Questions on the special mathematical functions implemented in Mathematica.

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1
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1answer
178 views

My code uses `ClebschGordan` but `Mathematica` is using `ThreeJSymbol`

I'm using a function that calculate CG coefficients with the function ClebschGordan but instead, I've got the following error because Mathematica is using ...
0
votes
1answer
70 views

Solving and Plotting Infinite sums over solutions to transcendental equation involving Bessel functions

I am trying to use Mathematica to plot the following equation $E(q,\Delta)$ vs $qa$, for set values of $ D \Delta/a^2$ and $\bar\rho a/D$. I have looked at other stackexchange posts which deal with ...
7
votes
2answers
136 views

`FindSequenceFunction` for sum of hypergeometric terms?

The built-in function FindSequenceFunction is quite good at recognizing hypergeometric terms, i.e. terms $c_k$ for which $c_{k+1}/c_k$ is a rational function of $k$....
4
votes
1answer
107 views

Series expansion of $(x;x)_\infty$ at $x=1^-$?

The so called Euler function is implemented in Mathematica as QPochhammer[x, x] I have been trying to obtain its leading behavior for $x\to 1$ from below. ...
1
vote
0answers
108 views

Plotting the Schwarz D minimal surface

With the aid of the following paper I'm trying to plot the Schwarz D minimal surface: paper. So far I have followed all the instructions to the best of my abilities and have come up with the following ...
6
votes
3answers
217 views

Why doesn't Lambert function (ProductLog) simplify?

I have Simplify[ProductLog[x*Exp[x]]] By the definition of the Lambert function, this should be simply x. But Mathematica outputs this: ...
1
vote
2answers
134 views

How to convolve the unit box function and the modified Bessel function of the second kind in 2D?

In 1D the convolution of the unit box function and the modified Bessel function of the second kind $K_0(x)$ works very well. ...
13
votes
2answers
231 views

$\tt SphericalHarmonicY$ does not seem to be an eigenfunction of the spherical harmonic equation

I applied the spherical harmonic equation on the SphericalHarmonicY functions like this: ...
3
votes
2answers
1k views

Irregular Confluent Hypergeometric Functions (Spherical Coulomb Wavefunctions)

I want to program in the regular and irregular spherical Coulomb wavefunctions $F_\ell(\gamma,kr)$ and $G_\ell(\gamma,kr)$, respectively, which are defined in terms of the regular and irregular ...
9
votes
2answers
277 views

How can I get the solutions of $x^n-x-t=0$ in hypergeometric form?

$x^3-x-t=0$ has three roots that can be expressed in hypergeometric form, for example $$x_1=-1-\frac{t}{2}{_2F_1\left(\frac{1}{3},\frac{2}{3};\frac{3}{2};\frac{27t^2}{4} \right)}+\frac{3t^2}{8} {...
1
vote
1answer
58 views

Derivative wrt to order of MacDonald function

I'm trying to get the following result confirmed in Mathematica: $$ \left.\frac{\partial\mathop{K_{\nu}}\nolimits\!\left(x\right)}{\partial\nu}% \right|_{\nu=\pm\frac{1}{2}}=\pm\sqrt{\frac{\pi}{2x}}\...
2
votes
1answer
94 views

How is it possible that PolyLog[2,1.1] returns an imaginary number?

Given that: PolyLog is defined as li[n_, z_] := Sum[z^k/k^n, {k, 1, ∞}], ...
0
votes
0answers
96 views

Spherical Hankel and Bessel in Explicit form

This is probably an easy question, when I type the spherical Hankel (first kind) and the Bessel function into Wolfram Alpha, it gives back an explicit form, the one you would get if you were to do it ...
1
vote
3answers
268 views

NIntegrate and Integrate of a Spherical Bessel function

I am trying to integrate a spherical Bessel function. I have used both the Integrate and NIntegrate functions in Mathematica but ...
11
votes
2answers
119 views

Why NIntegrate is badly-behaved on $J_{\frac{9}{2}}(x)$ by default?

A friend of mine showed me this example: Plot[BesselJ[9/2, x], {x, 0, 1}, PlotLabel -> Style["The integrand seems to be simple", 14]] ...
1
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0answers
36 views

How to invert an Elliptic function where the elliptic nome is a function of an independent variable?

