Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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1
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3answers
54 views

When is a symbol a Symbol? Is there an easy mathematica function that tests if an object is a symbol something like SymbolQ?

Yes I know there is no built in function SymbolQ[...] (but JavaScript does) but could one be coded to work for most cases? I often rely on ...
2
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1answer
51 views

The NIntegral can not give a correct answer by mathematica

Here I want to compute this integral using the Mathematica $\int_0^\infty \int_{-\pi}^{\pi}s^{\frac{1}{2}+2} \exp (-s) \cos \left(\frac{f}{2}\right) \exp (-i k s) \exp \left(-\frac{\sqrt{8 s u}}{\cos \...
2
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1answer
47 views

Plotting Stability of damped mathieu equation

I am trying to create the stability diagram of the damped Mathieu equation using Mathematica.The Mathieu equation is $$D(y)+(a-2q \cos(2t))y=0$$ where D(y) is the ...
3
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2answers
248 views

Code that produces plot in V5 doesn't work in later versions

I have problem in plotting Integral function. I can compute/plot the graph of this integration below in Mathematica 5.0, but it is not possible to plot it in higher Mathematica versions. My code is: <...
3
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0answers
74 views

Problems with Erf for large arguments with small imaginary part [closed]

While performing some higher precision calculation, I noticed that Mathematica 12.1 killed the kernel and cleared all prior definitions. The problem was with Erf ...
1
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0answers
17 views

Is there a way to count quantity of evaluations (this preferred) or edits in a cell since last notebook save?

Abstract The goal is to use this to create an autoSave[evals_:5]:=Module[{}, save after set number of evaluations or edits have occured...]. I asked a similar ...
4
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0answers
74 views

AppellF1 unevaluated [duplicate]

I have very long symbolic expressions, which I have to evaluate numerically later on. They contain the AppellF1 function, which stays unevaluated for the specific ...
4
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1answer
115 views

Verification of a general solution to d'Alembert equation

I solved a nonlinear differential equation (d'Alembert one) by hand. Mathematica gives the same answer. But I am not able to get Mathematica to verify the solution due to branch cuts. Any one knows of ...
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0answers
45 views

Alternative to Coefficient List

I was using CoeffientList to get the probabilities of this generating function... ...
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0answers
48 views

Branch cut of generalized hypergeometric function (error in Mathematica documentation?)

I am trying to compute the discontinuity around $x=1$ (equivalently, the branch cut) of generalized Hypergeometric functions ${}_{q+1}F_q(a_1,\dots,a_{q+1};b_1,\dots,b_q;x)$. Two formulae are given in ...
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0answers
62 views

What makes ListPlot better than N?

I wanted to numerically verify the validity of the formula for the first Stieltjes constant $$\gamma_1=-\frac12\sum_{n=0}^\infty\frac1{n+1}\sum_{k=0}^n\binom{n}{k}(-1)^k\log^2(k+1)$$ ...
3
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0answers
75 views

rotation of spherical harmonics using Wigner D-matrix

A very stupid question as I am very confused: I have a surface charge density which is a function of spherical harmonics $\sigma_{l,m}=Y_{lm}$ (only the real part). Now I need to rotate the particle, ...
5
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1answer
63 views

Sine vs Sinc vs SphericalBesselJ in NIntegrate

I'm evaluating an oscillatory integral numerically, and ran into a weirdness with NIntegrate, which I've boiled down to a simple case for this question. Consider ...
0
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1answer
31 views

Problem plotting finite sum involving HurwitzZeta function

I'm having problems regarding to the following code: ...
1
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1answer
62 views

Find analytic solution for integral only defined for even integers

I would like to calculate the integral $$\int_0^{2\pi} dx \sin^6\left(\frac{x}{2}\right) F\left(\frac{4-n}{2}, \frac{4+n}{2}, \frac{1}{2}, \cos^2 \frac{x}{2} \right)^2$$ where $F$ is the ...
3
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3answers
315 views

How can I compute Erf of large numbers to more precision?

I would like to compute Erf[80/3] to enough precision to know the order of magnitude of 1 - Erf[80/3] How can I do that? I ...
3
votes
1answer
50 views

Problem with a finite sum involving HurwitzZeta Function

I'm trying to reproduce some plots from the analytical expression: $f(\xi)=\left(\frac{2}{\beta^2}-1\right)+\left(\frac{\theta\,e^{-\beta}}{2}-\frac{1}{2}\right)\xi+\sum_{n=1}^{60}\left[\frac{(-\beta)...
2
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1answer
55 views

how to plot an approximate Riemann Zeta function beyond $t=180$?

