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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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.
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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
...
- 11
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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[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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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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4
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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:
$$\phi(t) = \frac{\mu \tanh \left(\frac{\mu \left(t-t_0\right)}{\sqrt{2}}\right)}{\...
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How can I output all zero-point data by graphing?
How can I output all zero-point data by graphing?
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How to test a recursive formula using Mathematica?
Through taking the derivative of $\binom{n}{k}$ w.r.t $n$ repeatedly, I found the recursive formula:
$$\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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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:
$$
z\,(1-z)\,\psi''(z)-2 z\,\psi'(z)-\frac{3}{4}\...
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Finding the roots of Abs[Hypergeometric1F1[1/4 (3 - (2 x)/I), 1.5, I]]=0
I want to solve for the first 100 or more zeros of Abs[Hypergeometric1F1[1/4 (3 - (2 x)/I), 1.5, I]]=0;
The following code is my simple attempt：
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spherical bessel function derivative
i want to evaluate differentiation of spherical Bessel function at r = 0 but i am not able to get a value for it. Any kind of help is appreciated
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I want to find differential of spherical besssel function at r=0 [closed]
I want to find an differential of a spherical Bessel function at r=0.
This is my reduced radial wave wavefunction.
u(r) = c*r*SphericalBesselJ[0, (b*r)/L]
c,b are ...
2
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Integrating and plotting solutions of a nonlinear dynamical system
So I have an ODE
$k^2\; u'(\theta)^2 = -\frac{1}{3}u^3 + \frac{\omega}{k}u^2-2B\;u+2A$
where $B,A$ are constants.
I attempted to directly integrate in Mathematica with
...
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Integration by parts for deriving gamma function
Well, there is an integral that has quite a lot to do with $n!$, and that is the following :
$f(n)$ = Integrate[x^(n - 1)/E^x, {x, 0, Infinity}] =$\int_0^{\infty } \...
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Best way to handle numerical integration and power series with large numbers
Due to the fact that almost everything I do in my research is analytic, I am quite unfamiliar with numeric calculus, so I was wondering if anyone could give me some advice on the most efficient way to ...
2
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Radial integral involving WhittakerM and Hypergeometric0F1
I am trying to find the definite integral of $$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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Hypergeometric function plot [closed]
I can't plot this??!! if you can check this thank you.
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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
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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:
...
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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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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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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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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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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].
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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
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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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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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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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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 ...
3
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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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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)$ ...
5
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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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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}$$
however for some input ...
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Convergence of a complete elliptic integral
How to prove that the following integral
assuming $k^2 < 1$ $$\int_{0}^{1} \frac{d x}{\sqrt{1-x^2}\sqrt{1-k^2 x^2}}$$ is convergent?
...
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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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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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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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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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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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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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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:
...
3
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0
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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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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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