Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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 ...
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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 \\ -\...
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  • 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 ...
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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 ...
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-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...
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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 ...
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  • 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{...
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  • 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, ...
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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 ...
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  • 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 ...
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-3 votes
1 answer
85 views

how to get the best Best Fit Parameters of a special-function nonlinear fitting and how to find a good starting value? [closed]

...
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  • 33
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 ...
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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 ...
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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 ...
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  • 269
0 votes
1 answer
65 views

Plotting a long and complex airy function

I try to plot an Airy function ...
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  • 269
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 ...
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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?) ...
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  • 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{...
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  • 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 ...
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  • 135
0 votes
2 answers
109 views

Why can't the two variables' equations be solved?

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  • 33
2 votes
1 answer
159 views

Why can't the data be fitted? [closed]

...
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  • 33
0 votes
1 answer
72 views

The inverse Laplace transform alters parameter constraints

I have this Laplace transform: $$\left( w \frac{L}{L+s}+(1-w) \frac{Q}{Q+s}\right)^n \ for \ L>0, Q>0,0<w<1.\ (1)$$ ...
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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. ...
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  • 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 ...
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  • 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 <...
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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?!! ...
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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)}{\...
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  • 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? ...
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  • 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}{\...
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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\...
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  • 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: ...
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  • 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 ...
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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 ...
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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 ...
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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 } \...
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  • 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 ...
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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)}{\...
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1 vote
1 answer
78 views

Hypergeometric function plot [closed]

I can't plot this??!! if you can check this thank you. ...
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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 ...
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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: ...
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  • 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 ...
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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 ...
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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 ...
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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] ...
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-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{...
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-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.
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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]. ...
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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 ...
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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 ...
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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 $\theta$...
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