Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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0
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1answer
49 views

Hypergeometric function not giving a numerical value [closed]

I tried to run the following code: ...
1
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0answers
57 views

Solution of a Bessel-like Equation with Boundary Condition at Infinity

I am trying to solve the following differential equation: $$x^{2}P''(x) + 2x P'(x)+[\omega^{2}x^{2}+2]P(x)=0\,,$$ defined for $x>x_{0}>0$ and with the following boundary conditions $$f(x_{0})=...
9
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1answer
148 views

Possible bug involving derivative of BesselI

In Mathematica 12.0, I run the following code: f[x_] = BesselI[0, x]; f'[x] which returns BesselI[1, x] as expected. But if I ...
2
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1answer
61 views

Wigner 3j-symbol in mathematica

I am looking for a nice and workable formulation of the 3j-symbol in terms of hypergeometric functions. On the wolfram webpage I found: http://functions.wolfram.com/HypergeometricFunctions/...
3
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2answers
140 views

How to speed up this code with Bessel functions

I need to perform a ContourPlot on a function with BesselJ included. But my code is very slow, especially when I set the PlotPoints to be 100. I would like to know how to speed up this code. Any ...
0
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0answers
64 views

Teach Mathematica analytical continuation of the gamma function

If I ask Mathematica to compute the gamma function for me Integrate[Exp[-s] s^(a - 1), {s, 0, Infinity}] It dutifully returns to me ...
3
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0answers
43 views

Evaluating Lauricella functions of Third kind numerically in Mathematica

Let's consider the Lauricella function of third kind, denote as $F_{C}(a,b;c_1,...,c_n;x_1,...x_n)$ in MathWorld. Is anyone aware of an algorithm that allows to numerically evaluate such a function ...
4
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3answers
217 views

How to simplify this formula?

$$ \int \frac{1}{\sqrt{1-2 x^3}} \, dx $$ Integrate[1/Sqrt[1 - 2 x^3], x]// FullSimplify The result is very complex. Then I want to differentiate it with the ...
0
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2answers
80 views

Why do both Hypergeometric1F1Regularized and Hypergeometric1F1 explode for large x?

Hi I am wondering why the increasing solution to eq0 below explodes for large x. Mathematica presents this as LaguerreL[-1-q,m,x], which is the same thing as a multiple of Hypergeometric1F1. Both ...
1
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1answer
112 views

Confluent Heun function

Does this function (confluent Heun function) exist in Mathematica? I try to solve the following equation, ...
3
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1answer
67 views

Negative Harmonic Series?

I was doing some work and the following expression came up in one my computations... ...
3
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0answers
47 views

Series expansion of Lerch transcendent still buggy?

This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent, ...
4
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3answers
418 views

Why does Mathematica refuse to evaluate my integral?

Here is the integral for which I want a symbolic result: Integrate[x^(z - 1)PolyLog[2, x]/(1 + x), {x, 0, 1}] But the output is the same as the input without any ...
3
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1answer
60 views

Finding Coefficients involving Bessel Functions

I have a differential equation: $x^2 \frac{d^2 y}{dx^2} +(x^2-3)y=0$ I solved it and obtained: y[x] -> Sqrt[x] BesselJ[Sqrt[13]/2, x] C[1] + Sqrt[x] BesselY[Sqrt[13]/2, x] C[2] Now I'm trying to ...
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0answers
65 views

DSolve function is not working for Hypergeometric Function

I am trying to get the solution of second order differential equation, but it is not working. Any idea about it will help me a lot. ...
0
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0answers
37 views

Radial Mathieu functions, divergence problem

I am working on a project that requires the utilization of even Mathieu functions. This is the definition of my functions: Even Mathieu function: ...
0
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1answer
38 views

Transforming a sum of products of binomial coefficients gives only partially determined expression

On 11.0.1.0, Sum[Binomial[n + 3, i] Binomial[n, k - i] 2^i, {i, 0, k}] gives ...
1
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0answers
52 views

Is there a built-in function for Multiple Zeta Values?

