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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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7 votes
1 answer
418 views
+50

Using NDSolve on the Painlevé equations

In an earlier question of mine, I was looking for a way to handle certain kinds of singularities (poles) when using NDSolve. Michael E2's answer, which relied on ...
5 votes
1 answer
247 views

Quantum Field Regularization: Choice between ExpIntegralE and Gamma for Casimir Effect

For the Casimir effect with exponential regularization, we compute the vacuum energy between the plates (somewhat simplified) with: $$\omega = c\ \sqrt{q^2+k_z^2}$$ $$ E = \sum_{k_z=n\pi/a} 2 \int\...
3 votes
0 answers
44 views

Dirac delta identity using MeijerG

In https://functions.wolfram.com/14.03.26.0002.01, there's an identity given for DiracDeltausing MeijerG which even has a code ...
2 votes
1 answer
107 views

WeberE definition from Wikipedia doesn't match the definition from Wolfram Language?

According to Wikipedia - Struve function: Relation to other functions the WeberE function should return this, if the first argument is a negative integer: $$ \...
3 votes
0 answers
69 views

Generalized Lambert W Function?

The Lambert W Function (defined as the inverse of $x e^x$) is implemented in Mathematica as ProductLog. Has anyone made any progress implementing the Generalized Lambert W Function defined as the ...
5 votes
4 answers
390 views

Is it possible to have the asymptotics of this function?

Working the problem of $$I_n=\int_0^1 \frac{\tan ^{-1}\left(x^n\right)}{\sqrt{1-x^n}} \, dx$$ which I have not be able to compute with Mathematica. A tedious work gave the result $$I_n=\frac{\sqrt{\pi ...
9 votes
0 answers
415 views

Does anyone know what is this number? [migrated]

Answering this question, @user64494 came with a very nice answer. My problem is that the front factor is HypergeometricPFQ[{1/2, 1/2, 1, 1}, {3/4, 5/4, 3/2}, -1] ...
2 votes
1 answer
113 views

Is it possible to express $\text{Li}_2(-\frac18(-i+\sqrt{15})+\text{Li}_2(\frac18 (i+\sqrt{15})$ as an explicit real expression (not numeric)? [closed]

I obtain this expression in my calculations, and numerically I am sure that it is a real number. ...
0 votes
1 answer
58 views

HypExp package throwing errors when loading

I am trying to load the package HypExp by using the instructions given on the site. The package throws an error ...
2 votes
2 answers
163 views

Are there any commands besides NSolve for solving equations which involve product logarithm?

I have two complicated equations and NSolve cannot give the complete set of solutions: ...
34 votes
2 answers
5k views

Visualizing vector spherical harmonics

I have painstakingly derived the vector-spherical harmonics $\mathbf{V}_{J,\,M}^\ell(\theta, \phi)$, which are the generalization of ordinary spherical harmonics $Y_\ell^m(\theta, \phi)$ to vector ...
2 votes
0 answers
427 views

How to simplify this HypergeometricPFQ function

I want to simplify this expression HypergeometricPFQ[{-12 + x/2, x/2, x/2}, {-11 + x/2, x}, 1] I tried to use FunctionExpand ...
18 votes
1 answer
712 views

Fast Hankel Function in Mathematica

I am working on a project that requires repeated calls to HankelH1[0, r] for $r$ spanning the full real axis. When I use the mathematica routine, it can be as much ...
0 votes
1 answer
54 views

Finding value of a function at limit zero

I want to evaluate the below function at limit zero. I tried with a function which has double differentiation for which it is giving me a finite value. For triple differentiation it is consistently ...
1 vote
0 answers
49 views

Is there a way of plotting elliptic rational function offline?

