Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,162
questions
1
vote
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
votes
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
vote
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
votes
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
vote
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
votes
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
vote
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
votes
1answer
46 views
1
vote
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
votes
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
votes
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
vote
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
votes
0answers
44 views
Simplifying hypergeometric function 4F3
I am trying to make Mathematica simplify the following expression below.
...
0
votes
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
vote
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
votes
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
vote
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
votes
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
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 ...