# 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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1 vote
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### Possible bug: Wrong limit for EllipticE?

I encountered problem with computing limits of Elliptic functions E and F. For example, ...
1 vote
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### 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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### 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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### 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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### 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 ...
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,y'==1}, y[x],x] I use ...
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### Plotting a long and complex airy function

I try to plot an Airy function ...
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 ...
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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### 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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### 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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### 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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### Bugs in hypergeometric functions with negative integer lower parameters

If you were to evaluate these expressions, Mathematica returns the value shown. ...
1 vote
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### 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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### A strange behaviour with function $\,_2F_1$

I posted today a question on MSE 1 u=(5*(49*Pi^2*Zeta - 558*Zeta))/(77*Pi^2*Zeta - 930*Zeta) f[k_]:= Hypergeometric2F1[1/2, -k, 3/2, u] Computing <...
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### 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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1 vote
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### Hypergeometric function plot [closed]

I can't plot this??!! if you can check this thank you. ...
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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### 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: ...
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 ...
1 vote
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### 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 ...
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 ...
1 vote
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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### 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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### 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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### 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]. ...
1 vote
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### 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 ...
I have the following two functions xs[u,v] and ys[u,v] defined through numerical integration of $\alpha$ and $\beta$ and $\theta$...