Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,496 questions
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How do I efficiently generate, store and read in Gaunt(-like) coefficients from a file?
I'm calculating the mode-mixing between various spin-weighted spherical harmonics (SWSHs) $_sY_{l}^{m}(\theta,\varphi)$. In what follows the indices can take the following values, $l_{max}$ being some ...
- 33
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46
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Ability of Integrate[ ] to try changes of variable on its own
I have a question about Mathematica's ability to try changes of variable when performing symbolic integration. My example is
$$ \int_0^\infty dx\ x^n \exp \left( -h \sqrt{x^2 + a^2} \right) = \frac{2^{...
- 528
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1
answer
96
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Converting HurwitzZeta function to PolyGamma function
A generalization produces a result in terms of Zeta[s,a] function, which can be converted to PolyGamma[s-1,a] using the ...
- 645
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3
answers
123
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How to compute the Jacobian matrix using Mathematica [duplicate]
Apologies if this is posted to the wrong forum. Have been using Maple and Maxima for 20+ years, and have only recently started using Mathematica (v. 14). Having a heck of a time getting Mathematica to ...
- 101
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85
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Issue with Reduction of Complete Elliptic Integral of the Second Kind
I am attempting to reduce the following equation:
y ==
I’ve entered it to be reduced as such, where L = 3.95:
By what means may this be properly reduced for y?
Both WolframAlpha and Desmos provide ...
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1
answer
59
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Compute integrals in singular integral equation
I'm looking at this paper
https://arxiv.org/abs/2404.07307
and in particular I am interested in eqs. (16) (17), meaning I'd like to check the validity of (16) by insert it in (17).
So I'd like to ...
2
votes
1
answer
88
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How to force Mathematica to evaulate some values of LerchPhi function?
Mathematica cannot give LerchPhi[1,0,1]=-1/2, LerchPhi[1,1,1]=0, and LerchPhi[1,1,1/2]=0, as ...
- 645
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91
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A hypergeometric series function
Maple and Mathematica are very efficient to find closed form of a hypergeometric function as for example:
$$\sum _{k=0}^{\infty } \frac{(-4)^k z^k \Gamma \left(1 k+\frac{1}{6}\right) \Gamma \left(\...
- 15
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1
answer
238
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How to force Mathematica to evaulate LerchPhi[1,0,1] to -1/2
Mathematica fails to evaluate LerchPhi[1,0,1] (it gives ComplexInfinity). Based on the relation LerchPhi[1,q,1]=Zeta[q], we ...
- 645
3
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2
answers
270
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Solving transcendental equations involving Bessel functions
I am unable to solve an equation of type that explicitly involves a Bessel function in it. I want to extract the value of $t$ from this equation.
Whenever I try ...
- 119
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171
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Hypergeometric Function Integration Using Mellin-Barnes Representation
I have the following integral:
$$
I=\int_0^1 d \alpha d \beta d \gamma r(\gamma) s(\alpha, \beta) T(\alpha, \beta, \gamma)
$$
where I define
$$r(\gamma)=(\gamma(1-\gamma))^{ -1 / 2+\epsilon / 2}$$
and,...
- 139
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2
answers
84
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Simplification of hypergeometric functions
I am trying to calculate the variance of the mean parameterised beta-binomial distribution. The probability mass function is given by
...
- 13
3
votes
1
answer
151
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Why is ComplexExpand[HeavisideTheta[0]] == 1 while HeavisideTheta[0] is left undefined? [closed]
The following code:
ComplexExpand[HeavisideTheta[0]]
HeavisideTheta[0]
returns
1
HeavisideTheta[0]
Why when using ...
- 179
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vote
1
answer
198
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Integration and expansion of hypergeometric function
I am trying to reproduce the evaluation of the expansion in $\epsilon$ for the function in Appendix B of the paper arxiv.org/abs/0705.0676v3, which involves the hypergeometric function ${}_2F_1$:
\...
- 139
5
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2
answers
195
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Solve cannot find solutions if integer parameters are assumed
This is a very simple toy problem that illustrates the problem.
Start with a fresh kernel.
