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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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1
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1answer
51 views

Dynamic influenced by a countdown Clock ignores actions in second argument

I stripped a Dynamic down to its bare essentials. Dynamic doesn't execute any of the functions asked for in second argument. ...
3
votes
1answer
187 views

Closed Form of Integration [duplicate]

I have tried this integral in Mathematica, Assuming[m > 0,Integrate[Cos[m k] Exp[ Cos[k]], {k, 0, 2 Pi}]] And Mathematica failed to perform this. Whereas if any ...
2
votes
1answer
138 views

Integrate of HypergeometricPFQ gives the wrong result

Good morning, I computed the following integral using Integrate in version 12.2 and it gives the wrong result. Can you help me understand what I am doing wrong? Here is the integral: ...
0
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0answers
16 views

Is there a general unifying formula for the Laplace transform of a general weibull density?

It seems that the Laplace transform of weibull density has different formulas for rational shape parameters, depending on the degree of the numerator/denominator https://idp.springer.com/authorize/...
1
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1answer
66 views

Efficient/Fast Complex Airy function implementation on Mathematica (Ai and Ai')

Is there a speedier compiled version for an efficient implementation for the complex Airy function (Ai and Ai') on Mathematica?
8
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1answer
306 views

Zeta zeros bug in Mathematica 12.0.0.0

In Mathematica 12.0.0.0 when I input one of these: Log[2/Im[ZetaZero[127]]] Log[2/Im[ZetaZero[128]]] Log[2/Im[ZetaZero[127]]]//N Log[2/Im[ZetaZero[128]]]//N ...
1
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2answers
48 views

Using NSolve to get a solution for Bessel-related functions

I want to get a solution of a equation using NSolve. $BesselI[1,x]/(x*BesselI[0,x])=0.2$ So I plugged this equation to NSolve: ...
0
votes
1answer
106 views

Elliptic Integral simplification

Integrate[Sqrt[(roh^2 - r^2)/(r^2 - rb^2)], {r, rb, roh}, Assumptions -> r > rb \[And] roh > rb \[And] roh > 1] Outputs to: ...
0
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1answer
27 views

How to resolve problem during summation of functions?

I am trying to find out the output of this basic problem but getting an error. If anyone can resolve this will be helpful. ...
1
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0answers
57 views

New symbollicaly integral operator [closed]

I have an u=u(x,t) that is an unknown function. Suppose that the function u can be written as follows: ** ...(Eq. 1) ** In here, $\Psi(x)$ is an Nx1 known vector, and C is an NxN matrix. (N is an ...
0
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1answer
46 views
1
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1answer
103 views

Associated Legendre polynomials in Mathematica

I'm trying to look at the Associated Legendre Polynomial, so I plugged it into Mathematica to see the values for different input. From wikipedia: $$ p_l(x)=\frac{1}{2^l l!}\frac{\partial ^l}{\partial ...
3
votes
1answer
73 views

MeijerG function: numerical evaluation of its derivative

Mathematica does not evaluate the derivative of the following MeijerG function. ...
0
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0answers
24 views

ComplexInfinity only encountered with ColorFunction when plotting HurwitzZeta

I am using Mathematica 11, I am plotting HurwitzZeta[2,x] and its curvature on x$\in$[1,4]. Everything is fine when I try to plot the two. ...
2
votes
2answers
108 views

Unpacking a Mathematica expression returned by DSolve

I was trying to solve a system of differential equations in Mathematica and had troubles understanding what the solution looked like. So I wanted help to unpack it. I had a system of two coupled ...
4
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1answer
81 views

Issues with an equation involving Logs and PolyLogs

I want to check a claimed identity that involves Log and PolyLog functions. The relation can be found on this paper and more ...
4
votes
2answers
263 views

How does mathematica numerically evaluate the Meijer G-function?

The Meijer G-function is defined as a contour integral in the complex plane. Mathematica is able to numerically evaluate such a function quickly and accurately. How does she do that?
1
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2answers
66 views

How do I come up with an equation that will describe the results of other integrals with changes to a constant?

