Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
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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-...
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Should expression evaluation depend on the choice of a variable name?
I am verifying the series representation of the Sonine polynomial or the associated Laguerre polynomial, which is
$$
L_m^{(n)}(x)=\sum\limits_{l=0}^n\left(-1\right)^l\binom{m+n}{m-l}\frac{x^l}{l!}=S_{...
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Mathieu Floquet solution
Mathematica provides the MathieuC[a,q,z] and MathieuS[a,q,z] functions - as well as some other Mathieu-related functions. Maple ...
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Euler angles and WignerD, a question of signs
Express a 3D point as a linear combination of Spherical Harmonics, then rotate that point to a new position and find the new expansion in SH :
...
- 1,333
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Elliptic theta function wont evaluate [closed]
It appears that Mathematica wont evaluate the Jacobi theta functions when the last argument has a magnitude greater than 1, for example: EllipticTheta[1, 1.5, 0.9] ...
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Simplify inverse of function
This would be a noob question, but I need help simplifying the inverse of an expression
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Solving a non-algebraic equation at the symbolic level [closed]
Versions 10, 11, 12, 13.0.0.0 and 13.2.0.0 solves the following system of equations
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Time independent perturbation theory for solving coupled differential equations
The eqexact1 and eqexact2 are the coupled differential equation of motion with g lets say a repulsive factor, that I choose.
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Series expansion message with special functions
I am trying to expand the following expression in alp,eps. Since the expression involves Hypergeometric2F1, I am using HypExp.
<...
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Can the Debye functions be implemented using built-in functions?
It is claimed in the comments here that the Debye functions can be implemented using built-in special functions. This is clearly true for some Debye functions, e.g., $D_n^{(1)}(x)$ for $n = 1, 2, 3$ (...
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Evaluate the time average of Mathieu functions
I defined a function composed of Mathieu's periodic functions:
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Derivative[0, 1, 1][QPolyGamma] cannot be calculated numerically?
During a more complicated calculation, I got an expression
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Dirac Delta does not converge
I have trouble evaluating a simple integral in Mathematica. I have the code:
...
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Derivatives in the SpinWeightedSpheroidalHarmonics package
Hello I'm using the SpinWeightedSpheroidalHarmonics package from the Black Hole Perturbation Toolkit
.
This package includes the SpinWeightedSphericalHarmonicY, ...
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Table of integrals involving modified Bessel function of the second kind
I need to compute integrals of the following form as accurately as possible (possibly with extended precision):
$$
I_{nl}(\omega)=\int_0^{1/2}\left(1-x^2\right)^{1/4} \sin (2 \pi l x)\; \sin (2 \pi ...
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Plot of some complicated functions containing special functions under both Sum and Product
I am trying to plot the $|f(\theta,\phi)|^2$, where the expression of $f(\theta,\phi)$ is as under:
$$ f =\sum_{n_j}\Pi_{j_x,j_y,j_z}P_{n_j}\sum_{l}A_s \text{e}^{-ia(l_1q_x+l_2q_y+l_3q_z)}\times
\...
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Finding the coefficients of a decomposition of complicated expression into products of special functions
I have the following expression which arises while studying minimal models in CFT. The 4-point amplitude of scalars is given by
$$G(z,\overline{z}) = ((1-z)(1-\overline{z}))^{\frac{m-1}{m+1}}\mathcal{...
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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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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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Implementing recurrence relation for an integral
I would like to implement the following recurrence relation,
$$I_{n+1}=-\log(2)I_n-\sum_{k=1}^n(-1)^k\left(1-\frac{1}{2^k}\right)\frac{n!\zeta(k+1)}{(n-k)!}I_{n-k}$$
with initial conditions,
$$I_0=\...
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Evaluation of an integral using Mathematica or otherwise
I need to find a closed form (in terms of known functions) using Mathematica or otherwise of $$\Re\left(\int_{\frac{1}{2}}^{1}\frac{\tan^{-1}\left(\frac
{1-x}{\sqrt{-i-x^2}}\right)}{\sqrt{-i-x^2}}\ dx\...
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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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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 ...
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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....
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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}]
...
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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 ...
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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 ...
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No result from FoxH
Why the following code shows no result?
FoxH[{{{0.5, 0.5}, {1, 1}}, {{-0.75, 0.75}, {0, 0.5}}}, {{{0, 1}, {0.5, 0.5}}, {{-0.75, 0.75}, {1., 0.5}}}, 1]
Tested on <...
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Closed form for a sum involving Bernoulli numbers
I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}(2^{2n}-1)\pi^{2n}B_{2n} {2n+4 \choose m}x^{2n+4}}{(2n)!}$$
where $0<x<1$, $B_{2n}$ denotes Bernoulli numbers , $m\in\mathbb{...
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How to prove an identity involving a hypergeometric function?
How to prove with the help of Mathematica the following statement?
$$ {}_2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};-1\right)=\frac{\pi -3 \sqrt{3} \log \left(\sqrt[3]{2}-1\right)-6 \tan^{-1}\left(\...
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Evaluating ${}_5F_4\left(1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right)$
Using Mathematica, how can I find a closed-form expression (in terms of elementary functions) of $$ {}_5F_4\left ( 1,1,1,1,1;\frac{3}{2},2,2,2;-\frac{1}{4} \right ),$$ where ${}_5F_4$ represents the ...
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Evaluating $\int_{1/\phi^2}^{1}\frac{\log^2x}{1-x}\ \log^2\left(\frac{1}{x}\left(\frac{1-x}{1+x}\right)^2\right)\ \mathrm{d}x$
I want to calculate the following integral:
$$I=\int_{1/\phi^2}^{1}\frac{\log^2x}{1-x}\ \log^2\left(\frac{1}{x}\left(\frac{1-x}{1+x}\right)^2\right)\ \mathrm{d}x$$
where $\phi=(\sqrt{5}+1)/2$ is the ...
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Inverse Laplace Transform of $\frac{\tan ^{-1}\left(\sqrt{s}\right)}{\sqrt{s}}$
How do I obtain the Inverse Laplace transform of the following expression?
$$\frac{\tan ^{-1}\left(\sqrt{s}\right)}{\sqrt{s}}$$
Using Mariusz Iwaniuk recipes with Feynmann trick and "Mellin ...
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Why can't NSolve solve for the obvious zeros?
Bug introduced in 13.2 or earlier and persisting through 13.2.1 or later
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How to verify the Riemann hypothesis up to a given height?
According to the Wikipedia article on Riemann hypothesis,
The number of zeros of the zeta function with imaginary part between $0$ and $T$ is given by
$$N(T)=\frac{...
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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 $f$...
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Weird expression for function Series-Expansion with Gamma function for different values of gamma coefficient
I extract the function jin[r] by solving eqsynin, and then I develop the function's series (around zero) to generate an equation for m1in and m2in based on esyn and gamma, knowing that the function ...
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Solving an implicit equation involving Elliptic integral
I have the following equation:
EllipticE[1/(1 + 16*(ja*me + jb*mo)^2)] +
16*(ja*me + jb*mo)^2*EllipticK[1/(1 + 16*(ja*me + jb*mo)^2)] == 0
I want to find values ...
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Approximation of the Fabius function with a quotient of exponentials
Approximation of the Fabius function $f(x) = \text{FabiusF}[x+1]\cdot \text{HeavisideTheta}[1-x^2]$ - FabiusF[x] doesn't work in Wolfram-Alpha
I am looking to figure out how well the displaced version ...
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