Questions on the special mathematical functions implemented in Mathematica.

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1
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0answers
119 views

Mathematica not evaluating q derivative of Jacobi theta function

Jacobi theta functions, $\theta_a(u,q)$ for $a=1,2,3,4$ are defined in the unit disk $|q|<1$. For some reason that I would like to understand, Mathematica does not evaluate numerically the $q$ ...
11
votes
1answer
689 views

Numerically solve 2nd order differential equation with singularity

Consider a second order differential equation with a potential that diverges at some generic value in the variable. For example: $$-y^{\prime\prime}(s)+\frac1{\mathrm{cn}{(s\mid k^2)}}y(s)=0$$ where ...
6
votes
2answers
148 views

Number of divisors visualized with the QPochhammer function, how improve performance of code?

I have this code that is originally Jeffrey Stopple's code for the Riemann zeta function in the complex plane. Because I discovered yesterday that the number of divisors can be generated with the ...
0
votes
0answers
61 views

JacobiCN inaccuracy [on hold]

For my calculations I need JacobiCN[s,k] for s near 0 from the left and k near 1 from the ...
3
votes
3answers
251 views

3D Gamma function

I try to plot the following function: Plot3D[Gamma[1+0.5*(n+m)]/Sqrt[Gamma[1+n]*Gamma[1+m]],{n,0,1000},{m,0,1000}] I expect that for m=n and near to it the value ...
28
votes
1answer
950 views

Fast Spherical Harmonics radiative transfer

This is a rather specific question and I apologize for spamming you with some lengthy code. But it could be interesting for some reader and maybe you can help out, so please bear with me. I am using ...
22
votes
8answers
5k 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 ...
7
votes
1answer
108 views

Eisenstein Series in Mathematica?

Mathematica doesn't seem to have built-in tools to deal with the Eisenstein Series: $E_{2}(\tau)= 1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n \tau}}{1-e^{2 \pi i n \tau}}$ $E_{4}(\tau)= 1+240 ...
7
votes
2answers
764 views

Mathematica 10 cannot solve definite integral [duplicate]

Bug introduced in 10.0 and fixed in 10.0.2 Mathematica 10 fails to solve the following integral, saying that it does not converge. ...
3
votes
1answer
112 views

Bug in associated Legendre Polynomials?

Mathematica's definition of the connection of associated Legendre polynomials with $m$ and $-m$ is: $P_l^{-m}=(-1)^m \frac{(l-m)!}{(l+m)!} P_l^m$. We also now that $|m|>l \Rightarrow P_l^m=0$. ...
2
votes
1answer
122 views

Equation involving hypergeometric functions

I want to solve this equation but nor NSolve nor Solve are able to do this. ...
1
vote
0answers
48 views

Use Solve[] with Bessel, gamma, and hypergeometric functions?

I need to find values of {a,b,c} such that the 0th, 2nd, and 4th order moments of f[x]=Exp[-ax^4 - bx^2 - c] will equal respectively {1,2,10}. I didn't really expect this to work, and it didn't: ...
3
votes
2answers
207 views

Storing Variables in “Loops” and Point Plotting

Given the function $y=\sin x$ defined over the region $-\pi \leq x \leq \pi$, I need to implement a "do loop" such that I sweep over 100 or so points $-1 \leq y \leq 1$ and find precisely the two $x$ ...
14
votes
1answer
346 views

How to enlarge Mathematica's knowledge about certain functions?

I'm often troubled with the following task. I need to carry out symbolical computations involving certain special functions. Let me take as an example Barnes gamma-function. It is included in ...
-1
votes
1answer
35 views

Inverse of a Digamma/ Polygamma Function

How one can find the inverse of a digamma/ Polygamma function in Mathematica 10?
5
votes
5answers
2k views

Series expansion in terms of Hermite polynomials

I am trying to expand a polynomial in terms of orthogonal polynomials (in my case, Hermite). Maple has a nice built-in function for this, ChangeBasis. Is there a ...
6
votes
2answers
59 views

Simplify expression to Integer

I have the following function: f[n_]:=(n-1) E Gamma[n,1]; I know that this expression always evaluates to an Integer, but Mathematica doesn't always output the ...
7
votes
2answers
204 views

