Questions on the special mathematical functions implemented in Mathematica.

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0answers
96 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
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3answers
123 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 ...
4
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2answers
79 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 ...
1
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2answers
74 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 ...
0
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0answers
8 views

Scaling properties of Airy functions. [migrated]

I'm specifically intereseted in how to rewrite $(\alpha\in \mathbb{R})$ $$\mathrm{Ai}(\alpha\cdot z)$$ as some constant times $\mathrm{Ai}(z).$
3
votes
1answer
54 views

Generating terms of the Stirling series

The Stirling series starts as follows: $$ n!=\left(\frac{n}{e}\right)^{n}\sqrt{2\pi n} \Bigl\{1+\frac{1}{12n}+\frac{1}{288n^{2}}-\frac{139}{51840 n^{3}}-\frac{571}{2888380 n^{4}}+O(n^{-5})\Bigr\}. $$ ...
3
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0answers
70 views

speed up evaluating a listable function

I apologize in advance for the vagueness, but I can't think of a more descriptive title. I am trying to find the average value of the square of BesselJ[0, x] ...
10
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4answers
448 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
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1answer
111 views

How can I find all solutions of a complicated expression in a desired interval with Mathematica?

I have an expression which is a combination of Bessel functions: ...
0
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0answers
51 views

Plotting complex roots of equation involving special functions

I consider an equation of the kind $u(k,\lambda)=0$, where $u$ involves special functions, ParabolicCylinderD, and $k$ and $\lambda$ are real parameter. By implicit ...
0
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0answers
55 views

PolyLog does not simplify

How come mathematica can not simplify a simple expression involving the PolyLog function? An example is: You know that ...
9
votes
2answers
203 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 ...
1
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0answers
71 views

Bessel functions and ListContourPlot [closed]

I'm trying to plot some eigenfunctions (for a semicircle) in Mathematica. Functions are of the following form: $$g_{ms}=J_{2m+1}(y_{ms}x) \sin ((2m+1)\varphi)$$ For example, for $m=0$ and $s=1$ it ...
0
votes
0answers
18 views

Not Simplify automatic some expression(values)

I try to calculate $$\text{res}\left(h(z) (\psi ^{(0)}(z)+\gamma )^2,\{z,-4\}\right)$$ the result is $$\frac{1}{6} \left(6 h'(-4)-25 h(-4)\right)$$ the result it is possible to write as $$2 \left(6 ...
2
votes
1answer
99 views

Mathematica and special functions

How is it possible that Mathematica doesn't recognize it's own definitions of special functions ? I tried as input: ...
0
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0answers
55 views

Solving a Transcendental Function of two variables numerically

I have a transcendental function F of two variables, i.e. F(x,y). The function F contains ...
16
votes
2answers
283 views

How to improve performance of BesselJ to the level of GSL?

Consider the following code: zs = N /@ Range[0, 12, 10^-5]; AbsoluteTiming[bessels = BesselJ[1, #] & /@ zs;] Length@zs I've tried to measure only computation ...
0
votes
1answer
71 views

What is a good way to check what kind of a function a function is?

i want a module or function or switch or If (whatever works), what asks if a function is an e function or a polynomial(a hole number one or fractorial), a exponential function, a logarithm function ...
0
votes
0answers
7 views

Some expectations of psi (digamma) function [migrated]

I want to derive an Expectation-Maximization algorithm for my model. But some expectations of psi (digamma) function is needed in the procedure. Assuming I have a Gamma distributed random Variable ...
0
votes
1answer
83 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 ...
7
votes
1answer
146 views

Wrong Limit with LaguerreL

Limit[(n! LaguerreL[n,-1]/(n^(n+1/4)/Sqrt[2]/E^(n-2 Sqrt[n]+1/2))-1) Sqrt[n], n->∞] Mathematica (wrong) output ...
3
votes
1answer
97 views

Roots of Whittaker W function

I am interested in finding the roots $u$ of the equation $$ W_{1,\imath b}(a)=0, $$ where $W_{\kappa,\mu}(z)$ denotes the Whittaker $W$ function, $a>0$ is a fixed parameter, $\imath=\sqrt{-1}$ and ...
1
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0answers
59 views

Orthogonality relations of Hermite polynomials

The Hermite polynomials are orthogonal. $$ \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2}\, \mathrm{d}x = \sqrt{ \pi} 2^n n! \delta_{nm} $$ Does Mathematica not use this relationship? Because running ...
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votes
1answer
121 views
0
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1answer
61 views

Problem with Erf near Infinity in NSolve

I have to solve a differential equation involving the error function Erf, but my code is not able to evaluate it in the range I need. I think I can formulate my ...
1
vote
1answer
78 views

Work around bugs in Summation, Hypergeometric function?

I'm so confused I don't even know how to phrase the question, so here's what's happening: I need an analytic form for this: ...
0
votes
1answer
59 views

Orthogonal Collocation Using Jacobi Polynomials

I'm trying to solve a PDE(diffusion-reaction in a spherical catalyst pellet) using Jacobi Orthogonal Collocation method. But at the stage of solving the system of ODEs(using ...
16
votes
0answers
190 views

Is MathieuC for moderately large imaginary arguments broken?

