Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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2answers
56 views

How to get the right approximation for a harmonic series?

The right numerical value of the closed form of $\sum_{n=1}^\infty\frac{4^n H_n^2}{{2n\choose n}n^2}$ is $40.66752074791188333...$. I tried to verify this result on Mathematica using the command: <...
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40 views

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....
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97 views

Does anyone have a Mathematica implementation of the standard $\arg\zeta(s)$ function required to evaluate $S(T)$?

This question is related to my question Is there an elegant exact formula for the zeta zero counting function? on Math StackExchange. Question: Does anyone have a Mathematica implementation of the ...
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2answers
125 views

How do I analytically-continue the dilogarithm function?

Here's the dilogarithm definition: $$\text{Li}_2(z)=\sum_{k=1}^{\infty} \frac{z^k}{k^2};\quad |z|<1$$ However, the function can be analytically-continued by several integral means for all $z$ with ...
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72 views

Computation of infinite series containing Zeta function

(this is my first question on this forum I'm totally inexperienced in mathematica) Consider the given alternating series: $$f(x) =\sum_{n=0}^\infty \frac{2a_n(x-1)^{2n+1}}{\zeta(-2n-1)}$$ Here, $a_n= (...
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2answers
60 views

Solving a recursive equation iteratively (including PolyLog function)

I have the following equation which I want to solve: $$ I_D = [Li_2(-e^{V_D-I_D})-Li_{2}(e^{I_D})] $$ Here $Li_2(x)$ is the PolyLog function of order $2$. Is there a way to solve this equation ...
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1answer
65 views

Asymptotic expansion at infinity given a branch cut

Basically, I have obtained the function $\rho (r)$ below as a result of integrating $$\rho(r)=\int_{b_0}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$ which results to ...
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2answers
67 views

Whats the mathematica command for the skew harmonic number?

Whats the mathematica code for the skew harmonic number: $$\overline{H}_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k}.$$ Wolfram expresses this number as $$\ln(2)-(-1)^n \text{LerchPhi}(-1,1,n+1).$$ I am ...
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1answer
49 views

How to get rid of defects in that ComplexPlot3D?

I mean the result of ComplexPlot3D[Sin[z]/MittagLefflerE[10,z],{z,-1-I, 8 + I},Exclusions->None, PlotPoints -> 50] Its blow-up ...
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37 views

Upper incomplete Gamma function numerical differences

I have the following issue. Using Mathematica 11.2 (with Rubi loaded) I find that e.g. In[504]:= Gamma[-1/3, 2] // N ...
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36 views

Binomial Expansion on Mathematica [duplicate]

Why when I typed Expand[Binomial[m-1,6]], it gives me Binomial[-1 + m, 6] instead of ...
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1answer
81 views

Does Mathematica have a problem with sums involving Stirling numbers of the second kind?

In one of my calculations, I run the command: Sum[(StirlingS2[k - 1, 4] + StirlingS2[k, 4])/6^k, {k, Infinity}] Surprisingly, ...
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42 views

Evaluation of a double summation invovlving hypergeometric and exponential functions

I am trying to compute the following double summation over the indices, $m$ and $n$, which involves the hypergeometric function, ${}_2 F_1$, an exponential function and, factorials as a part of a ...
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1answer
67 views

limit of an expression including BesselK function

i want to calculate the limit of the following expression when 'w' tend to zero. I have used the Limit function, it takes a lot of time for running without any result. could you please help me how to ...
5
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2answers
271 views

How to plot multifactorial function?

The multifactorial function can be extended to the reals (see TheSimpliFire answer) like so: It follows that we can extend the multifactorial function to the reals ...
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51 views

Numerical evaluation of InverseFunction

I am considering the inverse of the function rho[r]. I have used the InverseFunction to find the inverse, given by ry which is a monotonically increasing function of its argument $y$. Now for $1<q&...
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1answer
129 views

How does Mathematica evaluate these sum and integral?

How does Mathematica internally evaluate the following (interrelated) sum and integral, and how does it do the subsequent simplification? (I mean, based on what mathematical facts?) ...
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1answer
42 views

Help with NIntegrate settings to evaluate integral [closed]

I am trying to evaluate this integral: \begin{align*} \alpha_{2}=\int_{-\infty}^{1.645}\left[1-\Phi\left(\frac{\sqrt{25}}{\sqrt{15}} 1.645-\frac{\sqrt{10}}{\sqrt{15}} z_{1}\right)\right] \phi\left(z_{...
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1answer
71 views

Inconsistency in Asymptotic expansion of cylindrical functions

Context I am interested in asymptotic behaviour of Cylindrical functions which are solution to the differential equation $$ y''(x)+(x^2-1)y(x)=0\,. $$ I ask mathematica to find such solutions: ...
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52 views

Reduce only generically correct?

While it is clearly said in the Mathematica documentation that functions like FullSimplify yield only 'generically correct' results, I have found no mention of such restrictions for Reduce. However, ...
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32 views

Fourier transform in polar coordinates using built-in hankel transform of the function constant 1 [closed]

Like in the table of transforms https://en.wikipedia.org/wiki/Fourier_transform#Distributions,_one-dimensional the FT (Fourier transform) of $\delta$ is 1 and the FT of 1 is $\delta$, but in polar ...
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1answer
190 views

Program for efficient computation of given functional:

I need to plot the following functional with accuracy: $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1}, $$ Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
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1answer
49 views

Parametric plotting of molecular orbitals [closed]

I'm trying to plot a parametric curve, but I only get a blank plot. I don't usually use mathematica, but my end user needs to be able to do this using that software, so I'm really struggling on syntax ...
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2answers
83 views

Most efficient strategy for integrating over removable poles?

