Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,147
questions
1
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0answers
27 views
Is there a way to do Partial Fractions which don't work with apart function?
Is there a way to solve partial fractions on mathematica, other than 'apart' function. Maybe an add on function?
0
votes
1answer
57 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
67 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 ...
5
votes
1answer
204 views
+50
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)
$$\begin{array}{lll}
E_1 & = & \int_z^\infty \frac{e^{-...
0
votes
1answer
112 views
Closure of Bessel Functions of the first kind
I need to use the Bessel functions of the first kind to solve some initial value problem. For this I need the closure equation
$$
\int_0^\infty J_m(au)J_m(bu)u\,\text{d}u = \frac{\delta(a-b)}{a} \...
0
votes
0answers
39 views
Simplifying hypergeometric function 4F3
I am trying to make Mathematica simplify the following expression below.
...
0
votes
0answers
41 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
34 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
180 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
61 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
46 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
78 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
47 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
51 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
224 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
59 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
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0answers
56 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
313 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
50 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 ...
0
votes
1answer
43 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
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0answers
32 views
Jacobian elliptic function argument [closed]
I have a C++ code that computes jacobian elliptic sn, cn and ...
0
votes
0answers
39 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
105 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
56 views
3
votes
1answer
196 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
104 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:
$$
J_{m}\left( x \right) = \frac{ 1 }{ 2 \ \pi \ i^{m} } \int_{0}^{2 \pi} \ d\phi \ e^{i \left( x \cos{\phi} \ - \ m \ \phi \...
0
votes
0answers
58 views
Spheroidal Harmonics
There is a (scalar) field P varying on a spheroid (squashed) surface. This field is a function of four independent components ...
0
votes
1answer
44 views
EllipticPi argument is complex and can not be plotted. How to handle this problem?
inttau[r_]=-(1/Sqrt[0.0345106153943703 - ((37.3042 - r) (-25.578 + r) (62.8822 +
r))/(3000 r)])
This is my function of r, now I integrated it w r t r
...
0
votes
1answer
30 views
Search for terms contatining error functions
question might have an answer in this post (Efficient Search for specific Terms in symbolic Expression) but i don't understand how to convert it to my specific case
Through some definite integrals I ...
0
votes
1answer
88 views
Jacobi Elliptic Function Solution using DSolve
I am happy that I have a solution to a particular differential equation by hand:
...
2
votes
0answers
68 views
Understanding the behavior of HypergeometricPFQ
this is my very post here, so I apologize for any possible format issue.
I am using HypergeometricPFQ functions (more exactly $_3F_2$) as approximants for other more complicated functions. Here are ...
0
votes
0answers
57 views
Finding an analytic solution with a JacobiSD function
We are searching for an analytic solution to the given equation for $f_\text{n}(u)$, for $u \in [0, d/2]$ (this problem is a snippet from this paper here)
$$-\partial^2_{u} f_\text{n} + \left\lbrack 1 ...
1
vote
1answer
70 views
HurwitzLerchPhi
I am not sure why this is returned unevaluated:
HurwitzLerchPhi[1, 1, ∞]
Everything is returned unevaluated
...
4
votes
1answer
85 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
146 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
95 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:
$$\frac{2 \pi }{\sqrt{4 a b-c^2}...
0
votes
0answers
41 views
BellY function call fails
I would very much appreciate to get an answer about the reason of the message I get calling the BellY function.
...
2
votes
1answer
120 views
How to plot spherical harmonics using two primary colors?
I did go through Density plot on the surface of sphere where great examples are provided. What I am very much interested is in plotting spherical harmonics (real/imaginary or lets say just the assoc. ...
0
votes
1answer
49 views
NSolve missing solutions
I'm having trouble solving the transcendental equation. For some values of bi, NSolve obtains 6 roots, however when changing the value, it obtains 5. Graphing the function clearly shows that the ...
1
vote
2answers
69 views
Fittind data with shifted Chebyshev polynomials
I am trying to fit data from a simulation to a particular class of polynomials, according to least squares approach.
...
1
vote
3answers
138 views
How to approximate the harmonic sum $\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$
How to approximate $$\sum_{n=1}^\infty\frac{{4n\choose 2n}\overline{H}_{2n}}{n 2^{4n}} ?$$
Where $\overline{H}_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k}$ is the skew harmonic number.
The mathematica ...
1
vote
1answer
52 views
How to define distributivity of CenterDot on bras and kets
I am working on a code for a coupled quantum harmonic oscillator and found myself in a hiccup when trying to evaluate the inner product of linear combinations of bras and kets. I have initially ...
0
votes
1answer
46 views
How to find the exponent of Bessel function related infinite integral? [closed]
Please suggest how to find the power law exponent for curvature vs r, I am trying using exponet but it is not working.
...
0
votes
2answers
41 views
Warning when confluent hypergeometric function HypergeometricU is wrong?
From the definition as an integral (HypergeometricU/details), this function must be positive. However, it gives negative numbers in some cases with no warning of a potential error. For example,
In[...
2
votes
0answers
77 views
Series expansion of PolyLog[2, 1/z]
There is a well known identity involving the Dilogarithm:
$$
\mathrm{Li}_2(1/z) = - \mathrm{Li}_2(z) - \frac{\pi^2}{6} - \frac{1}{2} \log^2(-z)
$$
As far as I understand it should be valid for all $z \...
0
votes
0answers
34 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
49 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
votes
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 ...