On the definition of the associated Legendre polynomials

Mathematica computes for n = 1,2,...: (-1)^n (LegendreP[n, -1, -3]/Sqrt[2]) -I, -3 I, -11 I, -45 I, -197 I, ... Maple ...
9
votes
2answers
173 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}] Mathematica 9.0 is ...
3
votes
1answer
263 views

plotting hypergeometric functions

Does anyone know why a plot of a hypergeometric function turns out differently in Mathematica than in Maple? The function I'm plotting between x=-30 and +30 is: ...
4
votes
1answer
404 views

Strange result for the analytic integration leads to Hypergeometric2F1

The integration result for Integrate[1/(r^2 Sqrt[x/r^(4 - 2 \[Gamma]) + 1]), r] is: ...
2
votes
1answer
112 views

Unexpected Weierstrass $\wp$-Function Behavior

In Mathematica, the Weierstrass $\wp$-function is expressed by WeierstrassP[u + I v, {g₂, g₃}] where g₂ and ...
14
votes
4answers
529 views

How to plot Ramanujan's continued fraction in Mathematica?

I want to plot Ramanujan's continued fraction: $$R(q)=\cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ but I do not know how to define this function in Mathematica. How do I define and plot ...
2
votes
1answer
110 views

What are the terms of the sequence generated by Zeta(3s)/Zeta(s)?

The LiouvilleLambda function has Dirichlet generating function of Zeta[2s]/Zeta[s]. I am curious about an analogous function with Dirichlet generating function of Zeta[3s]/Zeta[s]. Can Mathematica ...
3
votes
1answer
88 views

Complex LogIntegral error

Going through Derbyshire's Prime Obsession & trying to take LogIntegral of 20^ZetaZero[1] & comes up with a value of ...
3
votes
1answer
127 views

Strange NSum behavior

If I do: NSum[(i + 1)/(i + 2) LegendreP[i, 0] LegendreP[i, 0], {i, 0, Infinity}] I get: 1.25216 If I do: ...
3
votes
2answers
132 views

ParametricPlot with 2 variables

I want to try plotting this: As you can see the first axis is v and the second is just dependent of the angle of $\Phi$ . The function is ...
0
votes
1answer
86 views

Why is the indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$ the upper Gamma function in Mathematica? [closed]

I have an indefinite integral $e^\frac{x}{2}x^\frac{g}{2}dx$, I try to solve in Mathematica. Solve[Int[e^{x/2}x^{g/2-1}dx ,x]] (* Gamma[g/2,x/2] *) How does ...
2
votes
2answers
308 views

How I can integrate $\int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$?

How can I get a solution to the integral given below ? $\quad \quad \int_0^{a} x e^{-\frac{b^2 x^2}{2 c}} J_0(n x) dx$ where $a,\,b,\,c$ and $n$ are constants and $J_0$ is a Bessel function of the ...
3
votes
3answers
252 views

Hypergeometric function with a matrix argument

I am looking for the evaluation of a Hypergeometric function with a matrix argument as for example in Koev and Edelman or as showcased in this Wikipedia article. From what I understand from ...
0
votes
0answers
62 views

Why is PolyLog[] giving weird answers for ordinary values? [duplicate]

Possibly related to this question, but it seems slightly different: Strange behaviour of PolyLog Function Wikipedia says that for real s, z<1 should be real. So I was confused when MMa returned: ...
4
votes
2answers
112 views

Imaginary terms in the derivative of Jacobi theta function (2) on the real line

I am trying to calculate/plot the derivative of the second Jacobi theta function $d\theta_2(0, e^{-\pi t} )/dt$. Calculating or plotting the function itself works fine: ...
2
votes
2answers
165 views

Define PolyLog so that positive reals evaluate on upper edge of branch cut

Generalizing the previous question: Define Log so that negative reals evaluate on lower edge of branch For real positive values $x>1$, Mathematica's polylogarithm function ...
4
votes
1answer
240 views

