Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,260
questions
4
votes
1
answer
100
views
Bivariate Hermite Polynomials
I know I can get the Hermite polynomials in a single variable with: HermiteH[n, x]
Now I need the bivariate Hermite polynomials.
I thought about building them with ...
- 151
0
votes
0
answers
46
views
How to transform this MeijerG function into a modified Bessel function $K_n(z)$
I have this MeijerG function
$$f(n,z)=\frac{z^2 \Gamma \left(n+\frac{3}{2}\right)}{2 \sqrt{\pi } K_2(z)}G_{1,3}^{3,0}\left(\frac{z^2}{4}|
\begin{array}{c}
n \\
-\...
- 601
6
votes
1
answer
177
views
Mathematica giving wrong answer for limit of LerchPhi[]
Mathematica is often frustrating to use, due to how it apparently get things wrong at times. It's giving me a totally wrong answer for this below limit (it gets it right though, if I pick a positive ...
- 61
1
vote
1
answer
119
views
Simplification of Elliptic integrals
Consider an indefinite "elliptic integral”,
$$
J_n(r):= \int^r_{r_0}\frac{x^n {\rm d}x}{\sqrt{f(x)}}, \quad
f(x):=(x-x_1)(x-x_2)(x-x_3),
$$
Here, $r_0>0, n=1,0,-1$, and $x_{1,2,3}$ are roots ...
- 361
-1
votes
0
answers
14
views
The complexity of convex hulls, Point-hyperplane duality [closed]
Do you know How to Solve this problem?
I don't even have an idea where to start...
1
vote
2
answers
184
views
How to use FindRoot to solve Hypergeometric1F1 imaginary number solution？
I want to use FindRoot to solve Hypergeometric1F1 imaginary number solution.
First of all, I try to use ...
- 177
2
votes
3
answers
103
views
Infinite Integral involving a Bessel function
I have trouble evaluating the following numerical integral,
$$
\int_{0}^{\infty} d k^{(yz)} \, k^{(yz)} J_{0} \left(d^{(yz)} k^{(yz)} \right) \frac{e^{-i d_x \sqrt{k_R^2 + {k^{(yz)}}^{2} }}}{ \sqrt{...
- 83
1
vote
0
answers
65
views
Possible bug: Wrong limit for `EllipticE`?
I encountered problem with computing limits of Elliptic functions E and F. For example,
...
- 535
1
vote
1
answer
68
views
Series Expansion of EllipticNomeQ differs from older Mathematica Version
I am trying to follow the numerical approach on how to calculate EllipticE and EllipticK following this paper. In there on ...
- 4,437
2
votes
0
answers
69
views
Evaluating and Plotting 3F2 is very slow
I am interested in certain hypergeometric functions
$ G_p(z) = \frac{1}{2} {}_3F_2 \left( \begin{matrix} p & p + 1 & \frac{1}{2} \\ 1 + i & 1 - i & \end{matrix} ;z \right) $
and would ...
- 121
-3
votes
1
answer
85
views
3
votes
1
answer
73
views
How to handle very numbers that are "too small to represent as a normalized machine number"?
I will preface this by saying that I am not very familiar with Mathematica whatsoever. That is why I am asking this here even though I have found a few forum posts that might provide the solution to ...
- 33
0
votes
0
answers
3
views
I have to proof that the integral converge for all p >0 [migrated]
$$
\int_{0}^{1} e^{-x} x^{p-1} dx
$$
I guess I should have used the sharing criterion but Im new to this and I dont know how it works, I understand the ...
3
votes
1
answer
131
views
Finding zeros of a complex Airy function
I want to find the zeros of the solution to this ODE
DSolve[{y''[x] - (I x - 2) y[x] == 0, y[0]==0,y'[0]==1}, y[x],x]
I use
...
- 269
0
votes
1
answer
65
views
2
votes
2
answers
180
views
Integral of orthogonal Bessel functions
This is the basic form of the problem i am solving.
The main difference with the integral on the photo and my example is that i have my integral defined from b to c instead of 0 to a, and Bessel ...
- 95
2
votes
1
answer
176
views
Goodness of fit [closed]
I want to evaluate the goodness of two sets or more fitting parameters, using Rsquared and RMSE (root mean square error), Then how to code?)
...
- 33
2
votes
1
answer
95
views
How to write a specific Bessel function in Mathematica
I want to plot the following function on Mathematica, and I gave it a go on wolframalpha.
