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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

0
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0answers
80 views

Projecting a surface onto a plane [on hold]

I have a three-dimensional function as ...
4
votes
5answers
217 views

How to convert list of digits to number? [closed]

I have a list such as: a = {1, 2, 4, 3}; and I want to get the 'number' of the list. For this example, I want to get this number: ...
1
vote
0answers
33 views

HypExp and HPL packages for hypergeometric functions: Evaluating a function HPL[{minus,plus},x]?

I am currently using the HypExp and HPL packages, which are useful for expanding hypergeometric functions in series around integer or half-integer values, as in common in dimensional regularization ...
0
votes
0answers
12 views

Inverting the asymptotic expansion of Gauss Hypergeometric Function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of the object rho[r_,b_,q_] below). Basically I want to series expand rho[r_,b_,q_] for large $r$ (i.e. as $r\...
4
votes
2answers
150 views

Expansion of the Meijer G Function

I'm trying to do the integral Integrate[ B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, a, ∞}] which I figured should be relatively simple as the integral of a Bessel ...
0
votes
0answers
34 views

Expression simplified for explicit int values, but not with FullSimplify/FunctionExpand and Assuming. What formula does Mathematica use?

I have a $_4F_3$ hypergeometric function (Mathematica 11.3) HypergeometricPFQ[{1, 3/2 + n, 1 + 2 m + n, 1 + 2 m + n}, {2 + n, 2 + n, 3/2 + 2 m + n}, z] If I ...
0
votes
0answers
33 views

How to plot a 4-dimensional relation in an animated way?

If we have an relation like this one: $$r(x_1,y_1) = (x_1^2 - y_1^2, 2 x_1 y_1) = (x_2,y_2)$$ How can we plot it on a 4-dimensional space? Where the axis are: $x_1,y_1,x_2,y_2$. And the variables ...
1
vote
1answer
49 views

Simplifying elliptic functions

I am working with (long) elliptic functions, which are rational functions in WeierstrassP and WeierstrassPPrime. As expressed in ...
-1
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1answer
42 views
0
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1answer
55 views

Plot $ g(r) = \int_0^\pi J_0(w\ r)\ \mathrm dw $

I want to plot $g(r)$ as a function of $r$ but $g(r)$ at each value of $r$ is an integral of the Bessel function over the the limits of $w$.
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0answers
38 views

Numerically integrating a highly oscillatory multi-dimensional Bessel function with good precision

This is a follow up question to Numerically integrating a highly oscillatory Bessel function with good precision. Now I am considering a three dimensional integral, ...
1
vote
2answers
94 views

Numerically integrating a highly oscillatory Bessel function with good precision

I am trying to evaluate the following integral Integrate[Csch[w]^2 (w BesselJ[1, w Sqrt[y]/π]^2), {w, 0, Infinity}] in an asymptotic large ...
9
votes
5answers
269 views

NDEigenvalues vs. FindRoot for finding the eigenvalues of Airy equation?

Say I am trying to find the first 5 eigenvalues of the differential equation $f''(x)=\lambda x f(x)$, on the interval [-1,0], with boundary conditions $f(-1)=f(0)=0$. I will try to do this 3 ways, ...
0
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1answer
46 views

Numerical Approximation of erfc

I have a crazy output. I already tried the N[] function. All it does is condense the inputs. Any idea of how to have a numerical output from the code?
0
votes
1answer
70 views

Solving two nonlinear equations in two unknowns

This is my code defining two nonlinear equations. I want to solve them simultaneously. ...
1
vote
2answers
169 views

Numerical integration with Dirac delta

I have some complicated function depending on many arguments $x,y,z$ and parameter $a$ multiplied by Dirac delta of another function, $$ \tag 1 f(a,x,y,z) = g(a,x,y,z)\delta(t(a,x,y,z)) $$ I want ...
1
vote
0answers
35 views
3
votes
0answers
75 views

Summation of the multipole expansions [closed]

Let $ \vec\Omega, \vec\Omega' $ be two unit vectors in $ \mathbb R^3 $ such that $ \vec\Omega\cdot\vec\Omega' = t $ and let $ r > 0 $. The multipole expansion for the exponential reads: $$ \...
1
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0answers
24 views

Inverse of Normal Distribution CDF incorrect for large value?

I have a function F which is the CDF of the standard normal distribution. The inverse of F should be infinity at 1. However, I ...
14
votes
4answers
273 views

Terrible accuracy of DawsonF

DawsonF[30.] returns 0. The correct value is 0.016676... At least it prints a warning message, ...
3
votes
2answers
86 views

Integration with Bessel function as result

When I try to calculate this integral, 1/(2*Pi)*Integrate[ Exp[I*2*θ]*Cos[1/2 (7 + Cos[θ])], {θ, 0, 2*Pi}] Mathematica is unable to compute it. But the result ...
3
votes
2answers
115 views

Wrong Output for Limit

I want to calculate the following limit: ...
0
votes
0answers
40 views

Analytic derivatives of LegendreP[a, z] w.r.t. a at a = 0?