Quit[]
Make some simple assumptions
...
- 8,448
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66
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How to get the root from a hypergeometric function and plot it in the complex plane
I'm trying to reproduce the plot(Figure 3) of this paper. Here, the strip length's expression is given by the Equation.13
$t_\pm(z)= A_\pm \pm \iota \frac{z_t}{d}(\frac{z}{z_t})^d \times
{}_2F_1\left(\...
- 115
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2
answers
68
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Same integral giving different results
I am trying to solve the following integral using Mathematica
$\int_{0}^1 dz \int_{0}^{1} dz' \exp(-k|z-z'|)\cos[\pi p z] \cos[\pi q z]$,
with $p,q\in \mathbb{Z}$. To do so, I am doing the following:
<...
- 585
6
votes
2
answers
207
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Iteration Limit for expression involving Gamma functions
I am using an old version (13.0.0 Windows) so this might be a bug that has already been fixed, unfortunately I don't have access to a more recent version.
On a fresh kernel,
(Plain text: ...
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56
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How to plot potential function with Heaviside step function [closed]
How to plot $\frac{V(\varphi)}{V_{0}}$ with respect to $\frac{\varphi}{\varphi_{0}}$ for the following $V(\phi)$, where $\lambda_{0}=-1.0\times10^{-14},e=2.718,N_{UV}=15,g_{0}=0.0217,k=(16\pi^{2})^{-1}...
- 39
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2
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366
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An integral using Mathematica or otherwise
Consider the unit square integral $$I=\int_{(0,1)^5}\frac{x(1-x)y(1-y)u(1-u)v(1-v)w(1-w)}{(1-(1-xyuv)w)^2}\ dxdydudvdw$$
Using Mathematica or otherwise I need a closed form of I, possibly in terms of ...
- 291
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58
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Problem with Plot3D in Mathematica for plotting functions involving parabolic cylinder, exp, and error functions
I have the following problem to make plots of two functions:
Eq.1
$$
\begin{align*}
A_1(a, \Delta, \omega_0)
&= 1 - \frac{2\Delta^2}{\omega_0} + \frac{a^2}{\omega_0^2}
- \left[1 + \frac{...
- 11
1
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0
answers
97
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Avoiding underflow errors in calculation of $e^{-z}j_n(iz)$
I would like some advice on how to handle underflow errors in the calculation of functions of the form $e^{-z}j_n(iz)$.
As an example, consider the following code:
...
- 10.3k
4
votes
0
answers
140
views
Limit of a PolyLog does not work [bug]
This is an issue I am having with Mathematica version 13.1.0.0. I tried the following code, but it does not work as expected:
...
6
votes
1
answer
277
views
Closed form of an integral using Mathematica or otherwise
Define $$I=-\int_0^1\int_0^1 \frac{x^2(1-x)y^2(1-y)(2(1-xy)+(1+xy)\log(xy))^3}{(1-xy)^7}\ dxdy $$
Now using Wolfram Alpha
$I\approx 0.00133186$.
Using Mathematica or otherwise, I need to find a ...
- 291
3
votes
1
answer
102
views
Many contradictory results for a single integral
I am interested in solving the integral $$
\int_{\mathbb{R}} dx \frac{e^{i a x^2}}{(x - b - ic)^d(x - b + ic)^d}
$$
for $a,b,c \in \mathbb{R}$ and $d \in \mathbb{N}$. As an early step, I have asked ...
- 131
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answer
363
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How to port Matlab/Python's multivariate FoxH implementation in Mathematica?
Ref code
MATLAB
Python
My implementation
Are there some mistakes in my Mathematica code? Any help would be greatly appreciated.
Attempt Version 1 (❌)
...
- 2,067
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0
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49
views
SixJSymbol not triangular in basis transformation
The basis transformation is given by
$$\left| J^{P}, jd \right\rangle = \sum_ {S} (-1)^{J + L + Sd + Sq}\sqrt {(2 S + 1) (2 jd + 1)}\, \biggl\{\begin {array} {ccc} L & J & jd \\ Sq & Sd &...
