Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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?
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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 ...
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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
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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^{-...
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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} \...
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0answers
39 views

Simplifying hypergeometric function 4F3

I am trying to make Mathematica simplify the following expression below. ...
0
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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
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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
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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{...
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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 ...
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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
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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{...
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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 ...
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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
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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
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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 ...
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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
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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. \...
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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
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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 ...
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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: ...
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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
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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
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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] ...
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0answers
56 views

Strange result of LaplaceTransform

I mean LaplaceTransform[BesselI[3, x], x, s] // Simplify ...
3
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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
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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
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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 \...
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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
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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 ...
0
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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 ...
0
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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
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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
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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
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1answer
70 views

HurwitzLerchPhi

I am not sure why this is returned unevaluated: HurwitzLerchPhi[1, 1, ∞] Everything is returned unevaluated ...
4
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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