Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,178
questions
2
votes
2answers
145 views
How to express Bessel Functions in terms of Hankel Functions
I have the following expression
BesselJ[\[ImaginaryI]*m, k*r]
FunctionExpand knows how to transform from ...
-1
votes
0answers
51 views
Calculating complex integral representation of the Hankel function of the first kind
I was wondering if any one can help me calculate this integral using Mathematica.
This is a form of integral representation of Hankel function of first kind:
...
0
votes
1answer
130 views
How do I avoid the singularities during the calculation of an integral?
I face a serious problem when I try to calculate the following integral. My code is the following:
...
0
votes
2answers
44 views
Error complex function ERFI(X): looking for alternative function representations?
I have some analytical results from a physics problem, where the Mathematica gives the results in terms of complex error function. I would like to explore another function representation using ...
4
votes
2answers
148 views
How to programmatically change CellStyle of all Cells from "Input" to "Code"?
I have a special notebook styled for only creating functions and their packages. I often lazily do Evaluate Notebook instead of finding and opening all cells I need ...
4
votes
2answers
113 views
Possible bug in asymptotic expansion of CoshIntegral and SinhIntegral at infinity
Consider this expansion
Series[CoshIntegral[z], {z, \[Infinity], 0}, Assumptions -> z > 0]// Normal
(* -((I [Pi])/2) + Cosh[z]/z^2 + Sinh[z]/z *)
Here a ...
3
votes
2answers
150 views
Differential Equation in Complex Plane
I asked this question 8 years ago, but the answer of @Jens was not satisfactory and it didn't reproduce the plot exactly. I'm wondering if new versions of Mathematica can handle it.
1
vote
0answers
54 views
Why does Mathematica only solve this indefinite integral for two values of n?
I'm trying to estimate this integrate:
where "a" and "b" are real constants and n> 0.
I use the code
...
1
vote
0answers
47 views
How to use only real values in a notebook?
I am wondering how to make Mathematica only evaluate real numbers by creating an initialization line in the beginning of a notebook.
I would like to prevent the following:
complex trigonometric ...
4
votes
3answers
433 views
Why can't Mathematica evaluate the limit of this hypergeometric function? Is there any way to find the answer?
I have the following function
...
1
vote
1answer
82 views
Strange result for an integral containing Meijer G-function
The question is about Mathematica's result for the integral
$$ \int_0^\infty \frac{16}{\pi^2} \left( \frac{\sqrt{\pi}}{4} r^2
%
G_{1,3}^{2,1}\left(\frac{r^2}{4}\Bigg|
\begin{array}{l}
0 \\
0,0,-\...
0
votes
0answers
56 views
Inverse of symbolic hyperbolic function
I have a function
$$u = 1/4*\cosh(x)^2$$
Entered in Mathematica as
1/4*Cosh[x]^2
Inverted using
...
3
votes
0answers
87 views
FullSimplify gives erroneous answer
FullSimplify gives a wrong expression when simplifying Hypergeometric functions:
f = Hypergeometric2F1[1/2 - n, -n, 1 - 2 n, -4 a];
g = FullSimplify[f];
The ...
1
vote
0answers
67 views
How does Mathematica numerically evaluate Gaussian Hypergeometric function $_2F_1(a,b,c,z)$ where $a,b,c \in \mathbb{C}$ [closed]
Gaussian hypergeometric function $_2F_1(a,b,c,z)$ usually takes $a,b,c \in \mathbb{R}$ and $z \in \mathbb{C}$. But using Barnes integral representation, one can analytically continue the $_2F_1(a,b,c,...
0
votes
0answers
50 views
Using generalized hypergeometrics in RegionPlot hangs the Kernel
This simple code, is just an example, hangs the kernel:
RegionPlot[
x + y + HypergeometricPFQ[{1, 2.5, 3.1}, {4.65, 3.28}, 0.2] >
0, {x, -1, 2}, {y, -2, 3}]
...
1
vote
1answer
98 views
Simplifying special functions via substitution
We have a well-known identity
$$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$
between beta function and gamma function.
I have a complicated expression involving the terms in the form $\frac{\...
1
vote
1answer
66 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
191 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
146 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
17 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
68 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?
9
votes
1answer
330 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
53 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
115 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
29 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
61 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
47 views
1
vote
1answer
107 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
76 views
MeijerG function: numerical evaluation of its derivative
Mathematica does not evaluate the derivative of the following MeijerG function.
...
0
votes
0answers
25 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
111 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
82 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
306 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
68 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
66 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
71 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)}$$
in $x$ leads to a very inaccurate jaggy graph possibly caused by ...
6
votes
1answer
285 views
Asymptotic inversion of ExpIntegralEi function
I'm looking at the small-x and large-x asymptotic expansions of the inverse of exponential integral $E_1$ (https://dlmf.nist.gov/6.2#E1)
$$\begin{array}{lll}
E_1 & = & \int_z^\infty \frac{e^{-...
0
votes
1answer
136 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
46 views
Simplifying hypergeometric function 4F3
I am trying to make Mathematica simplify the following expression below.
...
0
votes
0answers
45 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
41 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
201 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
74 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
51 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
84 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
61 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
55 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
237 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
117 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
63 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 ...
