$\prod_{j=0}^{k-1}(x-j)$ But is there a built-in function for falling factorial in Mathematica?
Remi.b
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5
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Is there a way to convert functions to hypergeometric functions?

For example, I have a function f[x_] = x*Sqrt[1 - x^2] + ArcSin[x] Is there a way to convert such functions to (a sum of) hypergeometric functions? In this case $f$ is an antiderivative of $2\sqrt{1-x^2}$, so it's possible. Funnily, for the Meijer…
Andrew
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Bivariate Hermite Polynomials

I know I can get the Hermite polynomials in a single variable with: HermiteH[n, x] Now I need the bivariate Hermite polynomials. I thought about building them with this procedure: basis = Flatten[TensorProduct[x^Range[0, n], y^Range[0,…
Enrico Detoma
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A strange behaviour with function $\,_2F_1$

I posted today a question on MSE 1 u=(5*(49*Pi^2*Zeta[3] - 558*Zeta[5]))/(77*Pi^2*Zeta[3] - 930*Zeta[5]) f[k_]:= Hypergeometric2F1[1/2, -k, 3/2, u] Computing N[f[10]] =-8.29107 N[f[10.]]=+3.02495 The second one is correct. Could any one…
Claude Leibovici
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How can I output all zero-point data by graphing?

How can I output all zero-point data by graphing? f[x_] := Hypergeometric1F1[ 1 + (-2 (2 - 2 Sqrt[2 x]) - 4 Sqrt[2 x])/(4 Sqrt[2 (-x)]), 2, 10 Sqrt[2 (-x)] ]; Plot[Re[f[x]], {x, 0, 10}] For example, in Hypergeometric1F1 above, by means of…
Vancheers
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Possible bug with EllipticPi

I am calculating the incomplete elliptic integral of the third kind in Mathematica 11.3 using EllipticPi. Since my range of phi goes beyond Pi/2 I tested my calculations before and realised that Mathematica evaluates EllipticPi[3,3*Pi/4,0.6] as…
Torben Frost
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5
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Appell series F3 on Mathematica

I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as $$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_1)_m(a_2)_n(b_1)_m(b_2)_n}{(c)_{m+n}m!\,n!}x^my^n.$$ I wonder if there exist something on…
ARWarrior
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