I'm studying the elliptic filter design and trying to plot the elliptic rational function, which is part of the filter equation. I have found out that there is an online way of plotting with the help ...
6 votes
1 answer
333 views

DSolve: unable to solve the conditions

I have a second order differential equation with $2$ boundary conditions and want to use the following code to solve it: ...
1 vote
0 answers
89 views

Differentiation of Parabolic Cylinder Functions and Hermite Polynomials w.r.t to Order

When I evaluated (ver. 12.3) parabolic functions below in the vanishing order limit: $u \rightarrow 0$, I got the following: ...
6 votes
2 answers
658 views

Spherical harmonics and Laplace operator

The spherical harmonic function $Y_l^m(\theta,\phi)$ is defined to be an eigenfunction of the angular part of the Laplace operator with eigenvalue $-l(l+1)$. In other words, it solves the PDE: $$\...
2 votes
0 answers
62 views

Abnormally long computation time using AppellF1 function

I am trying to use the AppellF1 function in Mathematica 13.3.1 on an Ubuntu machine with an i7 13700. The inbuilt function seems to be much slower in some cases ...
2 votes
0 answers
100 views

What is the exact formula Mathematica uses for Riemann zeta function? [closed]

I'm trying to find the formula of the zeta function used to plot this graph: This is taken from Wikipedia Riemann Zeta Function. It was created in Mathematica by ...
1 vote
2 answers
318 views

Spherical Harmonics Parity

In Mathematica's documentation, the Spherical Harmonics are said to be defined as follows, for $l \geq 0$: Furthermore, we know that $\cos(x)=\cos(-x)$, hence one can be led to believe that $Y_l^m(-\...
0 votes
0 answers
59 views

The Fourier-Legendre expansion of a plane wave

I want to perform a Fourier-Legendre expansion on the spatial part of a plane wave. $$ \mathrm{e}^{\mathrm{i}(\boldsymbol{k} \cdot R-\omega t)}=\mathrm{e}^{\mathrm{i}(k R \cos\theta-\omega t)} $$ $$ \...
3 votes
2 answers
159 views

The conversion equation between BesselI and BesselJ

The conversion equation between BesselI and BesselJ is as follows: $$ I_v(x)=J_v(i x)/i^v $$ $$ I_{-v}(x)=i^{-v} J_{-v}(i x) $$ ...
2 votes
1 answer
68 views

FoxH function format implementation issue

I am trying to code a particular Fox-H function in Mathematica, $H_{2,2}^{2,0}\left(z\left| \begin{array}{c}(1,1),(1,\alpha)\\(1,1),(1,2)\\\end{array} \right.\right)$. I've been trying to use the ...
2 votes
1 answer
148 views

InverseFourierCosTransform

As a following one of my previous question, I have a function about Z and take an InverseFourierTransform of it as: ...
2 votes
1 answer
67 views

Plotting real spherical harmonics with SphericalPlot3D - odd functions overlap themselves, even functions ok [duplicate]

I want to reproduce plots of real spherical harmonics as shown on Wikipedia. I used: ...
11 votes
3 answers
2k views

Are solid spherical harmonics implemented in Mathematica?

In certain applications, solid spherical harmonics can be very useful. They are essentially the usual, 'surface' spherical harmonics, with the appropriate power of the radius inserted: $$S_l^m(x,y,z)=...
2 votes
1 answer
42 views

Expressions in PlotLegends

When the order of the Bessel function is negative, it will automatically be evaluated in PlotLegends. ...
2 votes
2 answers
139 views

Circular function with arbitrary radius and center

I'm looking for an implementation in Mathematica of the generalized circular function $\text{circ}(x, y; R, c)$ of radius $R$ and centre $c=(c_1,c_2)$ such that $$\text{circ}(x, y; R, c)=\begin{cases}...
3 votes
2 answers
73 views

Using FindSequenceFunction to solve the integral of the product of Legendre polynomials and power functions

The integral of the product of Legendre polynomials and power functions: $I=\int_{-1}^1 x^n \mathrm{P}_l(x) \mathrm{d} x$ The calculation result from the textbook is: $$ \begin{aligned} I & =0 \...
1 vote
1 answer
85 views

How to integrate Legendre polynomials with parameters?

The orthogonality of Legendre polynomials: $\int_{-1}^1 \mathrm{P}_l(x) \mathrm{P}_k(x) \mathrm{d} x=0, \quad k \neq l$ But ...
1 vote
1 answer
108 views

Find the range of Legendre polynomials

The range of Legendre polynomials in the Reals domain is [-1, 1]. How can we calculate it using FunctionRange or other MMA code? ...
1 vote
2 answers
142 views

Mathematica cannot solve this complicated integration

first-time here. I am trying to use Mathematica to evaluate a solution from Duhamel's principle, the integration looks like $\int^t_0\frac{e^{-ks-\frac{r^2}{2(2Ds+\sigma^2+2D_pt)}}}{2Ds+\sigma^2+2D_pt}...
0 votes
0 answers
65 views

Plot of a complicated function under double summation

I wrote the following inputs in mathematica to plot the following expression: ...
4 votes
2 answers
168 views

How to compile inverse error function?