For a sequence of positive integers $s_{1},\ldots ,s_{k}$, all greater than or equal to $2$, let $$ \displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{{n_{1}>n_{2}>\cdots >n_{k}>0}}\ {\...
2
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2answers
120 views

Need help with using Bessel functions [closed]

I'm new to Bessel functions, especially those of the first kind. I'm working with a problem that goes as such: With that said, is my code for said problem correct? ...
4
votes
2answers
144 views

Overflow when using SpheroidalS2

I want to compute the value of the radial oblate spheroidal wave function of the second kind. However, I found the value at small arguments (for example $0.5I$ in the following code) cannot be ...
1
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2answers
104 views

These don't have solutions over the complexes?

I've tried W. Alpha for solving the equation involving the Lambert W function , but it seemed that W.A is either couldn't understand the expressions or it just simply doesn't have solutions over the ...
9
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3answers
376 views

Elliptic Integrals: Mathematica and Gradshteyn and Ryzhik

In Gradshteyn and Ryzhik, (specifically starting with the section 3.13) there are several results involving integrals of polynomials inside square root. These are given in terms of combinations of ...
1
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0answers
39 views

How to solve an Integral analytically using a predefined definition for Besselfunctions (phi part of angular spectrum representation)

I'd like to use mathematica to calculate an Integral that is dependent on phi and theta (to obtain the intensity distribution of a tightly focused TEM20 mode using the angular spectrum representation)....
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0answers
80 views

Verifying hypergeometric identity

By experiments, it's easy to convince ourselves that the following expressions are identical for positive integer $n$. ...
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0answers
51 views

A MeijerG function does not accep arguments wrapped in “*Form” for numeric evaluation [duplicate]

I get the following Meijer function as a result of a symbolic integration: expr = MeijerG[{{1, 3/2}, {}}, {{1, 1}, {1/2}}, x]. I try to calculate numerical ...
8
votes
2answers
147 views

Closed form of product of Gamma function

Mathematica recognizes this closed form \begin{align} \prod_{k=1}^{n-1}\sin(\pi k/n) &= 2^{1-n}\,n \end{align} just fine: but fails on this one despite that this expression also has a known ...
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1answer
83 views

Using the results from NDSolve in another equation

I want to use the results sol returned from NDSolve (values of f at different ...
0
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0answers
30 views

How to simplify this hypergeometric expression?

How can I rewrite this expression in terms of a single Hypergeometric function? Is it possible? I think it is possible to convert this expression in such a way that ...
3
votes
1answer
100 views

How can I inform Mathematica of an identity concerning Bessel functions?

I am doing some analytical work that includes the integral of $e^{i(n\, t - x \sin t)}$. I know the result of this integral is a Bessel function. $$J_n(x)=\frac1{2\pi}\int_{-\pi}^\pi e^{i(x\sin\tau-n\...
5
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3answers
255 views

How to evaluate theta function's derivative numerically?

I ran into this derivative that Mathematica won't evaluate: ...
0
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0answers
45 views

Verifying simple inequality involving FactorialPower function

It is obviously true that FactorialPower[k, m] <= k^m when k and m are both positive ...
2
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1answer
67 views

Problem with numerical evaluation of a Hankel identity

There an identity with the Hankel functions of both types (https://dlmf.nist.gov/10.11 eq. 10.11.4 or http://apps.nrbook.com/bateman/Vol2.pdf pg. 80 eq. 43): $$ \sin\left(\nu\pi\right){H^{(2)}_{\nu}}\...
1
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0answers
51 views

Defining dilogarithm with branch cut on $(-\infty,1)$

If my logarithms have a branch cut on $(0,\infty)$ then the dilogs constructed from these have a branch cut on $(-\infty,1)$. Is there a safe way to implement this in Mathematica? I tried with ...
0
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0answers
37 views

After activating, inactive integral, output is coming same as input

I am trying to plot s w.r.t r (0,10). But because of inactive integral I am not able to. When I activate inactive integral, output is coming same as input. When I am trying to plot graph w.r.t r(0 to ...
14
votes
4answers
269 views

How to convert this term to a Hypergeometric function?

term=8*(-1)^(1/4)*Sqrt[b]*q0^(3/2)*\[Kappa]* EllipticF[I*ArcSinh[((-1)^(1/4)*Sqrt[b]*r)/Sqrt[q0]], -1] This is a physical term and it is not convenient to appear ...
0
votes
1answer
55 views

How to force Mathematica to give answer in certain functional form?

