# Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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### 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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### 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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### 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 ...
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### 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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### Using the results from NDSolve in another equation

I want to use the results sol returned from NDSolve (values of f at different ...
23 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 ...
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### 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 ...
36 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 ...
260 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 ...
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### 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 ...
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### 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. ...
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### How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
154 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...
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### 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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### Simplifying sums and showing equality - limitations?

Is it possible to verify the following lhs,rhs involving the sums are equal, with Mathematica? I can verify it for individual values of $d$ variable: ...
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### Performing a contour integration in Mathematica for a contour starting at $1$ and ending at $-\infty$ while avoiding the origin?

I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica. I want ...
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### Trying to compute erfcx(x)? [duplicate]

The function erfcx(x) = exp(x^2)erfc(x) is sometimes provided in numerical packages to avoid numerical underflow for large values of ...
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### Explicit series notation for hypergeometric functions

Is there an automated way to express hypergeometric functions in series form using gamma functions, factorials, double factorials or rising factorials? For example using the formula (on the ...
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### How to solve this equation by Solve?

I have an equation to be solved. But Mathematica does not work for it. I hope the solution x can be expressed as a function of a and b ...
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### Calculation of Legendre Polynominals via Rodrigues Formula [closed]

I want to calculate Legendre Polynomials via Rodrigues formula for n=0,...,10 I wrote this down but how can I calculate n = 0,...,10 ? P_n(x)=\frac1{2^n n!}\frac{\mathrm d^n}{\mathrm dx^n}(x^2-1)^...
120 views

### Zeros of high degree polynomials

I am working with Hermite polynomials in Mathematica with the built-in function HermiteH. I want to compute the zeros of the polynomial ...
77 views

### Evaluating an integral combining a Bessel function with some other functions [closed]

How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?