Questions on the special mathematical functions implemented in Mathematica.

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2
votes
1answer
32 views

Replacing expressions with variables [closed]

I have a large matrix where the entries are rational in Jacobi elliptic functions e.g. I'd like to replace the elliptic functions by variables, e.g. ...
3
votes
1answer
67 views

Simplification of Gamma functions

I am having some trouble simplifying some Gamma functions. I have a large expression in which some combinations of Gamma functions appear, that can be simplified, but applying ...
0
votes
0answers
5 views

Meijer's G-function differentiation [migrated]

I am trying to calculate the derivative of the Meijer's G function, Based on wolfram function identities I have found in (07.34.20.0003.01) that the derivative is expressed asl: $\frac{d}{dx}G^{m,n}_{...
1
vote
2answers
75 views

Series of a hypergeometric function

Let $n>2$ be odd, and let $x\in [0,1]$. I would like to calculate the Taylor expansion of $$ x^{2-n} \, _2F_1\left(-\frac{n}{2}-1,-n;2-\frac{n}{2};x^2\right) $$ at $x=1$ leaving $n$ non specified....
1
vote
1answer
185 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 ...
4
votes
1answer
111 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. ...
6
votes
3answers
220 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: ...
0
votes
1answer
93 views
11
votes
2answers
125 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
vote
0answers
38 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
96 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
1answer
68 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 ...
2
votes
2answers
140 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 ...
2
votes
1answer
78 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 ...
1
vote
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. ...
4
votes
1answer
92 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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
1answer
52 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: ...
0
votes
1answer
73 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 ...
2
votes
1answer
95 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 ...
0
votes
2answers
103 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 ...
1
vote
0answers
43 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 ...
10
votes
1answer
251 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....
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 ...
3
votes
0answers
43 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) ...
1
vote
0answers
49 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
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
63 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
62 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) ...
3
votes
2answers
86 views

Integral invoving product of Whittaker functions

I'm trying to evaluate the integral $$ \int W_{0,a}(x)\,W_{1,b}(x)\,\frac{dx}{x}, $$ where $W$ denotes the Whittaker $W$ function, and $a,b\in[0,1/2]$. Using the Whittaker differential equations that $...
3
votes
2answers
151 views

How can I get LogIntegral[z] to be printed as “li[z]”?

I have no problem with the current formatting of the function, but for the sake of the reader less familiar with Mathematica functions, is there a way to define, say, ...
1
vote
1answer
81 views

Finding roots of Bessel function $y=3J_1(x)+xJ_1'(x)$ is returning inaccurate roots. Not Kernel bug [closed]

I can't figure out why Mathematica is returning the incorrect roots. The first five should be 2.9496,5.84113,8.87273,11,9561, and 15.0624 according to my textbook. ...
1
vote
1answer
241 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{\...
0
votes
1answer
58 views

NIntegrate[] Gamma Function

Gamma Function is known to be : Source first i plot the function z = 1; f[t_] := (t^(z - 1))/Exp[-t]; gamma = Plot[Gamma[g], {g, 1.0, 5.0}] then, i ...
4
votes
0answers
69 views

Huge difference after changing a fraction to decimal

I have a limit to calculate. With[{p = 1/2, q = 0.1}, Limit[a^q Integrate[Sin[x]/x^p, {x, a, Infinity}], a -> Infinity]] gives correct result ...
1
vote
0answers
77 views

Ramanujan's asymptotic formula for $p(n)$ [closed]

The following expression is the exact formula of partition function. Ramanujan's asymptotic formula for $p(n)$ is following $$p(n)\sim\frac{1}{4n \sqrt{3}}e^{\pi \sqrt{2n/3}}$$ Can I use ...
1
vote
1answer
83 views

NIntegrate gives two results for two forms of the same function. Which one to trust?

I am interested in evaluation the following integral numerically (since apparently there is no analytical solution) $$\int dx \,x^3 \left(e^{2 i c x }-i \text{erfi}\left(\frac{x +i c }{\sqrt{2}}\...
5
votes
1answer
76 views

ComplexInfinity for a convergent product

The infinite product involving the ratio of (n^2)! to its Stirling approximation ...
4
votes
0answers
86 views

Bug in Series[Pochhammer[1+n, n], {n,Infinity,1}]?

The function Pochhammer[1 + n, n] tends to infinity. We have ...
2
votes
0answers
44 views

Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]] (* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *) What is the reason the term ...
2
votes
1answer
101 views

Mathematica doesn't know about the absorption identity? [closed]

The well-known binomial absorption identity states that $$n\binom{n-1}{k-1} = k \binom{n}{k}$$ However, Mathematica gives ...
3
votes
1answer
27 views

Asymptotics of Bessel function for real arguments

I am trying to calculate the following asymptotic behaviour: Normal@Series[BesselK[1, r \[CapitalLambda]] / BesselK[1, \[CapitalLambda]], {r, 0, 1}] but for $\...
0
votes
1answer
109 views

Elliptic integral of the first kind [closed]

I want to plot the integral $$I(\phi) = \int_0^{\phi} \frac{\mathrm{d} \theta}{\sqrt{1 +\sin(\theta)^2}}$$ In Mathematica notation, it is a case of an elliptic integral of the first kind with $m=-1$, ...
8
votes
1answer
305 views

Show how Mathematica defines a function

Is there a way to show how Mathematica defines a function, such as In: Something[Sqrt], Out: Sqrt[x] -> x^(1/2) As far as I understand it the command ...
1
vote
2answers
79 views

Finding roots of a function that includes Bessel functions [duplicate]

I'm fairly new to Mathematica so forgive any stupid mistakes. Here's my function: ...
6
votes
0answers
100 views

SiegelTheta throws errors from calling Range with complex arguments

Bug introduced in 10.4 or earlier. This may or may not be related to the bug reported in this question. I was trying to verify the results of this challenge over on Code Golf with the following code:...