Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,119
questions
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44 views
Why is this integrand not integrating to a Bessel function?
I know from the identities of Bessel functions that the following is true:
$$
J_{m}\left( x \right) = \frac{ 1 }{ 2 \pi i^{m} } \int_{0}^{2 \pi} \,\mathrm{d}\phi \ e^{i \left( x\cos{\phi} - \ m \...
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0answers
49 views
Spheroidal Harmonics
There is a (scalar) field P varying on a spheroid (squashed) surface. This field is a function of four independent components ...
0
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1answer
43 views
EllipticPi argument is complex and can not be plotted. How to handle this problem?
inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 +
r))/(3000 r)])
This is my function of r, now I integrated it w r t r
...
0
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1answer
29 views
Search for terms contatining error functions
question might have an answer in this post (Efficient Search for specific Terms in symbolic Expression) but i don't understand how to convert it to my specific case
Through some definite integrals I ...
0
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1answer
81 views
Jacobi Elliptic Function Solution using DSolve
I am happy that I have a solution to a particular differential equation by hand:
...
2
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0answers
65 views
Understanding the behavior of HypergeometricPFQ
this is my very post here, so I apologize for any possible format issue.
I am using HypergeometricPFQ functions (more exactly $_3F_2$) as approximants for other more complicated functions. Here are ...
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0answers
52 views
Finding an analytic solution with a JacobiSD function
We are searching for an analytic solution to the given equation for $f_\text{n}(u)$, for $u \in [0, d/2]$ (this problem is a snippet from this paper here)
$$-\partial^2_{u} f_\text{n} + \left\lbrack 1 ...
1
vote
1answer
61 views
HurwitzLerchPhi
I am not sure why this is returned unevaluated:
HurwitzLerchPhi[1, 1, ∞]
Everything is returned unevaluated
...
4
votes
1answer
79 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 $\theta_3 (q)=1+2\sum_{n\ge 1}q^{n^2}$, $q=e^{\pi i\tau}$, $q^{\lambda}=e^{\pi i\tau \lambda}$, $q\in\mathbb{C}$, $|q|\lt 1$, $\tau\in\mathbb{C}$, ...
0
votes
1answer
142 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
vote
1answer
92 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:
$$\frac{2 \pi }{\sqrt{4 a b-c^2}...
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39 views
BellY function call fails
I would very much appreciate to get an answer about the reason of the message I get calling the BellY function.
...
2
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1answer
78 views
How to plot spherical harmonics using two primary colors?
I did go through Density plot on the surface of sphere where great examples are provided. What I am very much interested is in plotting spherical harmonics (real/imaginary or lets say just the assoc. ...
0
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1answer
47 views
NSolve missing solutions
I'm having trouble solving the transcendental equation. For some values of bi, NSolve obtains 6 roots, however when changing the value, it obtains 5. Graphing the function clearly shows that the ...
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2answers
63 views
Fittind data with shifted Chebyshev polynomials
I am trying to fit data from a simulation to a particular class of polynomials, according to least squares approach.
...
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3answers
130 views
How to approximate the harmonic sum $\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$
How to approximate $$\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
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1answer
48 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 ...
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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.
...
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2answers
37 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[...
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0answers
29 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 ...
2
votes
0answers
45 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
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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 ...
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1answer
46 views
4
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2answers
195 views
How to get the right approximation for a series involving the harmonic number?
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:
<...
5
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0answers
55 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....
3
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0answers
114 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 ...
5
votes
2answers
167 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$$
However, the function can be analytically-continued by several integral means for all $z$ with ...
1
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0answers
77 views
Computation of infinite series containing Zeta function
(this is my first question on this forum I'm totally inexperienced in mathematica)
Consider the given alternating series:
$$f(x) =\sum_{n=0}^\infty \frac{2a_n(x-1)^{2n+1}}{\zeta(-2n-1)}$$
Here, $a_n= (...
2
votes
2answers
71 views
Solving a recursive equation iteratively (including PolyLog function)
I have the following equation which I want to solve:
$$
I_D = [Li_2(-e^{V_D-I_D})-Li_{2}(e^{I_D})]
$$
Here $Li_2(x)$ is the PolyLog function of order $2$. Is there a way to solve this equation ...
1
vote
1answer
76 views
Asymptotic expansion at infinity given a branch cut
Basically, I have obtained the function $\rho (r)$ below as a result of integrating
$$\rho(r)=\int_{b_0}^{r}\frac{dx}{\sqrt{1-(b_{0}/x)^{1-q}}}$$
which results to
...
1
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2answers
99 views
Whats the Mathematica command for the skew harmonic number?
What's the Mathematica code for the skew harmonic number:
$$\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 ...
0
votes
1answer
51 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
...
0
votes
2answers
42 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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0answers
38 views
Binomial Expansion on Mathematica [duplicate]
Why when I typed Expand[Binomial[m-1,6]], it gives me Binomial[-1 + m, 6] instead of ...
0
votes
1answer
84 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, ...
0
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0answers
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 ...
1
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1answer
69 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
votes
2answers
275 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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0answers
56 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&...
1
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1answer
132 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?)
...
1
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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_{...
5
votes
1answer
80 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\,. $$
I ask mathematica to find such solutions:
...
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0answers
53 views
Reduce only generically correct?
While it is clearly said in the Mathematica documentation that functions like FullSimplify yield only 'generically correct' results, I have found no mention of such restrictions for Reduce. However, ...
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0answers
42 views
Fourier transform in polar coordinates using built-in hankel transform of the function constant 1 [closed]
Like in the table of transforms https://en.wikipedia.org/wiki/Fourier_transform#Distributions,_one-dimensional the FT (Fourier transform) of $\delta$ is 1 and the FT of 1 is $\delta$, but in polar ...
1
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1answer
196 views
Program for efficient computation of given functional:
I need to plot the following functional with accuracy:
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy,s) − F(x −\mathrm iy,s)}{\mathrm e^{2πy}-1},
$$
Where $ F(z,s) = \dfrac{1}{z^s\Gamma(\...
0
votes
1answer
54 views
Parametric plotting of molecular orbitals [closed]
I'm trying to plot a parametric curve, but I only get a blank plot.
I don't usually use mathematica, but my end user needs to be able to do this using that software, so I'm really struggling on syntax
...
2
votes
2answers
90 views
Most efficient strategy for integrating over removable poles?
I am finding many situations where I have to numerically integrate some function $f(x)$ of the form:
$$f(x)=f_{s}(x)-ax^{-n},$$
where $f_s$ is a special function with a finite-order pole that is ...
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0answers
32 views
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0answers
35 views
Check equality of two expression “With Conditions”
I want to check if these equation holds if only if $t>0$
$$\frac 1t\left(-\gamma+\operatorname{Chi}(t)-\ln(t)-\operatorname{Shi}(t)\right)=\frac 1t\left(-\gamma+\operatorname{Ei}(-t)-\ln(t)\right),\...
0
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0answers
49 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
...