I want Mathematica to evaluate $$0=\psi (x) \left(\frac{k x^2}{2}-\text{En}\right)-\frac{\hbar ^2 \psi ''(x)}{2 m}$$ and give me answer in terms of the Hermite polynomials but it gives me result in ...
1
vote
1answer
74 views

Strange evaluation of Bessel Functions near $x=730$?

I am doing a calculation which involves the numerical evaluation of the following function: $$f(x)=I_0(x)K_0(x)-I_1(x)K_1(x)$$ where $I_{\nu}(x)$ and $K_{\nu}(x)$ are the modified Bessel functions. ...
1
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0answers
113 views

How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
1
vote
1answer
82 views

Polar plotting Hankel Function with a lot of terms

I am trying to plot a normalized polar plot for the following function with different values of $a$ $$\left\lvert \sum_{n=1}^\infty i^n (2n+1) \frac {P_n^1(cos(\theta))}{\sqrt{\frac{\pi k a}{2}}[-H_{...
0
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0answers
100 views

Complicated Integral output with Unfamiliar Regularized Hypergeometric Function

I need the solution for following integral and it has output in MATHEMATICA as: ...
3
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1answer
144 views

Mathematica command to convert $\pi^{2n}$ to $\zeta(2n)$ [duplicate]

Is there a command on Mathematica that helps me to get the answer of some harmonic series in terms on $\zeta(2n)$ instead of $\pi^{2n}$? Let me give you an example: The command : ...
0
votes
1answer
72 views

Domain specifications for InverseFunction

I have difficulty implementing on how to specify the domain that I want for ry which is the inverse function of rho. The necessary condition is that, $\textbf{ry}$ must remain $\textbf{positive}$ for ...
0
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0answers
75 views

Inactive integral

I am trying to solve inactive integral but output is coming same as input. Any idea about how to solve inactive integral which is a function of 'r'. I am using version 12. The integration contains ...
0
votes
4answers
83 views

Solve for coefficients to express polynomial in terms of another polynomial

If I have a polynomial, say $p(x) = 6x^3 - x^2 + x$, and I want to express that in terms of a sum of other polynomials, how may I do that in Mathematica? Specifically I would like to say that $$p(x) = ...
3
votes
1answer
148 views

What does this superscript on HypergeometricPFQ mean?

I was messing around with some integrals and I got as output the following: What does the superscript on the last term in that expression mean? I looked at the documentation for ...
0
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0answers
38 views

How to solve a system of five second order differential equations with boundary conditions?

I want to solve system of five differential equations of second order with their respective boundary conditions. So, I create a function that depends on their solutions. Such as ...
1
vote
1answer
85 views

About some hypergeometrical formulas for roots of trinomial and quadrinomial

Please correct my interpretation of formulas by Pietro Majer from post. For equation $a=x+x^p$ one root is $x=a\sum\limits_{k=0}^{\infty}\frac{(-a)^{(p-1)k}}{(p-1)k+1} {{pk}\choose{k}}$ Wolfram ...
0
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1answer
86 views

Asymptotic solution of a second-order ODE containing InverseFunction

Essentially, I have a second-order differential equation given by ode below. In order to solve it, I need to obtain an asymptotic solution where $g(x)$ must vanish at infinity which will be used after ...
6
votes
1answer
158 views

Analytical form of 2d integrals relevant to graphene

This question is continuation of my previous post. Alex Trounev was very helpful in fixing a crucial typo in the analytic solution known from the article "Density Dependent Exchange Contribution to ∂𝜇...
4
votes
1answer
150 views

Elliptic integrals

In trying to reproduce results from one paper I stumbled upon a problem with definition of some elliptic integrals (this is my guess of what could be the problem). I will first present in a ...

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