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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

0
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0answers
45 views

Summation involving 2F2 hypergeometric function

Trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$, ...
0
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1answer
39 views

Complex Infinity of Hypergeometric+Gamma function

I am solving some integral which gives an hypergeometric function+gamma function . The point is that my values of n (see the code below) are integers, so ...
0
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0answers
43 views

Plotting the parameters of Mathieu equation for stability region

I am trying to plot stability regions of Mathieu equation with a and q parameters, I plot this and now i want that i fix the value of x between -5 and 5, and corresponding to each value of x,I get ...
0
votes
1answer
40 views

Using Mathematica to find series expansions for partial derivatives of the generalized Riemann zeta function

I am trying to use Mathematica to find a suitable series expansion for the expression $$ \zeta ^{(1,0)}\left(-1,1-\frac{i}{2}\right) - \zeta^{(1,0)}\left(-1,1+\frac{i}{2}\right),$$ which ...
0
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1answer
51 views

Finding the symbolic inverse of a function

Is there a way of inverting this function to obtain $r(\rho)$? ...
1
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4answers
115 views

Calculating the Dottie number using an infinite series

The "Dottie" number is the solution to the equation $\cos(x) = x$ It is approximately equal to $0.739085133215160641655312.$ This number can be expressed analytically in the following form (see ...
3
votes
2answers
73 views

Legendre polynomials that evaluated with huge difference

I'm dealing with Legendre polynomials, involving the first kind, second kind, and the associated ones. However, I found this: ...
3
votes
0answers
92 views

How can I do a faster integration?

I have this part of my code, which takes forever to run. Does anybody know how to make it faster? Using NIntegrate I face error: "NIntegrate::eincr: The global error of the strategy GlobalAdaptive ...
1
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1answer
49 views

Partial differential equation heat/diffusion equation 3d

I'm trying to solve the heat/diffusion equation in 3d in spherical symmetry $\partial_t f=D\Delta f$. I wrote : ...
0
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0answers
47 views

double Integral in complex number field

The following expression is an expansion of Hypergeometric2F1 function from the above expression with the help of 9.113(in 'Table Of Integrals, Series And Products'). but I got different results.WHY? ...
2
votes
1answer
68 views

Definition of WignerD function?

On Wikipedia, elements of Wigner's D-matrix are defined as $$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-...
5
votes
2answers
140 views

Does Mathematica gives us a wrong result for the integral of a function including elliptic functions?

Calculus tells us that the differentiation of the integral of a function should be itself, but at least in one case Mathematica answers NO. I feel very confused. The new figure seems to bifurcate at ...
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2answers
176 views
1
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1answer
44 views

Expansion of hypergeometric function with symbolic parameters

I just tried in Mathematica 11.3 ...
0
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1answer
88 views

Solving the spherical harmonics PDE using DSolve

I am trying to solve the spherical harmonics PDE in Mathemtica. My code is: ...
2
votes
2answers
94 views

How to solve a Bessel differential equation with a boundary condition at infinity?

I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use: ...
0
votes
2answers
64 views

DSolve - Unable to obtain plot of solution - 2nd order ODE

I am trying to solve the equation below with DSolve. The equation is that of a wave, expected to fall off exponentially as r approaches infinity. The solution is a combination of Spherical Bessel ...
0
votes
1answer
27 views

Solving an equation involving a determinant (including spherical recursive functions)does not compute

I'm trying to solve this matrix to get a resulting function that depends on the variable Q (or if impossible, H3). When I try to to that I get two results: if I try to solve for Q, it doesn't show ...
2
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0answers
41 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
2answers
153 views

Calculating the series expansion of a theta function

I have defined the q-theta function as follows: $$\theta(x;q) = \prod_{k=0}^{\infty} (1-q^k x)(1-q^{k+1}/x)$$ I want to calculating, using this, the series expansion of the following series: $$\...
0
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0answers
19 views

Inverting the asymptotic expansion of Gauss Hypergeometric Function

I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of the object rho[r_,b_,q_] below). Basically I want to series expand rho[r_,b_,q_] for large $r$ (i.e. as $r\...
4
votes
2answers
168 views

Expansion of the Meijer G Function

I'm trying to do the integral Integrate[ B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, a, ∞}] which I figured should be relatively simple as the integral of a Bessel ...
0
votes
2answers
64 views

How to deal with the loss of significant digits in this expression with Fresnel integrals?

I needed to solve this integral: $$\int_1^\infty \frac{dx}{\sqrt{x}} \cos \left(a x-\frac{\pi}{4} \right) \cos \left(b x-\frac{\pi}{4} \right) \cos \left(c x-\frac{\pi}{4} \right)$$ Coming from ...
1
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0answers
35 views

Expression simplified for explicit int values, but not with FullSimplify/FunctionExpand and Assuming. What formula does Mathematica use?

