Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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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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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  • 113
1 vote
2 answers
92 views

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

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2 votes
1 answer
142 views

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

...
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  • 31
0 votes
1 answer
68 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
106 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
81 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
61 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
230 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
150 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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5 votes
2 answers
139 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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  • 123
3 votes
3 answers
98 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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5 votes
3 answers
269 views

DSolve returning "Indeterminate"

Bug introduced in 13.0 or earlier and persisting through 13.0.1 or later I am trying to solve the following differential equation with Mathematica: $$ z\,(1-z)\,\psi''(z)-2 z\,\psi'(z)-\frac{3}{4}\...
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  • 457
6 votes
3 answers
244 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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  • 123
-1 votes
1 answer
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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
131 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
211 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
194 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
162 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
70 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
76 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
124 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
90 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
2 answers
240 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
83 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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2 votes
2 answers
88 views

Why does CapForm behave strange with different options? [closed]

This CapForm works well: Graphics[{CapForm["Square"], Thickness[.1], Line[{{-1, -1}, {1, 1}}]}, PlotRange -> 1.5, PlotLabel -> "Square"] ...
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  • 1,750
0 votes
0 answers
110 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
147 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
88 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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0 votes
1 answer
47 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
77 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
185 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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4 votes
2 answers
335 views

How do I get other representations of the Gamma function?

In my studymaterial I see different definitions of the Gamma function In Mathematica it is for example: ...
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1 vote
1 answer
52 views

Does anyone know how to fix the issue not having the hydrogen atoms accounted for when converting to a .mol file?

Hi I am using moleculeRecognize function to upload structures from images and I am attempting to convert the structures in .mol files. However every time I convert to .mol files the hydrogen atoms in ...
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3 votes
0 answers
61 views

Can I solve a differential equation with the (ir)regular Coulomb wavefunctions?

When I evaluate DSolve[w''[ρ]] + (1 - (2 η)/ρ) w[ρ] == 0, w[ρ], ρ] Mathematica returns me a solution in terms of the Hypergeometric1F1 and HypergeometricU ...
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  • 31
2 votes
1 answer
144 views

Evaluation of a summation involving hypergeometric functions

I need help in evaluating the following tricky summation mainly involving a product of two Kummer's confluent hypergeometric function, ${}_1 F_1(a;b;z)$. Is there some identity of ${}_1 F_1(a;b;z)$ ...
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5 votes
1 answer
167 views

How to get the expression of a special function?

How to get the expression of a special function? Like gamma function: Gamma[z]=Integrate[t^(z-1)/e^t, {t, 0, Infinity}]
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1 vote
0 answers
46 views

Calculate special function by conditional FunctionExpand

I want to calculate an expression with Whittaker's W function $$f(z)=\sum _{v,k=0}^{\infty } z^{\frac{k+v}{2}} W\left(\frac{k-v}{2},- \frac{k+v+1}{2},z\right),\tag{1}$$ however for some input ...
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0 votes
2 answers
147 views

Convergence of a complete elliptic integral

How to prove that the following integral assuming $k^2 < 1$ $$\int_{0}^{1} \frac{d x}{\sqrt{1-x^2}\sqrt{1-k^2 x^2}}$$ is convergent? ...
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  • 1,365
2 votes
2 answers
148 views

Why is the parametric plot failing?

I tried running the following code but it doesnt seem to be working, it seems to be stuck on running. I think it's got something to do with the way I am defining functions with a variable integration ...
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3 votes
1 answer
79 views

Inverse Laplace Transform of function containing ArcTanh

I'm after the numerical inverse Laplace transform of a function, so I type f[t_?NumericQ] := InverseLaplaceTransform[1/(1 + s + ArcTanh[1/(s - 1)]), s, t]; This ...
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0 votes
0 answers
59 views

My notebook backup function works fine. I am just asking for your critical review before submitting to FunctionRepository

Why the name and what does it do exactly? Youre probably asking yourselves why I named it NotebookBackupStack. All that is explained in ...
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  • 1,787
2 votes
1 answer
86 views

Mathematica code for q-Stirling numbers

In the paper [A new $q$-Analog of Stirling Numbers],(https://hal.archives-ouvertes.fr/hal-01372920/document)[PDF] J. Cigler defined $q$-Stirling numbers of the second kind as the following: He ...
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  • 121
1 vote
1 answer
42 views

Reduce relations with Fibonacci function never terminates,

i have a reduce operation which includes Fibonacci function. but the problem is it never terminates, i tried some modifications, for example removing f1 in my relation. sometimes it works and at least ...
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  • 351
3 votes
2 answers
209 views

Numerical values of Fox H function

I am playing around with the new function from 12.3, FoxH and found the result is not stable. It is interesting that the results are pretty different with the introduction of higher accuracy, and the ...
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0 votes
1 answer
153 views

Numerical integration of a function with Dirac delta

I have a little question about the numerical integration of a function that includes a Dirac delta. I have the following function: ...
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3 votes
0 answers
144 views

Using new FoxH function to solve a trinomial equation. Possible issue?

There is a possible issue on solving the following trinomial equation $$x^\alpha+x=y$$ for real $x,y,\alpha$. From Belkic [1], it is known that for $\alpha>1$ a solution is found using the ...
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2 votes
2 answers
173 views

Can the CopulaDistributions be fitted?

This is an immediate follow-up to DoAny--which I did consider editing, clarifying and correcting in some respects. But perhaps here I can put the immediate question at hand more directly (the emphasis ...
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