Numerical errors/inaccuracies in ProductLog

Context In cosmology, a fairly accurate model to describe the gravitational potential, $\psi(r)$ of dark matter halos is given by $\psi( r)=\log(1+r)/r$. ...
1
vote
0answers
58 views

Improve speed of evaluating a sum over four indices

I am trying to implement the Fox-H function with several variables as sum of residues. I have arrived at the following function; ...
10
votes
2answers
246 views

Asymptotics of $\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}}$

I am fairly sure that asymptotically $$\frac{\sum _{i=0}^{\lfloor n/2 \rfloor} {2(n-2i) \choose n-2i} {n \choose 2i} {4i \choose 2i}}{2^{3n - 1}} \sim \frac{2}{\pi n}.$$ I tried ...
3
votes
3answers
134 views

Calculating $\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$

How can we force Mathematica to compute this limit? $$\lim_{x\to 1} \, \int -\frac{i \text{Li}_2\left(x-x^2\right)}{\sqrt{3} \left(x-\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)} \, dx$$ It seems it ...
5
votes
2answers
246 views

When does the real part of Zeta vanish on the critical line?

This seems to be a quite a simple problem but I cannot make it work. I am trying to find all values within a given range for which the real part of the Zeta ...
1
vote
1answer
477 views

Plotting Fresnel function

I am trying to plot the partial sums and the Cesàro means of the function $\sqrt{|x|}$ and for $a_{n}$, I obtained the following code which contains FresnelS. ...
0
votes
2answers
128 views

How can I compute the real part of $\zeta^2$ numerically? [duplicate]

I want to compute and plot $\Re(\zeta(x+iy)^2)$ and $\Im(\zeta(x+iy)^2)$. How can I do that with Mathematica?
0
votes
1answer
360 views

How to use `FindRoot` to solve an equation containing a parameter?

I'm trying to derive some of the results of the following paper: Electrodynamics of semiconductor-coated noble metal nanoshells, JT Manassah - Physical Review A In the paper there is matrix $\mathbf ...
4
votes
2answers
97 views

Evaluating a Series expansion of PolyLog function

I am trying to evaluate an expansion of the following integral $$ \int_0^\infty \frac{p^4}{1+\exp\left({\frac{p^2}{2mT} - \frac{\mu}{T}}\right)}\, dp = A_0(m,\mu) + A_2(m,\mu)T^2 + \ldots $$ in terms ...
0
votes
1answer
201 views

How to calculate the unknown quantity in an infinite series?

I'd like to calculate x value in this equation. Basically, I tried to 2 types of method which are FindRoot and NSolve. But, I have failed the calculation caused by these errors up to now. If ...
5
votes
1answer
206 views

How to plot the result of this singular integral?

Please I open a new post here after this one : http://mathematica.stackexchange.com/a/59203/10158 Now I want to plot the function $f(a,b)$ as a function of $b$ for different values of $a$ : $a=0.5$ , ...
1
vote
1answer
211 views

Plotting Integral equation

I want to plot the following indefinite integral : $C_l^{CC}=\int k^2\mathrm{d}k\: [e^{-2k^{2}}P_{Cl}^2(k\eta)|\dot{h}(\eta)|^2]$ with k from 0 to some large value (considered to be $\infty$), ...
2
votes
1answer
156 views

Definite Integral over Bessel Function

Hello I am interested in evaluating the following integral. ...
4
votes
1answer
298 views

Fredholm integral equation of the second kind with kernel containing Bessel and Struve functions

I need to solve this Fredholm integral equation of the second kind: f[s]+integrate[f[t] K[s,t],{t,0,1}]=s where ...
1
vote
2answers
103 views

Find all roots of a function with parabolic cylinder functions in a range of the variable

I want to find all roots of a function involving Parabolic Cylinder Functions. In what follows, I define 2 variables $\xi1$ and $\xi2$, which in turn depend on $\omega$. My function is then defined as ...
3
votes
1answer
56 views

Generating terms of the Stirling series

The Stirling series starts as follows: $$ n!=\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n} \Bigl\{1+\frac{1}{12n}+\frac{1}{288n^{2}}-\frac{139}{51840 n^{3}}-\frac{571}{2888380 n^{4}}+O(n^{-5})\Bigr\}. $$ ...