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157
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Faster implementation with Mathematica for these p,q-analog functions
I have implemented some p-q analogue functions in Mathematica.
...
- 2,067
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1
answer
191
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Inverse Laplace transform does not give a soluton
I am trying to derive the inverse Laplace transform of the following Laplace transform:
$$
\mathcal{L}(d, \sigma; t) = \left(\sec \left(\frac{\pi d}{2}\right) \left(\left(\sigma ^4 t^2-1\right)^{-d/2}...
- 25
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2
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72
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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}] ...
- 181
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73
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Clebsch–Gordan coefficient calculation for L-S basis in system of three particles
I am trying to calculate eq (A3)
...
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2
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174
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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 ...
- 2,343
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1
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58
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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.
...
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0
answers
75
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 ...
- 51
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votes
5
answers
294
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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 ...
- 495
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3
answers
172
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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 ...
- 16.4k
1
vote
1
answer
77
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Mathematica can't handle expressions with Bessel functions in the limit of a large argument [duplicate]
Consider the following function:
...
- 5,963
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1
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103
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Question about System`MeijerGDump`* [closed]
Steps
Start a fresh kernel, then copy the below sentence, run it.
...
- 2,067
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answers
126
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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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How to transform this MeijerG $G_{3,3}^{2,3}$ into a hypergeometric function $ _2F_1 $
I want to transform the MeijerG function $G_{3,3}^{2,3}$ below into a hypergeometric function $ _2F_1 $:
$$G_{3,3}^{2,3}\left(\frac{\kappa -z}{z},\frac{1}{2} |
\begin{array}{c}
\frac{1-\kappa }{2},2-\...
- 878
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2
answers
205
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How to calculate this improper integral?
How to calculate the Integral involving MarcumQ function whose Integral interval is (0, +Infinity)?
Any help (code or reference) would be greatly appreciated.
...
- 2,067
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1
answer
81
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Bugs with `Integrate` when dealing with complex situation?
I'm trying to calculate $\int_0^{\infty } \exp \left(t \left(-x^3+(1+i) x\right)\right) dx$ with mathematica, and different assumptions on t give different results:
...
- 2,295
6
votes
3
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455
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Why do I get different results for the products of two identical expressions?
I have two expressions with the Gamma function that are identical:
...
- 3,369
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1
answer
56
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Laplace transform of special function
The Confluent hypergeometric function of first kind (aka Kummer's function) is defined as
$${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a%
\right)}\int_{0}^{1}e^{zt}t^{a-...
- 111
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1
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Calculation time too long for FoxH
Clear["Global`*"];
FoxH[{{}, {}}, {{{0, 0.5}, {-3, 1}}, {}}, 0.2]
Related:
No result from FoxH
- 2,067
2
votes
4
answers
127
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Specify the Method for `NIntegrate` to evaluate a integral of special functions
I have a special function as
$$f(x)=\frac{\Gamma \left(\frac{5}{3}\right) \, _2F_3\left(\frac{5}{12},\frac{11}{12};\frac{1}{3},\frac{1}{2},\frac{5}{6};-\frac{4
x^6}{729}\right)}{\pi }-\frac{x^2 \, ...
- 2,295
3
votes
2
answers
226
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How to calculate the PDF of product of two random variables from generalized gamma distributions?
Namely, we want to find the explicit formula of PDF of double Generalized Gamma distribution.
...
- 2,067
0
votes
2
answers
140
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Computing a highly oscillating Fourier Transform
I'm trying to compute the Fourier Transform of a diverging and oscillating function::
...
- 179
1
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0
answers
99
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Simplification of expression with PolyLog[4,2]
I am trying to simplify the following expression.
expr=1/2 \[Pi]^2 Log[2]^2-2/3 I \[Pi] Log[2]^3-4 PolyLog[4,2]+7/2 Log[2] Zeta[3]
This ...
- 1,907
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1
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83
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All possible tuples satisfying conditions
Let $\beta = (\beta_1, \beta_2, \ldots)$ be any given sequence of non-negative integers with all but finitely many $\beta_i$ zero. I want to collect all possible tuples $\beta^{\prime}= (\beta_1^{\...