Bessel[n,z] is the usual form, but I am not sure how to use this to compute the following plot:
\begin{...
- 269
10
votes
3
answers
2k
views
Can Mathematica calculate this elliptic, triple integral?
Integrate[(4 a b/Pi) (a^2 + b^2 - 2 a b Cos[c])^(1/2), {a, 0, 1}, {b,
0, 1}, {c, 0, Pi}]
I'm using basic plan, it gives me the result like that.
The ...
- 135
0
votes
2
answers
109
views
2
votes
1
answer
159
views
0
votes
1
answer
72
views
The inverse Laplace transform alters parameter constraints
I have this Laplace transform:
$$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$
...
- 187
6
votes
0
answers
133
views
Bugs in hypergeometric functions with negative integer lower parameters
If you were to evaluate these expressions, Mathematica returns the value shown.
...
- 61
1
vote
2
answers
83
views
How to plot the perimeter of an ellipse where the variable is its eccentricity?
I'm basically just trying to plot a function where the x value is the eccentricity of an ellipse and the y value its perimeter.
I've tried first defining the eccentricity
...
- 11
5
votes
2
answers
64
views
A strange behaviour with function $\,_2F_1$
I posted today a question on MSE 1
u=(5*(49*Pi^2*Zeta[3] - 558*Zeta[5]))/(77*Pi^2*Zeta[3] - 930*Zeta[5])
f[k_]:= Hypergeometric2F1[1/2, -k, 3/2, u]
Computing
<...
- 597
0
votes
3
answers
249
views
Computing the elliptic integral $\int \frac{1}{x \sqrt{f+\frac{a}{x^3}-\frac{k}{x^2}}} \, dx$
I don't know how to solve this with or without Rubi?!!
...
- 391
2
votes
4
answers
158
views
Solving $0=-\lambda \phi (t)^3+\mu ^2 \phi (t)+\phi ''(t)$
I happen to know that the equation
$$0=-\lambda \phi (t)^3+\mu ^2 \phi (t)+\phi ''(t)$$
has a simple solution:
$$\phi(t) = \frac{\mu \tanh \left(\frac{\mu \left(t-t_0\right)}{\sqrt{2}}\right)}{\...
- 1,478
5
votes
2
answers
144
views
How can I output all zero-point data by graphing?
How can I output all zero-point data by graphing?
...
- 177
3
votes
3
answers
101
views
How to test a recursive formula using Mathematica?
Through taking the derivative of $\binom{n}{k}$ w.r.t $n$ repeatedly, I found the recursive formula:
$$\frac{\partial^a}{\partial n^a}\binom{n}{k}=\sum_{j=0}^{a-1}\binom{a-1}{i}\frac{\partial^j}{\...
- 435
7
votes
4
answers
292
views
DSolve returning "Indeterminate"
Bug introduced in 13.0 or earlier. Fixed in 13.1
I am trying to solve the following differential equation with Mathematica:
$$
z\,(1-z)\,\psi''(z)-2 z\,\psi'(z)-\frac{3}{4}\frac{\psi(z)}{z+a^2(1+z)}=0\...
- 477
7
votes
4
answers
326
views
Finding the roots of Abs[Hypergeometric1F1[1/4 (3 - (2 x)/I), 1.5, I]]=0
I want to solve for the first 100 or more zeros of Abs[Hypergeometric1F1[1/4 (3 - (2 x)/I), 1.5, I]]=0;
The following code is my simple attempt：
...
- 177
-1
votes
1
answer
68
views
spherical bessel function derivative
i want to evaluate differentiation of spherical Bessel function at r = 0 but i am not able to get a value for it. Any kind of help is appreciated
...
3
votes
1
answer
132
views
I want to find differential of spherical besssel function at r=0 [closed]
I want to find an differential of a spherical Bessel function at r=0.
This is my reduced radial wave wavefunction.
u(r) = c*r*SphericalBesselJ[0, (b*r)/L]
c,b are ...
2
votes
2
answers
221
views
Integrating and plotting solutions of a nonlinear dynamical system
So I have an ODE
$k^2\; u'(\theta)^2 = -\frac{1}{3}u^3 + \frac{\omega}{k}u^2-2B\;u+2A$
where $B,A$ are constants.
I attempted to directly integrate in Mathematica with
...
