Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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1
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1answer
70 views

Optimizing a series expansion for high order in $x$

I would like to expand the following function at $x \sim 0$ up to some high $x_\text{max} = \Delta_\text{max}$: $$16 \sum_{\Delta=1}^{\Delta_\text{max}} \sum_{s=0}^{\Delta-2} f_{\Delta,s} \frac{(s+\...
6
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3answers
453 views

Integrating a real function I get a complex value, while after variable transformation the result is real. Bug?

I have the following integral: Integrate[1/Sqrt[0.7 + 0.3*(1 + z)^3], {z, 0, ∞}, Assumptions -> z ∈ Reals] >> -3.36354 - 3.85013 I The output is complex, ...
26
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3answers
2k views

Eisenstein Series in Mathematica?

Mathematica doesn't seem to have built-in tools to deal with the Eisenstein series: $$\begin{align*} E_{2}(\tau)&= 1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n \tau}}{1-e^{2 \pi i n \tau}}\\ E_{...
1
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2answers
217 views

Integrating an exponential with upper incomplete gamma functions

I would greatly appreciate calculating an integral consisting of an upper incomplete gamma function and an exponential function. ...
6
votes
3answers
304 views

Express MeijerG as integral

For definite integrals MMA gives identities in terms of Meijer G-functions, e.g. $\begin{align}\sqrt{\pi}\int_0^\infty \textrm{e}^{-4x/t^2-t}\ \textrm{d}t &= G_{0,\,3}^{3,\,0} \left( x\left. \...
1
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1answer
54 views

Meaning of ProductLog [closed]

I have the following equation $$\frac{2\kappa}{(k+\kappa)^2}=2i\ell e^{-2ik\ell} $$ with $\kappa, \ell \in \mathbb{R}$ and $k\in \mathbb{C}$ which I want to solve for $k$. Using ...
7
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1answer
154 views

Bug in Series[Pochhammer[1+n, n], {n,Infinity,1}]?

Bug introduced after 5.2, fixed between 12.0.0 and 12.1.1. The function Pochhammer[1 + n, n] tends to infinity. We have ...
1
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0answers
46 views

How to intelligently use FullSimplify and FunctionExpand to simplify complex sums

I am trying to find a compact form of some sums which is related with some Bayesian probability factor (not so relevant, if required further explanation please ask). The point is that I know that the ...
2
votes
1answer
225 views

Plot of Laguerre-Gaussian wavefront

I have tried to plot the wavefront of a Laguerre-Gaussian beam, I know that to do that I have to plot the set of all points where the wave has the same phase. I have try to use the conditional ...
0
votes
1answer
38 views

Recognizing the type of hypergeometric series based on the dominant terms

I am solving a (infinitely long) differential equation which has the solution $$ y(r)=-\frac{c}{5}+\frac{l^4c^3}{20r^5}+\frac{l^{6}c^5}{16r^9}+\mathcal{O}(l^8), $$ where I am not sure about the sign ...
1
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1answer
119 views

Spherical Harmonics Parity

In Mathematica's documentation, the Spherical Harmonics are said to be defined as follows, for $l \geq 0$: Furthermore, we know that $\cos(x)=\cos(-x)$, hence one can be led to believe that $Y_l^m(-\...
0
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1answer
41 views

Problem when applying FindRoot to a very complicated function [closed]

The error I get when evaluating my code is: FindRoot: The function value [...] is not a list of numbers with dimensions {1} at {q}={2.25}. I have read a lot of questions regarding this error, though ...
0
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0answers
67 views

How to change Machine Precision digits to meet the tolerances

I am trying to solve for Tcm and Mag by solving nonlinear equations using FindRoot command using following code: ...
4
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3answers
524 views

Calculating $\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$

How can we force Mathematica to compute this limit? $$\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$$ It seems it ...
1
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0answers
30 views

Jacobian elliptic function argument [closed]

I have a C++ code that computes jacobian elliptic sn, cn and ...
0
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0answers
38 views

How can I plot complex functions which take small values?

