Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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0answers
33 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
65 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
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2answers
141 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
81 views

Using the results from NDSolve in another equation

I want to use the results sol returned from NDSolve (values of f at different ...
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0answers
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 ...
3
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1answer
69 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\...
-1
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0answers
42 views

Chebyshev Generating Function on (x,y) with Arbitrary Unknowns

I'm trying to create a generating function in Mathematica for Chebyshev polynomials with arbitrary unknowns on x and y. So far I was able to come up with ...
5
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3answers
247 views

How to evaluate theta function's derivative numerically?

I ran into this derivative that Mathematica won't evaluate: ...
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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}}\...
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0answers
47 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 ...
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0answers
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 ...
14
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4answers
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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1answer
52 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 ...
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1answer
73 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. ...
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0answers
105 views

How do I solve the integral over four spherical harmonics?

I want to solve this integral ...
1
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1answer
79 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_{...
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0answers
95 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
140 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
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1answer
66 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 ...
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0answers
68 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 ...
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4answers
77 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
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1answer
143 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 ...
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0answers
37 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
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1answer
78 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 ...
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1answer
77 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
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1answer
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...
4
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1answer
140 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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1answer
45 views

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: ...
1
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1answer
73 views

Inversion of a hypergeometric function

I have been trying to invert the hypergeometric function $$\rho(r)=\frac{2b}{1-q}\sqrt{1-\left(\frac br\right)^{1-q}}\,_2F_1\left(\frac{1}{2},1-\frac{1}{q-1};\frac{3}{2};1-\left(\frac br\right)^{1-q}\...
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2answers
95 views

Solving of Equation which contains Hypergeometric Function 2F1

I am trying to solve this equation where I need the solution of K in term of v ...
1
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2answers
94 views

What series does Mathematica use for Hypergeometric1F1?

I'm trying to get an analytical expression for Hypergeometric1F1[-a, 1/2, X] Provided a is an integer number. I tried adding ...
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0answers
47 views

What is the representation of the Harmonic Number being used by Mma in this result?

The Fourier Transform of the function F[x_] = (m/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*m)/Sqrt[2]] where all variables are real, and $m>0$ is given by (Mma 11.0) ...
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2answers
136 views

Plotting the inverse function of a complicated function

So I have a function F[x_] = Assuming[{Element[x, Reals], -1 < x < 1}, Integrate[1/Sqrt[(x^2 - 1)^2 + alpha*x], x]] I'm now interested in the ...
2
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3answers
146 views

Is there a way to speed up this Fourier transform and optimise the results?

Can anyone suggest a method of speeding up the evaluation of the following Fourier transform? FourierTransform[UnitStep[t] Exp[-t/τ] Cos[(m t + ω0 ) t], t, ω] I'...
1
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0answers
70 views

Plot of Laguerre-Gaussian wavefront

I have tried to plot the wavefront of a Lagherre-Gaussian bean, I know that to do that i have to plot the set of all points where the wave has the same phase. I have try to use the conditional If ...
0
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0answers
55 views

How's analytic continuation done in Mathematica?

I have the following expression: $$I_1=\frac{\sqrt{a} c^{1-n} \Gamma (n-1) \, _2F_1\left(1,n-1;\frac{1}{2};\frac{b^2}{a c}\right)-\sqrt{\pi } b c^{\frac{1}{2}-n} \Gamma \left(n-\frac{1}{2}\right) \...
0
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1answer
95 views

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 ...
0
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1answer
75 views

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 ...
1
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2answers
70 views

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 ...
4
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1answer
125 views

Appell series F3 on Mathematica

I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as $$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_1)_m(a_2)_n(b_1)_m(b_2)_n}{(c)_{m+n}m!\,n!}x^...
1
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0answers
25 views

Computing Hypergeometric Funtion of Matrix Argument [duplicate]

I'm new to Mathematica and unsure of how to compute functions or set up definitions. I'd like to do some computations with the $_1F_1$ hypergeometric function of matrix argument as in the Koev and ...
0
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1answer
78 views

How to get solutions for recursive relations using RSolve?

I have $$a(x,t+2) - a(x,t) = -\cos{\theta} [a(x+1,t+1) + a(x-1,t+1)]$$ Setting $t' = -t\cos(\theta)$ fetches under the continuum approximation $$2 \frac{\partial a(x,t')}{\partial t'} = a(x-1,t') - ...
0
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1answer
49 views

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 ...
2
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1answer
69 views

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)^...
3
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3answers
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 ...
3
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1answer
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 ∞?
1
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1answer
40 views

Limit of an integral and hypergeometric function

I want to evaluate the following integral: Integrate[Sin[θ]^(D1 - Nc - 1)/(A Cos[θ] - I ϵ)^(N1 - Nc), {θ, 0, π}, Assumptions -> A > 0 && ϵ > 0] ...
0
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0answers
23 views

Problem with integrating special function under assumption

i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible. In fact, if ...