I have one equation: 10*(1-x)(x^n)+(x)(1-x^n), to which I must find the integral of from [0,1], with n ranging from 1 to 100. I want to then plot the results with the x value as the value of n and the ...
0
votes
1answer
61 views

Microfluctuations analysis and Power spectrum

I measured a parameter over time and obtained the values below: 0,627896 0,205004 0,259237 1,059125 0,832184 0,587992 0,565537 0,527323 0,460228 0,471958 0,26696 0,75367 ...
0
votes
1answer
70 views

Errors when plotting $\operatorname{Re}\sum_{n=1}^{200} \frac{(a\ln x)^n}{n!\, n\zeta (n+1)}$ in $x$

Let $a=1/2-30.424876126i$ ($i^2=-1$). Then trying to plot $$\operatorname{Re}\sum_{n=1}^{200} \frac{(a\ln x)^n}{n!\, n\zeta (n+1)}$$ in $x$ leads to a very inaccurate jaggy graph possibly caused by ...
6
votes
1answer
269 views

Asymptotic inversion of ExpIntegralEi function

I'm looking at the small-x and large-x asymptotic expansions of the inverse of exponential integral $E_1$ (https://dlmf.nist.gov/6.2#E1) $$\begin{array}{lll} E_1 & = & \int_z^\infty \frac{e^{-...
0
votes
1answer
125 views

Closure of Bessel Functions of the first kind

I need to use the Bessel functions of the first kind to solve some initial value problem. For this I need the closure equation $$ \int_0^\infty J_m(au)J_m(bu)u\,\text{d}u = \frac{\delta(a-b)}{a} \...
0
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0answers
44 views

Simplifying hypergeometric function 4F3

I am trying to make Mathematica simplify the following expression below. ...
0
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0answers
42 views

How to create and store a plot for each iteration of a Do loop

I have created the following code below. Its purpose is for me to divide the plot into grids and for each grid I reterive the center point's co-ordinates. ...
0
votes
1answer
39 views

Converting HyperInt Hlog to Polylogtool G [closed]

how do I replace efficiently HyperInt's Hlog (in maple expression) {Hlog(Y,[1]),Hlog(X, [1, -Y + 1]),Hlog(X,[1,1,1])} in an expression into PolyLogTool G: {G[1,Y],G[1,-Y+1,X],G[1,1,1,X]} Basically, ...
5
votes
1answer
191 views

Airy function zeros, conflict (error?) between Wolfram Functions vs. Mathematica

According to functions.wolfram.com, the zeros of the Airy function $\operatorname{Ai}(z)$ occur at $z_k=f\left(\tfrac{3\pi}{8}(4k-1)\right)$ for $k\in \mathbb{N}$ where $f(d)=-d^{2/3} \left(1 + \frac{...
3
votes
0answers
72 views

Spherical harmonics Y (l,m,theta,phi) for general l, m

I am trying to solve integrals involving spherical harmonics Y(l,m, theta, phi) and their derivatives. I do not have any particular l,m, theta, phi values. I need to solve it for general l,m. When I ...
1
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0answers
48 views

Reduce the time-complexity of an algorithm using BellY

The code below is critical for my computation. I have to evaluate it for n around 100 and 200...
3
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2answers
82 views

How to integrate product of Bessel and exponential fucntion

I have obtained the following solution for inhomogeneous Helmholtz equation \begin{align*} W(u) = \dfrac{i}{2 \lambda} e^{i \lambda u} \int_{0}^u J_{n}(\lambda u^{'})e^{-i \lambda u^{'}} du^{'} \end{...
0
votes
1answer
57 views

Finding a mapping between two types of (generalized) hypergeometric series

I am given two functions, one is of the form $2F1(a,b,c;z)$, where $2F1$ is a hypergeometric series. The other one is a generalized hypergeometric series $3F2(d,e,f;g,h;w)$, where the characters are ...
1
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2answers
54 views

Problem with plotting (resp. expanding) the Hurwitz Zeta function

I expected the two plots to be identical. Can anyone confirm that the discrepancies show a bug? ...
0
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2answers
228 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. ...
1
vote
1answer
83 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 ...
1
vote
0answers
61 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 ...
6
votes
3answers
318 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. \...
0
votes
1answer
45 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
votes
0answers
71 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: ...
1
vote
0answers
32 views

Jacobian elliptic function argument [closed]

I have a C++ code that computes jacobian elliptic sn, cn and ...
0
votes
0answers
41 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. ...
2
votes
1answer
109 views

Fourier transform of DawsonF not recovered by using Erfi

Bug introduced in 8 or earlier and fixed in 12.2 I wish to compute the Fourier transform related to the Dawson function: FourierTransform[1/u DawsonF[1/u], u, x] ...
0
votes
0answers
57 views

Strange result of LaplaceTransform

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

Is this a bug in mathematica for integrals of multiple error functions?

I'm scratching my head over the the following result in Mathematica (v11.3) I'm considering the function B = Erfc[x] Exp[-x^2/2] + Sqrt[2] Erfc[x/Sqrt[2]] Exp[-x^2] ...
1
vote
1answer
107 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
vote
0answers
62 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 \...
0
votes
0answers
61 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
votes
1answer
46 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
votes
1answer
31 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
votes
1answer
93 views

Jacobi Elliptic Function Solution using DSolve

I am happy that I have a solution to a particular differential equation by hand: ...
2
votes
0answers
71 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
votes
0answers
60 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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