Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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30 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 ...
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3answers
248 views
+50

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. \...
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1answer
37 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 ...
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1answer
40 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 ...
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0answers
66 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: ...
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0answers
30 views

Jacobian elliptic function argument [closed]

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

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
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1answer
100 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 ...
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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 \...
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56 views

Spheroidal Harmonics

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

HurwitzLerchPhi

I am not sure why this is returned unevaluated: HurwitzLerchPhi[1, 1, ∞] Everything is returned unevaluated ...
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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 $\theta_3 (q)=1+2\sum_{n\ge 1}q^{n^2}$, $q=e^{\pi i\tau}$, $q^{\lambda}=e^{\pi i\tau \lambda}$, $q\in\mathbb{C}$, $|q|\lt 1$, $\tau\in\mathbb{C}$, ...
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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 ...
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1answer
94 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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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. ...
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1answer
104 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. ...
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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 ...
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2answers
64 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
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 ...
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1answer
51 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
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[...
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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 ...
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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. ...
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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

How to make FindRoot work with PolyLog functions

I have the following code: ...
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2answers
198 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
57 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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0answers
115 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
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2answers
173 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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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
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2answers
73 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
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1answer
78 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
102 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 ...
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1answer
52 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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2answers
47 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 ...
1
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0answers
39 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
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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, ...
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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
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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
59 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
134 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_{...

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