Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1,341
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How to convert a Whitaker function to a Coulomb function?
$$
\frac{d^2F}{dx^2}+\left[ 1-\frac{2n}{x}-\frac{l\left( l+1 \right)}{x^2} \right] F=0
$$
The solution of the above equation is the Coulomb function.
$$
\frac{d^2w}{dz^2}+\left[ -\frac{1}{4}+\frac{k}{...
- 319
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3
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Evaluate using Mathematica or otherwise $\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx$
Denote $T_n(x)$ as Chebyshev polynomial of the first kind (see here). Then I need to evaluate for $n$ a odd natural number $$\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx $$
I am requesting a code with ...
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0
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0
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On a formula in Wolfram|Alpha [migrated]
I need to understand a formula for the Hypergeometric function $_3F_2(a,b,c;d,e;1)$
I searched and got the following from Wolfram alpha ( see:- second formula in https://functions.wolfram.com/...
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Simplify or FullSimplify doesn't work on Gamma function [migrated]
I have this expression:
Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I]
which returns real values for all $n$. However, neither Simplify nor FullSimplify can ...
user91875
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2
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4
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Evaluate using Mathematica or otherwise the value of $f'(0)$
If we have$$f(a)=_3F_2\left(\frac{1}{2},\frac{a+1}{2},\frac{a+2}{2};
\frac{a+3}{2},\frac{a+3}{2};1\right)$$
where $a\geq 0$ and $_3F_2$ is the Hypergeometric function with unit argument.
Evaluate ...
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$ using Mathematica
Consider the integral $$I(a)=\int_{0}^{\pi/2} \cos^a (x)\sin(ax) dx$$
where $a\geq 0$
I am trying to evaluate the above integral using Mathematica using the following code :
...
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0
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Why does this finite integral blow up in Mathematica?
I am having some trouble understanding the output of a particular integral. I think I am misunderstanding something about how Mathematica is used, or about how the Integrate function works, and I ...
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Evaluate $\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$ using Mathematica [closed]
Consider the integral $$I(a,b)=\int_{0}^{\pi/2} \cos^a (x)\sin(bx) dx$$
where $a\geq 0$ and $b\geq 0$
I am requesting a code in Wolfram Mathematica so as to write the non negativity restrictions of $...
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How to take the derivative of a function in `Im[]`?
I tried to take the derivative of a long expression containing $\text{Re} \text{Li}_3(e^{2\pi ix})$, the trilogarithm while $x$ is real. Since the real part function is linear ,it should be like this
$...
- 153
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51
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Solving differential equation using substituion [duplicate]
I've been trying to solve an differential equation using substitution.
I have the equation
...
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How to use Slater Type Orbitals as a basis functions in matrix method correctly?
This question is a continuation of my previous series of questions about basis functions.
I would like to find the minimum energy of Coulomb potential motion using matrix method.
$H=-\frac{1}{2}\Delta-...
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Why two the same integrals give different values?
Why two the same integrals give different values?
...
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Use two derivative rules and iterate several times to get the simplest expression of higher derivative
I have 2 rules of recursive relation of the derivative, I want to use it several times get the higher derivative on [\Theta] of ...
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Calculating Length of an Elliptical Arc [closed]
I am trying to calculate the arclength of a section of an ellipse, given the correspoding parametrization angle $\phi$. While doing so, I am running into problems with elliptic integrals. The ...
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Proving orthogonality of Hermite polynomials
The orthogonality relation for Hermite polynomials is:
$$\int\limits_{-\infty}^\infty e^{-x^2} H_n(x) H_m(x)\ dx = 2^n n! \sqrt{\pi} \delta_{nm}$$
where $\{ n, m \} \in \mathbb{Z}_{\geq 0}$, $H_n(x)$ ...
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Bessel Function Numerical Integration: Invalid Limit of Integration
I am new to the Mathematica and trying to compute numerically value that involves Bessel function and its derivate.
Here ae+- (in code it's Xplus and ...
2
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2
answers
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How can I calculate complicated infinite sums with FindIntegerNullVector (or related methods)?
I've recently been very interested in the wonderfully complex world of Euler sums, i.e. (convergent) infinite sums that, roughly speaking, consist of some rational polynomial combination of ...
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Implementation of elliptic gamma function
I need to perform series expansion in p, q (to order order) of products/ratios of the so-called elliptic gamma functions, ...
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SeriesCoefficient stops working on EllipticTheta in v13.2
In v12, the following SeriesCoefficient computation gives the expected result,
...
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Buggy behavior of EllipticE[0,k] with arbitrary precision input
I am having an issue where if I provide the EllipticE function with a first argument of zero and a second argument with a precision lower than that of machine precision, the kernel crashes.
For ...
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Polar graph of the Riemann zeta function
I would like to plot a complex graph of the Riemann zeta function on the Argand diagram $ς(s)$, where $s = \frac{1}{2} + i t $, and the value of $t$ is varied to get a graph in the polar form.
Can ...
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Using DSolve to solve the Hypergeometric ODE in terms of Legendre Polynomials
I have been curious about if is it possible to "Force" DSolve to get solutions of differential equations in terms of another related functions.
Eg. The ...
- 36
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0
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66
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Siegel modular forms in Mathematica
Is there a convenient way to work with Siegel modular forms in Mathematica? I am interested in doing analytic computations using the $\chi_{10}(\Omega)$ Siegel modular form, where $\Omega$ is the $2\...
- 41
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votes
2
answers
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Stop Mathematica from expanding Dirichlet beta and eta functions
It seems that Mathematica treats DirichletBeta[s] and DirichletEta[s] merely as symbols for ...