I'm trying to plot MathieuC[-3,0.3,I x] for $x\in[0,10]$, and here's what I get even with arbitrary precision arithmetic (here I use ...
0
votes
1answer
59 views

Using Assumptions in Expressions that Evaluate to be Elliptic Integrals

I have the following integral that I am trying to evaluate in Mathematica: $\int \sqrt{\alpha + m g l \cos(q)} dq$. If $\alpha > m g l$, then the result is a complete elliptic integral of the ...
0
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1answer
42 views

Having trouble interpreting the results DSolve gives for the Laguerre equation

I am trying to solve a second order ODE using Mathematica. Before I get into solving my (more complicated) problem, I am trying to use DSolve on known ODEs to check that the answer that Mathematica ...
3
votes
1answer
72 views

Why minimization does not work with symbolic array as arguments

If I try to minimize with constraints a function of several variables, with Gamma regularized function involved in the constraints it seems to works, as shown below (this is just a dummy example ...
7
votes
1answer
98 views

How to define $\operatorname{Mc}$ and $\operatorname{Ms}$ Mathieu functions in terms of MathieuC and MathieuS?

Reading multiple books about Mathieu functions, I always come across notation like $\operatorname{ce}_r(z,q)$, $\operatorname{se}_r(z,q)$ for angular functions and $\operatorname{Ce}_r(z,q)$, ...
3
votes
1answer
86 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: ...
3
votes
1answer
86 views

Why is the Spherical Bessel Function acting strangely at this point?

I'm doing some computation that requires the use of Spherical Bessel Functions of the 1st kind, at high orders and values. So, I managed to find this, while running it over a wide range of values. I ...
1
vote
1answer
76 views

Computational Complexity of HypergeometricPFQ

What is the computational complexity of HypergeometricPFQ? I got a result of a product of a multinomial and HypergeometricPFQ and I was wondering if that would be considered a closed form solution ...
7
votes
1answer
89 views

Sum over Binomials and Gammas

Given the function, ...
18
votes
4answers
439 views

Expansion of a hypergeometric function takes ages with Mathematica 9 and 10 (regression?)

Mathematica 8 (Linux version) can evaluate AbsoluteTiming[Series[Hypergeometric2F1[1, 1 - eps/2, 3 - eps, 1/2], {eps, 0, 0}]] in no time. On one of the ...
-1
votes
2answers
76 views

Complete definition of special functions in Mathematica [closed]

How can one display the actual integral involved in special functions like this MarcumQ[2, a, b] in Mathematica?
2
votes
2answers
236 views

Perturbation theory with Mathematica: Definite integral of polynomial times exponential times hypergeometric function of imaginary argument

I would like to ask also Mathematica users about my question from the math forum. To expand, I'm adding the code which calculates the full double integral for $n=0$ and $\mu=0$ (the second in the ...
1
vote
1answer
46 views

Defining functions that depend upon other functions which depend upon other functions and so on

The problem is that I have to evaluate functions, that take output from previous functions, and combine them to form new functions, and do some operations and evaluate the value at new function. But, ...
1
vote
1answer
100 views

Is there an easy way to let mathematica print out every Erfc and InverseErfc as F and F^{-1}

Mathematica uses complementary error function and its inverse as functions for example when integral of a Gaussian is taken. Therefore, all output expressions of Mathematica involve Erfc and ...
1
vote
2answers
95 views

Trouble using Solve and NSolve with functions involving Erf

I have the following functions: R[k_, x_, t_] := -.5*(k - x)*(1 + Erf[-(k - x)/t]) L[k_, c_, x_, t_] := .5*c*(k - x)*(1 + Erf[(k - x)/t]) I'm interested in ...
1
vote
1answer
97 views

How to find and verify relationships between functions?

MMa gave me a complicated result involving Hypergeometric0F1's. It was much less complicated after I discovered this identity: ...
0
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0answers
55 views

NSolve can not handle a PolyGamma equation

I would like to solve the following equations numerically: ...
0
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0answers
83 views

Computing Poincaré symbolic solution for an arbitrary integer order polynomial

In the 1880s, Poincaré created functions which give the solution to the nth order polynomial equation in finite form. These functions turned out to be "natural" generalizations of the elliptic ...
1
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0answers
49 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 ...
0
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0answers
55 views

Does Mathematica support Laguerre Polynomials of Matrix Argument?

I am wondering whether Mathematica can also compute the Laguerre polynomials of matrix argument as the appear in multivariate statistics? For example, they appear in this paper here: ...
0
votes
2answers
109 views

NIntegrate and Integrate of a Spherical Bessel function

I am trying to integrate over a spherical bessel function. I have used both the Integrate and NIntegrate functions in Mathematica but the values given by each do not match. Any reason why this ...
2
votes
2answers
129 views

Probability of multivariate normal being positive on each coordinate

How can I find the probability that each coordinate of a specified multivariate normal distribution is positive? I tried the following, which I believed should work ...
0
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1answer
52 views

EllipticF reduction

Is it possible to reduce the following Z to Legendre form Elliptic integral of the First kind? $$ \int \dfrac {\sec u\; du } {\sqrt{ (1- {(\nu \tan u)}^2 }} ..(1*)$$ With ...