- 151
2
votes
2
answers
202
views
Integration by parts for deriving gamma function
Well, there is an integral that has quite a lot to do with $n!$, and that is the following :
$f(n)$ = Integrate[x^(n - 1)/E^x, {x, 0, Infinity}] =$\int_0^{\infty } \...
- 649
5
votes
2
answers
167
views
Best way to handle numerical integration and power series with large numbers
Due to the fact that almost everything I do in my research is analytic, I am quite unfamiliar with numeric calculus, so I was wondering if anyone could give me some advice on the most efficient way to ...
2
votes
1
answer
75
views
Radial integral involving WhittakerM and Hypergeometric0F1
I am trying to find the definite integral of $$I=\int_{0}^{\infty}dx\left(\frac{\sqrt{n^{2l-1}}M_{n,l+\frac{1}{2}}(\frac{2x}{n})}{\Gamma(2l+2)}\right)x\left(\frac{(2r)^{l_f+1} {}_0F_1(2l_f+2;-2r)}{\...
- 23
1
vote
1
answer
78
views
Hypergeometric function plot [closed]
I can't plot this??!! if you can check this thank you.
...
- 391
2
votes
2
answers
130
views
Simplifying a Sum to GammaRegularized
It doesn't seem Mathematica will automatically simplify
Sum[Exp[-x] x^m / m!, {m,0,5}]
to
GammaRegularized[6,x]
Is there a way ...
4
votes
1
answer
101
views
How to calculate the sum of the series of Hermite polynomial?
I want to calculate the infinite sum of Hermite polynomials, which works fine with older versions of Mathematica, but with version 13 it doesn't.
The infinite sum is:
...
- 43
0
votes
3
answers
297
views
Solving elliptic integrals in Mathematica
I have an integral
$$\int_{a2}^{a1}\frac{dx}{\sqrt{(a1 -x)(a2 - x)(a3 - x)}}$$
And I'm trying to integrate it with
...
- 151
1
vote
1
answer
90
views
Plotting the Jacobi Elliptic functions
I have a function
$$f(x) = a_2 + (a_3 - a_2)\text{cn}^2\left(\frac{\sqrt{a_3 - a_1}}{\sqrt2};m \right)$$
Where $m$ is the modulus, given by $m = (a_3-a_2)/(a_3-a_1)$.
How can I plot this function in ...
- 21
0
votes
0
answers
111
views
How to Solve a set of equations
I would like to solve the following two equations. For that I tried Findroot and solve function
...
- 19
1
vote
1
answer
148
views
Hypergeometric function limit to Cosh[] or Sin[] [closed]
I have this hypergeometric function I woluld like to see in which condition goes to Cosh[] o Sin[]
Sqrt[1/a] x Hypergeometric2F1[1/2, 1/b, 1 + 1/b, -(f x^b)/a]
...
- 391
-2
votes
1
answer
89
views
How to find the Mathematica command for the function $a_k?$ [closed]
I am trying to find the $n$-th derivative of $\csc(m\pi)$, so I took few cases:
for simplicity let $x=\cot(m\pi)$ and $y=\csc(m\pi)$,
$$\frac{d^0}{dm^0}\csc(m\pi)=\pi^0(\color{red}{1}x^0y^1)$$
$$\frac{...
- 435
-1
votes
1
answer
52
views
what is the Mathematica command for the Euler numbers $E_k?$ [closed]
We know that the Euler numbers $(E_r)$ has many integral and series representations but I am wondering if there is a simpler Mathematica command.
- 435
2
votes
1
answer
76
views
How to calculate an `InverseMellinTransform` up to its definition in Mathematica?
I have a question about how to calculate Inverse Mellin Transformation's in Mathematica, one way or the other.
Look at these findings.
The following integral results in Gamma[s].
...
- 9,357
1
vote
1
answer
81
views
I want to integrate spherical bessel function but it is not coverging
L = (1/.197)10;
p = Table[i, {i, 1, 50}];
Roots of spherical bessel function
...
0
votes
1
answer
79
views
InverseFunction fails at some parameter
Currently, I am faced with the difficulty of evaluating the inverse of the function below for certain negative decimal values of the parameter q. Specifically for ...
- 171
4
votes
2
answers
186
views
Plotting functions defined via NIntegrate. Too slow
I have the following two functions xs[u,v] and ys[u,v] defined through numerical integration of $\alpha$ and $\beta$ and $\theta$...
- 555