I have a Jacobian elliptic function as a function of two independent variables $x$ and $y$. The elliptic parameter $m=m(y)$, $0 \leq m \leq 1$, is also a function of the variable $y$, and thus the ...
2
votes
1answer
74 views

Weird behavior of `HypergeometricPFQ`

Consider the following function: fun = HypergeometricPFQ[{1, x1-y1, 1-x1-y1-2u}, {x2-y1, 3-x2-y1-2u}, 1]; Let us try and evaluate this function on the following ...
2
votes
2answers
139 views

NIntegrate fails in a very strange way

I found a strange behavior in Mathematica when trying to evaluate the integral $$f(n) = \int_{1}^2 \frac{\Gamma(n)\Gamma(x)}{\Gamma(x+n)}{\rm d}x$$ I evaluate this using ...
0
votes
1answer
64 views

Clothoid (using Fresnel Integrals) [closed]

A clothoid is represented by The Problem Given the following functions, use Mathematica's Fresnel Integrals to plot the curve. My Attempt ...
0
votes
1answer
1k views
4
votes
1answer
91 views

Calculating sum of BesselJ[n, x]

My friend has a sum in his research paper that looks like this $$ \sum_{n=-\infty}^{\infty}\frac{J_n^2(x)}{n-\kappa}. $$ He was able to calculate this sum analytically, by substituting the denominator ...
1
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0answers
64 views

How to do this integral of Hypergeometric functions in Mathematica?

I've tried to integrate by part, but it seems that Mathematica is still not able to integrate. ...
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0answers
47 views

How to get ContourPlot3D to run involving elliptic functions?

I need to plot zeroes of a function of three variables that involves elliptic functions. For example, I have ...
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1answer
40 views

Determine parameter from which on there is no more root for a given function

Let $\gamma>0$ be a real number and $\Phi(r)=\frac{1}{r}-\frac{\pi}{4}\left(H_0(r/2)-Y_0(r/2)\right)$ defined on $[0,\infty)$, where $H_0$ is the Struve function of order zero and $Y_0$ is the ...
2
votes
1answer
91 views

Round off in Mathematica Built-in functions [duplicate]

Is there a way to force Mathematica to use its Built-in functions instead basic functions? For instance, the Hypergeometric1F1[a,b,x] function has a exponential form when its firsts parameters are ...
1
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1answer
48 views

How to solve an iterative integral equation for a function in Mathematica 8.0?

I would like to solve the following iterative integral equation in order to find the functional form of $P_{0}(k)$ numerically and then use it subsequently for the rest of my code: ...
3
votes
1answer
80 views

Interval arithmetic for DawsonF (or other special functions)

I am currently trying to estimate a complicated expression involving DawsonF using interval arithmetic. The interval arithmetic is partially supported by ...
7
votes
2answers
303 views

Can your Mathematica do the following integral?

Backslide introduced in v10 and fixed in v10.4.1. I was trying to do the following integral in Mathematica: Integrate[(z-2) PolyLog[2,z]Log[1-z]/z^3,z] What I ...
0
votes
2answers
101 views

Integral over real valued function becomes complex

I've tried to solve the following integral, but I get a complex solution even tough it should only have a real part. $$f(x)=- \frac{10^{-20} x}{0.99005- e^{10^{-12}x}}$$ Now I want to calculate the ...
10
votes
1answer
247 views

Vastly incorrect answers obtained by increasing WorkingPrecision with modified Bessel functions

Bug introduced in 7.0 or earlier and persisting through 10.4.1 This is a follow-up to this question regarding numerical instabilities occurring with modified Bessel functions. In trying to explore J....
1
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0answers
41 views

`QPochhammer` function simplification?