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How to write expression of the motion functions in Coulomb field so that it can be used as a basis in the matrix method for finding eigenvalues?
I would like to find the energy eigenvalues of the corresponding Hamiltonian (H = -1/2Laplacian -b/r * Exp[-a*r] , a, b - numbers) in the matrix way. For this, it is necessary to have a complete set ...
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Why does an error occur in one case, but not in the other, and how to fix it?
Why do errors appear for this Kx[11, 15] when this Kx[11, 14] is considered without error? How to fix it?
...
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How do I speed up this numerical integration with Bessel functions? (Tried changing Working Precision and Integration Strategies)
I need to compute the function P[b] given as follows:
...
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How can we change the EoS (to PR using the Flags function) for REFPROP used within Mathematica?
I would like to change the EoS used by REFPROP to the PR EoS for several calculations.
It's stated that the Flags function must be used.
"
Peng-Robinson or PR 0: Turn off the Peng-Robinson ...
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Finite result under "Integrate" while infinite under "NIntegrate" of a complicated integral with "NSum"
I have been trying to compute the following complicated integral along with summation, details codes/function of which is given below:
...
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1
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44
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Solving an integral over gaussian function in spherical coordinates (or an intermediate function involving BesselI)
I am trying to solve an integral from the following function (int) and set of assumptions (as):
...
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How to get Mathematica to integrate over multivariate constrained ranges [closed]
I am trying to get an answer to:
Integrate[Exp[k*Cos[\[Theta] ]], {\[Theta], 0, 2*Pi}]
which I happen to know is proportion to $\mathbb{I}_0(k)$, which is the ...
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2
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Prevent the use of a specific function in an output
I am trying to asymptotically integrate a function with the code
AsymptoticIntegrate[Exp[I k x + I/5 k^5], {k, 0, Infinity}, x -> Infinity]
and the output is ...
- 123
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66
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Non-sensible hypergeometric function being generated
I was evaluating a sum, and Mathematica gave the following result:
$${}_3 F_2\left( -n, 1+\frac{a}{2}, -\frac{a}{2};1,-b;1\right)$$
where ${}_3 F_2$ is a generalised hypergeometric function, and the ...
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A fraught with incorrect results ODE
I mean the following ODE
$$y''(x)+y'(x)=\exp (-2 x) y(x)^3.$$
Trying to solve it in version 13.1 on Windows 10 by
...
- 21k
3
votes
2
answers
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Fourier transform of hypergeometric function only returns where FT vanishes
I have a probability distribution of a random variable $X$ with support between, roughly, $[-3, 3]$. I do not know the probability distribution, but I can simulate instances from it approximately, and ...
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Convergence of Integral of ParabolicD (Exp[-x^2] HermiteH[n,x]
This is a question about non-convergence of an integral that I am fairly certain has a symbolic and convergent solution.
...
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Evaluating an integral symbolically seems impossible
I have an integral of general form
$$ i_n(z)=\dfrac{z^2}{K_2(z)}\int_1^{\infty} \dfrac{\left(x^2-1\right)^{n+\frac{3}{2}}}{x^{2n+1}} \exp(-zx) dx $$
$K_2(z)$ is the bessel function and $0.01 \...
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Consecutive neighbours of Hypergeometric ${}_1 F_1(a,b,z)$
I am working with the hypergeometric function ${}_1 F_1(a,b;z)$, where $a\in \mathbb{N^+}$, $b=2$, and $z\in \mathbb{C}$. The Wolfram function repository lists the following relation
$$\begin{equation}...
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Make expression at certain points regular
Is there a way to find representations that are regular at certain specified points? For example the following both equations are identical, but the first is not regular at point $\{i=2,v=1,k=0\}$, ...
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3
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General limit of a function with Pochhammer for any natural number
The function
f[z_]:=2^(i-k/2-v/2) (k+v)! Gamma[1+i,z] Pochhammer[-v, i]/(i! k! Pochhammer[-k-v,i])
for $i,v,k \in \mathbb{N}_0$ and $z\in \mathbb{R}^-$ is for some ...
- 690
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Dedekind Zeta Function in Mathematica (at least for quadratic number field)
Does there exist some way to use Mathematica to compute the Dedekind Zeta function for an arbitrary algebraic number field? Or does there exist some package to do this?
I am actually only interested ...
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45
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How to double sum or triple sum up over the list of functions?
I'm dealing with a summation problem by now using a list of function $F(x; \theta_i)$. The function is defined as
\begin{align*}
F(x; \theta) = 1 - \Phi[\Phi^{-1}(1 - \frac{x}{2}) - \theta] + \Phi[-\...
- 121
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vote
2
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Limit of hypergeometric series and gamma function
MMA does not calculate the limit, however the limit exists as seen in the approximative plot.
Limit[HypergeometricPFQ[{-1/2}, {1/2, m/2}, x]/Gamma[m], m -> 0]
...
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How does Mathematica compute certain "bad" cases of the Meijer G function?
The Meijer G function $G^{m,n}_{p,q}\left(\begin{matrix}(a_1,\dots,a_p) \\ (b_1,\dots ,b_q)\end{matrix}\bigg|~z\right)$ is defined via a certain contour integral in the complex plane. In this ...
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Roots of expressions involving the complementary error function
I have an expression as follows
...
- 1,228
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How to numerically integrate this function near x = 0?
I'm trying to NIntegrate[] the following integration: Let's define a function
$F_\delta(x) = 1 - \Phi[\Phi^{-1}[1 - x] - \delta], \ \ x \in (0, 1)$
where $\delta > 0, \Phi(x)$ is the cdf of the ...
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