I have the following code: ...
1 vote
2 answers
133 views

Should expression evaluation depend on the choice of a variable name?

I am verifying the series representation of the Sonine polynomial or the associated Laguerre polynomial, which is $$ L_m^{(n)}(x)=\sum\limits_{l=0}^n\left(-1\right)^l\binom{m+n}{m-l}\frac{x^l}{l!}=S_{...
6 votes
3 answers
703 views

Implementation of a complex recurrence relation for polynomials

I would like to implement the recurrence relation for the polynomials $U_n(x)$ that appear in the large-order asymptotics of the modified Bessel functions. The recurrence in question is $$U_{n+1}(x)=\...
1 vote
0 answers
85 views

Strange behavior of error function series expansion at infinity

Evaluating Normal[Series[1/ (1 + xi Erf[xi]), {xi, ∞, 2}]] Normal[Series[1/( xi (1/xi + Erf[xi])), {xi, ∞, 2}]] gives ...
1 vote
1 answer
131 views

A request on the kernel of the function ”SpheroidalEigenvalue“ in Mathemitica 11

I want to understand the algorithm of the function "SpheroidalEigenvlaue" in Mathematica 11, especially the code of the following command. Then I will try to use this algorithm to reproduce ...
5 votes
1 answer
334 views

Solving third order DE from fluid dynamics

I am trying to use DSolve to solve a differential equation from G. Batchelor: An Introduction to Fluid Dynamics, eq. 5.12.4: [...] equation now reduces to $$\boxed{...
9 votes
3 answers
527 views

Why can't NSolve solve for the obvious zeros?

Bug introduced in 13.2 or earlier and fixed in 14.0 ...
1 vote
1 answer
102 views

Limit of Hypergeometric Functions

Mathematica (12) seems to have a lot of trouble with taking limits involving hypergeometric functions. A simple example I am interested in is the following ...
3 votes
0 answers
75 views

PrimeZetaP evaluation in different versions of Mathematica

PrimeZetaP was introduced in version 7.0. I suspect there were made some changes in the definition of this function in subsequent versions. Is there any user that ...
1 vote
2 answers
226 views

Error complex function ERFI(X): looking for alternative function representations?

I have some analytical results from a physics problem, where the Mathematica gives the results in terms of complex error function. I would like to explore another function representation using ...
1 vote
1 answer
132 views

What formula does Mathematica use for PolyGamma function of complex order?

For instance, PolyGamma function in Mathematica gives different values than the similar Psi function in Maple, which uses the ...
0 votes
1 answer
151 views

PartitionsP for non-integer arguments [closed]

Is there a reason why the function PartitionsP[] does not return anything for non-integer arguments, although there is a way to calculate using Rademacher’s ‘exact’ ...
4 votes
1 answer
139 views

Strange result simplifying higher order BesselJ [duplicate]

Consider the following integral: $$\int_0^1 Z_n^m(r)\ J_m(\rho r) r dr$$. The solution of this should contain a single Bessel function: $$(-1)^{(m-n)/2}\ J_{n+1} (\rho)/\rho$$ (see https://www.osti....
5 votes
1 answer
436 views

HypExp and HPL packages for hypergeometric functions: Evaluating a function HPL[{minus,plus},x]?

I am currently using the HypExp and HPL packages, which are useful for expanding hypergeometric functions in series around integer or half-integer values, as in common in dimensional regularization ...
1 vote
0 answers
86 views

Heaviside function in NDSolve

I have: ...
2 votes
0 answers
63 views

Expanding Pochammer symbols/Gamma function for simplifying expressions

TLDR: How to expand gamma functions or Pochammer symbols in an arbitrarily long product? Some context I am trying to find out a closed-form expression for $\langle r^\alpha\rangle$ for the non-...

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