I would like to know, how can I plot some functions that have oscillating behavior each function can be plot independent. The goal is to show both oscillations even is they are very small values. ...
0
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0answers
54 views

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
1
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1answer
102 views

Solve using PolyGamma function

I am trying to solve for the value of y. My code is: x = 0.165; f = -Log[y] - PolyGamma[0.5 + 0.2*(x/y)] + PolyGamma[0.5]; Solve[f == 0, y] Running gave me ...
1
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0answers
61 views

Why is this integrand not integrating to a Bessel function? [duplicate]

I know from the identities of Bessel functions that the following is true: $$ J_{m}\left( x \right) = \frac{ 1 }{ 2 \ \pi \ i^{m} } \int_{0}^{2 \pi} \ d\phi \ e^{i \left( x \cos{\phi} \ - \ m \ \phi \...
7
votes
1answer
172 views

Solve`DirInf[] — Meaningful value or just a bug?

Bug introduced in 4.1 or earlier and persisting through 12.2.0. or later To the problem below, I get four independent, incomplete solutions, three in terms of ...
0
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0answers
57 views

Spheroidal Harmonics

There is a (scalar) field P varying on a spheroid (squashed) surface. This field is a function of four independent components ...
0
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1answer
44 views

EllipticPi argument is complex and can not be plotted. How to handle this problem?

inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 + r))/(3000 r)]) This is my function of r, now I integrated it w r t r ...
2
votes
0answers
67 views

Understanding the behavior of HypergeometricPFQ

this is my very post here, so I apologize for any possible format issue. I am using HypergeometricPFQ functions (more exactly $_3F_2$) as approximants for other more complicated functions. Here are ...
0
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1answer
30 views

Search for terms contatining error functions

question might have an answer in this post (Efficient Search for specific Terms in symbolic Expression) but i don't understand how to convert it to my specific case Through some definite integrals I ...
0
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1answer
84 views

Jacobi Elliptic Function Solution using DSolve

I am happy that I have a solution to a particular differential equation by hand: ...
4
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1answer
171 views

HypExp and HPL packages for hypergeometric functions: Evaluating a function HPL[{minus,plus},x]?

I am currently using the HypExp and HPL packages, which are useful for expanding hypergeometric functions in series around integer or half-integer values, as in common in dimensional regularization ...
0
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0answers
55 views

Finding an analytic solution with a JacobiSD function

We are searching for an analytic solution to the given equation for $f_\text{n}(u)$, for $u \in [0, d/2]$ (this problem is a snippet from this paper here) $$-\partial^2_{u} f_\text{n} + \left\lbrack 1 ...
4
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1answer
80 views

Force EllipticTheta to “use” $(e^{\pi i\tau})^{\lambda}=e^{\pi i\tau \lambda}$

Define $\theta_2 (q)=2\sum_{n\ge 0}q^{(n+1/2)^2}$ and $\theta_3 (q)=1+2\sum_{n\ge 1}q^{n^2}$, $q=e^{\pi i\tau}$, $q^{\lambda}=e^{\pi i\tau \lambda}$, $q\in\mathbb{C}$, $|q|\lt 1$, $\tau\in\mathbb{C}$, ...
1
vote
1answer
68 views

HurwitzLerchPhi

I am not sure why this is returned unevaluated: HurwitzLerchPhi[1, 1, ∞] Everything is returned unevaluated ...
0
votes
1answer
145 views

Integrate real function returns complex function [closed]

I want to compute the integral $$ \int_0^c \exp(-cx+x^2) \mathrm{d}x, $$ where $c>0$ is an unknown constant. In Mathematica Version 12.2.0 ...
1
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1answer
95 views

Evaluating this generalised integral

I have the following integral $$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\exp \left(a u^2+b v^2+c u v\right) \; dvdu,$$ which returns the following solution: $$\frac{2 \pi }{\sqrt{4 a b-c^2}...
0
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0answers
39 views

BellY function call fails

I would very much appreciate to get an answer about the reason of the message I get calling the BellY function. ...
2
votes
1answer
107 views

How to plot spherical harmonics using two primary colors?