I am interested in finding analytic expressions for various derivatives of the LegendreP[a,z] w.r.t. a at ...
4
votes
1answer
88 views

Tips for Complicated Indefinite Integrals

I would like to have an analytical expression for some complicated indefinite integrals (example below). I am able to solve these integrals numerically, and I am able to plot them in Mathematica, ...
6
votes
1answer
275 views

What is wrong with this code? (Usage of FunctionInterpolation and how to make code efficient)

I am trying to create an interpolation out of a function that comes from an improper integral. The end goal is to make a 3d plot over a specific region. Specifically, I have the following code: ...
5
votes
2answers
62 views
1
vote
2answers
126 views

Why Mathematica does nothing with the expression Gamma[2 z]? [closed]

Could someone tell me why Mathematica just returns the input with this expression: Gamma[2z] Gamma[2 z] ? I expected this result: $Γ(2z)=\frac{2^{2z-1}Γ(z)Γ(...
0
votes
1answer
86 views

Analytical form of the answer of a definite integral

I am trying to find out the analytical form of the answer of the following integration, $$I=\int_0^l r \, dr \int_0^{2\pi} \, d\phi (iks)[ \frac{\exp [ikx_0(N+r^2)^{1/2}]}{[x_0(N+r^2)^{1/2}]}\frac{\...
-1
votes
1answer
65 views

Numerical Inversion of an incomplete beta function expressed as gauss hypergeometric function using Mathematica

I am currently working with this hypergeometric function ${_2}F_1$, $\rho(r)=\frac{2b}{1-q}(1-(\frac{b}{r})^{1-q})^{\frac{1}{2}}{_2}F_1(\frac{1}{2},1-\frac{1}{q-1},\frac{3}{2},1-(\frac{b}{r})^{1-q})$ ...
0
votes
2answers
87 views

Product of large number with a very small number returns zero because Mathematica sets the very small number equal to zero [closed]

I have a product Exp[-I*Pi*x]*BesselK[-1, 2.43*Ix]. Now, Exp[-I*Pi*x] grows larger and larger as $x$ increases for imaginary ...
1
vote
1answer
76 views

Integral of modified Bessel function is wrong

A simple integration of a modified Bessel function gives: ...
1
vote
0answers
134 views

Approximate the relationship between 6 nonlinear functions involving elliptic integrals

I am trying to solve a physics-related problem, which results in approximating a relationship between 6 symbolic functions $F_1(\alpha,\beta ,x_0,y_0),F_2(\alpha,\beta ,x_0,y_0),...,F_6(\alpha,\beta ,...
3
votes
1answer
40 views

Strange automatic Hypergeometric2F1 simplification

On this Wolfram Functions page we can find the following identity: ...
3
votes
1answer
90 views

Problem with the function N with second argument

I was asked to look at a complicated definite integral that could be integrated analytically, but numerically did not behave as expected. After a lot of simplifications I arrived at the following: <...
1
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0answers
40 views

Inverse Laplace transform of powers with an arbitrary index

Tried for Inverse Laplace transform (ILT) for the following: L[s] = (L /(L + s) w + Q /(Q + s) (1 - w))^n $L[s]$ can also be written as $$L[s]=\sum_{k=0}^n\...
4
votes
3answers
154 views

Finding Root of BesselJ [closed]

When I work on some physical problem I needed to know how to get all of first 100 roots of BesselJ[n,x] function -which is a quasi periodic function-, as a List. I ...
2
votes
0answers
40 views

Expansion of HypergeometricU

I try to get the first three terms of expansion of HypergeometricU in Mathematica. My input is ...
0
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0answers
54 views

Question regarding multidimensional integral

I would like to compute the following iterated integral in 3 dimesions: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \left|\int_{-\infty}^\infty f(y)f(y-t)e^{-i 2 \pi t \xi} dy\right|dt d\xi $$ ...
4
votes
0answers
60 views

Series of LegendreP[a,b,x] takes ages in Mathematica 11

My institution just upgraded to Mathematica 11.3 (from v10) and I'm experiencing a problem that is absent in Mathematica 10. Namely, ...
2
votes
0answers
43 views

How does one generically expand the argument of a gamma function? [closed]

I have expressions of the form Gamma[2 (56 + k)] which I would like to convert to Gamma[112 + 2 k] But ...
0
votes
0answers
129 views

Plotting a function with Dirac delta function and its derivative

I am trying to plot the following expression which is a function of delta and its derivative: \begin{equation} y(t)=sin[\alpha t]+\frac{1}{\alpha^2}f(t)[(1-cos[\alpha t])+(\frac{t}{\alpha}-sin[\alpha ...
3
votes
0answers
100 views

Issues with the series expansion of Nielsen generalized polylogarithms in Mathematica 11.3

In Mathematica 9, 10.3 and 11.0 I can easily expand PolyLog[2, 2, x] around $x=1$ using ...
2
votes
1answer
99 views

How to take the curl of a vector function involving hypergeometric functions?

I have a vector function involving a hypergeometric function as its inner constituent. I need to take the curl of this vector and when I do, Mathematica prompts this array of errors: ...
0
votes
1answer
75 views

Evaluate integral containing Erfc, Exp and Log [closed]

I'm trying to find the solution to the integral below. It runs in Mathematica but does not produce any output. Every other part of the code works fine except the integral involving Exp, Erfc and Log ...
3
votes
2answers
75 views
3
votes
1answer
87 views

Incorrect Series with EllipticTheta under 11.3

The following program gives the correct coefficients under versions 7, 10.2 or 11.0 ...
3
votes
1answer
126 views

How to make Mathematica substitute exact numerical values of (derivatives) 2F1

I have a long output that contains Taylor expansion of some function. Some of the coefficients are combinations of the (derivatives of) 2F1 at nice integer values of the coefficients. I can see that ...
0
votes
1answer
70 views

Is there an error pertaining to HypergeometricU function on functions.wolfram.com?

If one clicks on the last of the nine formulas on http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/17/02/07/ one obtains Hypergeometric2F1[a, b, c, z] == HypergeometricU[a, b, a +...
1
vote
1answer
40 views

Simplification of a JacobiCN Integral

Consider the following indefinite integral: Assuming[A > 0 && B > 0 , Integrate[1/(1 - A JacobiCN[B x, k]^2), x] // FullSimplify] -((...