I have a $_4F_3$ hypergeometric function (Mathematica 11.3) HypergeometricPFQ[{1, 3/2 + n, 1 + 2 m + n, 1 + 2 m + n}, {2 + n, 2 + n, 3/2 + 2 m + n}, z] If I ...
2
votes
1answer
56 views

Simplifying elliptic functions

I am working with (long) elliptic functions, which are rational functions in WeierstrassP and WeierstrassPPrime. As expressed in ...
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0answers
80 views
-1
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1answer
48 views
2
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1answer
79 views

Plot $g(r) = \int_0^\pi J_0(w\ r)\ \mathrm dw $

I want to plot $g(r) = \int_0^\pi J_0(w\ r)\ \mathrm dw $ as a function of $r$, but $g(r)$ at each value of $r$ is an integral of the Bessel function over the the limits of $w$.
0
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0answers
48 views

Numerically integrating a highly oscillatory multi-dimensional Bessel function with good precision

This is a follow up question to Numerically integrating a highly oscillatory Bessel function with good precision. Now I am considering a three dimensional integral, ...
1
vote
2answers
108 views

Numerically integrating a highly oscillatory Bessel function with good precision

I am trying to evaluate the following integral Integrate[Csch[w]^2 (w BesselJ[1, w Sqrt[y]/π]^2), {w, 0, Infinity}] in an asymptotic large ...
9
votes
5answers
295 views

NDEigenvalues vs. FindRoot for finding the eigenvalues of Airy equation?

Say I am trying to find the first 5 eigenvalues of the differential equation $f''(x)=\lambda x f(x)$, on the interval [-1,0], with boundary conditions $f(-1)=f(0)=0$. I will try to do this 3 ways, ...
0
votes
1answer
48 views

Numerical Approximation of erfc

I have a crazy output. I already tried the N[] function. All it does is condense the inputs. Any idea of how to have a numerical output from the code?
0
votes
1answer
74 views

Solving two nonlinear equations in two unknowns

This is my code defining two nonlinear equations. I want to solve them simultaneously. ...
1
vote
2answers
241 views

Numerical integration with Dirac delta

I have some complicated function depending on many arguments $x,y,z$ and parameter $a$ multiplied by Dirac delta of another function, $$ \tag 1 f(a,x,y,z) = g(a,x,y,z)\delta(t(a,x,y,z)) $$ I want ...
1
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0answers
38 views

Performance of the nested functions with Integrate

I have functions similar to this: ...
3
votes
0answers
84 views

Summation of the multipole expansions [closed]

Let $ \vec\Omega, \vec\Omega' $ be two unit vectors in $ \mathbb R^3 $ such that $ \vec\Omega\cdot\vec\Omega' = t $ and let $ r > 0 $. The multipole expansion for the exponential reads: $$ \...
1
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0answers
27 views

Inverse of Normal Distribution CDF incorrect for large value?

I have a function F which is the CDF of the standard normal distribution. The inverse of F should be infinity at 1. However, I ...
14
votes
4answers
290 views

Terrible accuracy of DawsonF

DawsonF[30.] returns 0. The correct value is 0.016676... At least it prints a warning message, ...
3
votes
2answers
91 views

Integration with Bessel function as result

When I try to calculate this integral, 1/(2*Pi)*Integrate[ Exp[I*2*θ]*Cos[1/2 (7 + Cos[θ])], {θ, 0, 2*Pi}] Mathematica is unable to compute it. But the result ...
3
votes
2answers
120 views

Wrong Output for Limit

I want to calculate the following limit: ...
0
votes
0answers
43 views

Analytic derivatives of LegendreP[a, z] w.r.t. a at a = 0?

I am interested in finding analytic expressions for various derivatives of the LegendreP[a,z] w.r.t. a at ...
4
votes
1answer
106 views

Tips for Complicated Indefinite Integrals

I would like to have an analytical expression for some complicated indefinite integrals (example below). I am able to solve these integrals numerically, and I am able to plot them in Mathematica, ...
6
votes
1answer
279 views

What is wrong with this code? (Usage of FunctionInterpolation and how to make code efficient)

I am trying to create an interpolation out of a function that comes from an improper integral. The end goal is to make a 3d plot over a specific region. Specifically, I have the following code: ...
5
votes
2answers
72 views
1
vote
2answers
129 views

Why Mathematica does nothing with the expression Gamma[2 z]? [closed]

Could someone tell me why Mathematica just returns the input with this expression: Gamma[2z] Gamma[2 z] ? I expected this result: $Γ(2z)=\frac{2^{2z-1}Γ(z)Γ(...
0
votes
1answer
88 views

Analytical form of the answer of a definite integral

I am trying to find out the analytical form of the answer of the following integration, $$I=\int_0^l r \, dr \int_0^{2\pi} \, d\phi (iks)[ \frac{\exp [ikx_0(N+r^2)^{1/2}]}{[x_0(N+r^2)^{1/2}]}\frac{\...
-1
votes
1answer
68 views

Numerical Inversion of an incomplete beta function expressed as gauss hypergeometric function using Mathematica

I am currently working with this hypergeometric function ${_2}F_1$, $\rho(r)=\frac{2b}{1-q}(1-(\frac{b}{r})^{1-q})^{\frac{1}{2}}{_2}F_1(\frac{1}{2},1-\frac{1}{q-1},\frac{3}{2},1-(\frac{b}{r})^{1-q})$ ...
0
votes
2answers
93 views

Product of large number with a very small number returns zero because Mathematica sets the very small number equal to zero [closed]

I have a product Exp[-I*Pi*x]*BesselK[-1, 2.43*Ix]. Now, Exp[-I*Pi*x] grows larger and larger as $x$ increases for imaginary ...
1
vote
1answer
79 views

Integral of modified Bessel function is wrong

A simple integration of a modified Bessel function gives: ...
1
vote
0answers
135 views

Approximate the relationship between 6 nonlinear functions involving elliptic integrals

I am trying to solve a physics-related problem, which results in approximating a relationship between 6 symbolic functions $F_1(\alpha,\beta ,x_0,y_0),F_2(\alpha,\beta ,x_0,y_0),...,F_6(\alpha,\beta ,...