I am finding many situations where I have to numerically integrate some function $f(x)$ of the form: $$f(x)=f_{s}(x)-ax^{-n},$$ where $f_s$ is a special function with a finite-order pole that is ...
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31 views

Check equality of two expression “With Conditions”

I want to check if these equation holds if only if $t>0$ $$\frac 1t\left(-\gamma+\operatorname{Chi}(t)-\ln(t)-\operatorname{Shi}(t)\right)=\frac 1t\left(-\gamma+\operatorname{Ei}(-t)-\ln(t)\right),\...
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48 views

Discrepancy in the series expansion of BesselK[ν, z]

I am trying to expand the modified Bessel function of the second kind $K_{\nu}(z)$ for small values of the argument $z$ keeping $\nu$ fixed. Mathematica 12.2.0 says that it is ...
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1answer
129 views

How to expand Lie characters?

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-...
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2answers
122 views

Is there a built-in for “circle function”?

Is there a built-in function in Mathematica for this function? \begin{equation} {\displaystyle \operatorname {Circ} (r)=\left\{{\begin{array}{rl}1,&{\text{if }}r<{1}\\{\frac {1}{2}},&{\text{...
2
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1answer
56 views

Verifying Summation form of Derivative of Hypergeometric1F1

First. Please read my code: ...
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0answers
61 views

How to do a fast numerical computation of an oscillatory integral including HeunC function using Mathematica?

I am trying to numerically compute the following integral in Mathematica ...
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2answers
94 views

Computing the sign of an expression

Can you please help with the following. I would like to compute the sign of an expression: ...
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1answer
286 views

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 ...
3
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1answer
59 views

Too small to represent and Chop in inaccessible algorithm

Overview I want to apply a specified Chop function to every step in an function call in Mathematica (in my case LegendreP), ...
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2answers
50 views

Convert GiNaC harmonic polylogarithms to HPL packages' notation in Mathematica

A general output from GiNaC (https://www.ginac.de/) of harmonic polylogarithms is H(a,b,c...,x). We want to convert it to Mathematica format (https://www.physik....
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0answers
34 views

Plotting Radiation Pattern for Hermite-Gaussian beams for different modes

I want to plot the Radiation Pattern for Hermite-Gaussian beams (HG beam) as given in the Figures Ref Research Paper Fig 2, where u and v be the beam modes for the Hermite-Gaussian beams The coding ...
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0answers
62 views

How to simplifying the following integral that involves Bessel functions and Exponential integral function?

I have obtained the following as a solution of heat conduction equation of a semi-infinite model in cylindrical coordinates which is initially subject to non-homogenous initial condition and isolated ...
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1answer
75 views

How to simplify the following integral to be in terms of Bessel functions?

I have evaluated the following integration using Mathematica. I obtained a solution in terms of Meijer G function. I wonder if it can be simplified to be in terms of Bessel functions. ...
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48 views

How to evaluate the following integrals using Mathematica?

I have the following integrals obtained during solving heat diffusion equation for semi-infinite system that is subject to non-homogenous initial condition in Laplace domain. I want to simplify the ...
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3answers
192 views

How to prove the following integration identity?

I have the following integration that I want to evaluate it using Mathematica. ...
5
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1answer
112 views

How to show these two expressions are the same?

I think Can be simplified to $-x^2$ I tried Reduce, FullSimplify, FunctionExpand, and ...
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1answer
87 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(-\...
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1answer
108 views

Keeping Phase Factors in Sqrt

I am trying to plot certain holomorphic functions that contain square and higher roots. In the complex analysis sense, the function $f:z\mapsto z^\alpha$ for some $\alpha\in\mathbb C$ has a phase ...
2
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2answers
112 views

Plot[Zeta[x], {x, 2, 20}, ScalingFunctions -> “Log”] is not logarithmic [closed]

The code Plot[Zeta[x], {x, 2, 20}, ScalingFunctions -> "Log"] produces the following image, which is a plot of the ...
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1answer
52 views

Clebsch-Gordan coefficients: General Expression Does Not Match Specific Expression

The expression ClebschGordan[{2, 0}, {4, 0}, {2, 0}] yields the correct result of Sqrt[2/7]. However the expression ...
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2answers
304 views

Calculating a double integral

I want to calculate the following integral: $$\int^{10}_{0}\int^{\pi}_{0}\sqrt{(37-\frac{45\cdot37\cdot x^2}{74\cdot 150})^2\cdot \sin(t)^2-(40-\frac{27\cdot37\cdot x^2}{16\cdot 150})^2\cdot \cos(t)^2}...
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1answer
38 views

Integration of Bessel function including specific rings

Now, I am interested in the rings of Bessel function. So, basically, I used cylindrical coordinate to make it easy to figure out rings of Bessel function. ...
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1answer
105 views

Cannot understand the meaning of Derivative[1, 0][BesselK][-M, 2]?

When I do the following integration Integrate[(Log[x]/x)*x^M*Exp[-x-1/x],{x,0,\[Infinity]},Assumptions->Element[M,PositiveIntegers]] Mathematica return a very ...
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3answers
308 views

Numerically evaluating parameter derivatives of a hypergeometric function

I am unable to obtain the numerical value of the derivative of the hypergeometric function. Please note that the (2,4,0,0) is the derivative with respect to the first and second argument ...
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1answer
100 views

Mathematica definition of Hermite polynomials: fractional index

I am doing some calculations in Mathematica. I have solved a differential equation and noticed that the solution contains the Hermite polynomial $H$. The strange thing is that it appears as ...

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