Consider the function which in Mathematica is denoted as QPochhammer[a,q,n], and its infinite product cousine: which in Mathematica reads ...
0
votes
2answers
72 views

Wrong numerical results from LegendreP

{Cos[Pi/180] // N, LegendreP[46, 0.9998476951563913`], LegendreP[46, Cos[Pi/180]] // N} give ...
5
votes
3answers
328 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ appearing in the large order asymptotics of the Bessel functions. The recurrence in question is: $$U_{n+1}(x)=\frac{1}{...
11
votes
3answers
257 views

Integrating a BesselJ integrand to obtain the same result as Maple 16

I would like to check the following integration: Integrate[y Integrate[1/x BesselJ[1, x Exp[I π/4]] BesselJ[1, x Exp[-I π/4]], {x, 0, y}], {y, 0, r}] ...
33
votes
8answers
7k views

About multi-root search in Mathematica for transcendental equations

I have some questions for multiroot search for transcendental equations. Is there any clever solution to find all the roots for a transcendental equation in a specific range? Perhaps ...
3
votes
4answers
416 views

Mathematica not able to confirm its own solution to differential equation

I type the following into Mathematica: DSolve[q''[x] + 2 x/(x^2 - 1) q'[x] - 4*q[x]/(x^2 - 1) == 0, q[x], x] It gives me the result ...
5
votes
2answers
243 views

Symbolically Expanding One-Variable $q$-hypergeometric Series

In general, how do I get Mathematica to symbolically expand various infinite series to an arbitrary degree of accuracy? I deal with $q$-hypergeometric series quite frequently, and specifically want ...
9
votes
4answers
463 views

Numerical instability in cosh and sinh - integral functions [duplicate]

I'm trying to calculate the function: CoshIntegral[x] Sinh[x] - Cosh[x] SinhIntegral[x] Unfortunately Mathematica seems to hit a point (x~20) and things become ...
5
votes
2answers
262 views

Strange phenomenon occurring in analytic integration result involving Bessel functions

For the following integral, Integrate[x^2 Exp[-a x^2 - b x^4], {x, -∞, ∞}, Assumptions -> {a > 0, b > 0}] Mathematica gives the following analytic ...
0
votes
1answer
249 views

Asymptotic forms of Bessel function

I want to replace Bessel functions by asymptotic forms, so the question is: can I find the best ones with help of Mathematica? And if it's possible, how can I do it? Update How can I get with ...
5
votes
0answers
110 views

MacDonald formula for Modified Bessel Functions

How can I make Mathematica understand these two integrals? $$\int_0^{\infty} e^{-x \cosh{\xi}} d\xi = K_0(x)$$ $$\int_0^{\infty} e^{-\frac{1}{2} \Big( \frac{x y}{u} + u \frac{x^2+y^2}{x y} \Big) } K_{...
3
votes
0answers
41 views

Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) ...
2
votes
0answers
46 views

Solution of the Lagrange's minimal surface equation [closed]

Can I use Mathematica to solve the Lagrange's partial differential equation? $$(1+f_y^2)f_{xx}-2f_xf_yf_{xy}+(1+f_x^2)f_{yy}=0$$ I tried: ...
1
vote
1answer
240 views

Why Integral does not converge?

I am trying to solve for $\Omega$ this nonlinear integral equation: $$1+\dfrac{z}{k^2}-\dfrac{z^2}{K_2(z)} \dfrac{\Omega}{k^3} \displaystyle\int_{1}^{\infty} \gamma^2\, \text{ArcTanh} \left(\sqrt{\...
1
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0answers
47 views

High Precision Plots of Eisenstein Series [closed]

When plotting the Eisenstein Series (great information here Eisenstein Series in Mathematica?) you observe highly non-trivial branch cut behavior close to the real axis. This makes the numerics break ...
3
votes
0answers
61 views

Are there any Mathematica programs on or about the LMFDB Archive?

I would like to explore the LMFDB Archive with L-functions using Mathematica. Is there any Mathematica sample code available to get me started?
0
votes
1answer
59 views

integral involving square of exponential integral

I'm trying to compute the integral $$ \int_{a}^{\infty}\frac{e^x}{x}[\mathrm{Ei}(-x)]^2\,dx, $$ where $\mathrm{Ei}$ is the exponential integral, and $a>0$. The obvious ...
1
vote
1answer
78 views

Solving two equations with modified Bessel functions

I am trying to solve two equations with Bessel functions in them, 1) C1*BesselK[0, 3.7268*10^-4*x] == 1.3*10^-6 2) ...