I did go through Density plot on the surface of sphere where great examples are provided. What I am very much interested is in plotting spherical harmonics (real/imaginary or lets say just the assoc. ...
0
votes
1answer
48 views

NSolve missing solutions

I'm having trouble solving the transcendental equation. For some values ​​of bi, NSolve obtains 6 roots, however when changing the value, it obtains 5. Graphing the function clearly shows that the ...
1
vote
2answers
66 views

Fittind data with shifted Chebyshev polynomials

I am trying to fit data from a simulation to a particular class of polynomials, according to least squares approach. ...
5
votes
1answer
175 views

How can I inform Mathematica of an identity concerning Bessel functions?

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function. $$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
1
vote
3answers
134 views

How to approximate the harmonic sum $\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$

How to approximate $$\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$$ Where $\overline{H}_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k}$ is the skew harmonic number. The mathematica ...
1
vote
1answer
52 views

How to define distributivity of CenterDot on bras and kets

I am working on a code for a coupled quantum harmonic oscillator and found myself in a hiccup when trying to evaluate the inner product of linear combinations of bras and kets. I have initially ...
0
votes
1answer
106 views

Solve Equation trinomial Nine degree!

I am trying to solve equation: x^9+p*x^4−q==0, {p, q} ∈C using functional analysis. Ηow do I solve this equation and with help Mathematica program for case p=π, q=...
0
votes
1answer
45 views

How to find the exponent of Bessel function related infinite integral? [closed]

Please suggest how to find the power law exponent for curvature vs r, I am trying using exponet but it is not working. ...
3
votes
1answer
297 views

Find RSolve solution reflecting special structure of DifferenceRoot[Function[{\[FormalY], \[FormalN]}

RSolve yields a large (multi-page) solution (LeafCount=25891) containing a number of 7F6 (and higher) hypergeometric functions when applied to ...
0
votes
2answers
39 views

Warning when confluent hypergeometric function HypergeometricU is wrong?

From the definition as an integral (HypergeometricU/details), this function must be positive. However, it gives negative numbers in some cases with no warning of a potential error. For example, In[...
1
vote
3answers
133 views
6
votes
1answer
90 views

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 $E(k)$ respectively: $E(k)K(\...
2
votes
0answers
48 views

Angular functions defined using Wigner D-functions [closed]

How do I properly implement angular functions using Wigner D-functions in Mathematica? Angular functions are commonly used in light scattering calculations and can be defined using Wigner D-functions. ...
0
votes
0answers
30 views

How to nicely expand a Gauss Hypergeometric function?

Does anybody know how to obtain the z->1 expansion for the Gauss Hypergeometric 2F1(a,b;c;z) on Mathematica as shown here ? I tried to use Series with the assumption c-a-b non-integer, but the ...
0
votes
2answers
58 views

Error in Integration of special functions using mathematica 12.0

When I try to integrate the following, Integrate[-GegenbauerC[22,-1/2,x]/(1+k*x),{x,-1,1}] where -1<k<1 and k!=0, Mathematica gives different results if I ...
3
votes
2answers
281 views

Code that produces plot in V5 doesn't work in later versions

I have problem in plotting Integral function. I can compute/plot the graph of this integration below in Mathematica 5.0, but it is not possible to plot it in higher Mathematica versions. My code is: <...
19
votes
1answer
4k views

Creating a Mathieu stability diagram

I am attempting to re-create a Mathieu stability diagram like the one shown in a paper by Leary and Schmidt [1]: I expected that I could use MathieuC to generate ...
1
vote
1answer
46 views

How to make FindRoot work with PolyLog functions

I have the following code: ...

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