Riemann zeta function $\zeta(s)$ is related to Riemann Xi function $\Xi(z)$ via: $$s=\frac12+ iz,\qquad \Xi(z):=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\tag{1}$$ We found the following function $\...
0
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1answer
44 views

Simplify Jacobi Polynomials

How can I force Mathematica to use the identities satisfied by Jacobi polynomials $$ (1-\cdot)P_n^{(\alpha+1,\beta)} = \frac{2}{2n+\alpha+\beta+2}\left((n+\alpha+1)P_n^{(\alpha,\beta)}-(n+1)P_{n+1}^{(\...
3
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1answer
27 views

Writing generalized hypergeometric function ${_nF}_{n-1}(\cdot)$ as a function of $n$

How can we code the hypergeometric function $$ f(n,k,z)={_nF}_{n-1}{\huge(}{1-k,\overbrace{2,\dots,2}^{n-1\ \text{times}} \atop \underbrace{1,\dots,1}_{n-1\ \text{times}}};z{\huge)} $$ as a function ...
1
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0answers
38 views

Dirichlet L-function associated to Kronecker symbol

The Fourier coefficients of the genus 2 Eisenstein series on the Siegel upper half-space are given by sums over Cohen functions. The Cohen function contains as a multiplicative factor a Dirichlet L-...
1
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1answer
66 views

Integration of a complicated oscillatory function

I've tried the answers in similar posts but they don't seem to work. As per title, I need to double integrate a complicated quickly oscillatory function. I've checked and there are no poles, the ...
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1answer
65 views

Is this convergence error?

...
0
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1answer
67 views

Plotting a sum of Bessel functions

I would like to plot the Nth partial sum, given τ for different values of "n". I'm really new at this so i don't even know how to begin. I have to make a graph of ξ vs ϕ and find the minimum of n so i ...
1
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1answer
85 views

How do you find the Inverse of Elliptic Integral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer. ...
3
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2answers
360 views

Why doesn't Integrate evaluate an elliptic integral?

My code is Integrate[ 1/Sqrt[(x - a) (x - b) (x - c) (x - d)], {x, a, ∞}, Assumptions -> 0 < d < c < b < a] I know this can be ...
5
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2answers
171 views

Evaluate integrals with parameters

I'm new to Mathematica so I apologize if the answer to my question is trivial. I need to calculate the following integral $$ \int_{-\infty}^{0}\frac{dx}{\sqrt{(x-a)(x-b)(x-c)(x-d)}} $$ with $$ 0 < ...
1
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1answer
42 views

Is there any way to prioritize Gamma function?

In Mathematica, if you type Gamma[1, 0, -a] It automatically simplifies to 1 - E^a. I wonder though if there is a way to stop ...
1
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0answers
31 views

Slow convergence of Fresnel integral evaluation

I am trying to solve a problem from optics, a so-called Fresnel integral. The argument of integration is defined with a function: ...
1
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1answer
49 views

difference equation and continued fractions

I'm interested in solving the following difference equation: $x[k-1]+(k^2+k+a)/x[k]=b$, $k=1,2,\ldots$, where $a,b$ are fixed positive numbers; let's say $x[1]=c>0$. Mathematica's ...
4
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1answer
63 views

How to Force the PolyGamma[0, x] function or the HarmonicNumber[ x ] function

I'm crunching some infinite summations, and sometimes Mathematica generates results that have the PolyGamma[0, x] function (which is the Digamma) and sometimes the <...
13
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2answers
542 views

Looping through all functions defined in Mathematica

Is there a way to loop through all the Functions (Elementary and Special functions) that exist in Mathematica? I want to construct a table of some identities and maybe I can discover something ...
0
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0answers
63 views

Radial wave-function integration with mathematica

I'm new in using Mathematica. I wanted to do some integrations with the radial wave function for Hydrogen atom. The radial wave function is given by, $$R_{nl}(r)=2^{l+1} e^{-\frac{r}{a n}} \sqrt{\frac{...
1
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0answers
27 views

Help in Better contour plot

In the following piece of code, I am trying to plot a function: fun[x,y] for some set of given parameters. but on the y axis of contour plot, after $\pm 4.5$, ...
12
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5answers
324 views

Possible bug in finite sum over inverse squares $\sum\limits_{i=1}^n \frac{1}{(x (n-i)+i)^2}$

Revisiting the problem Limit of partial sums involving inverse squares I found another difficulty with Sum[] Consider this sum ...
0
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0answers
63 views

Numerical comparison of two integrals and a function :

Consider the following integral: $$S(q)=\int_{x=2}^q\sin^2\left(\frac{π\Gamma(x)}{2x}\right)dx$$ And consider the functions : $$R(q)=\frac{q}{\log(q)}$$ $$T(q)=\int_2^q\frac{1}{\log(x)}dx$$ I ...
1
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1answer
70 views

Pochhammer Simplification

Assumption:: m and n are integers and greater than zero. $Assumptions = {Element[m, Integers] && m >= 0, Element[n, Integers] && n >= 0} ...
1
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2answers
50 views

How to do series expansion for functions which have symbolic parameters?

I would like to find the series expansion of (E^(x^k/k!) Gamma[k, x])/Gamma[k] for $k$ being a positive integer, up to the order of $x^{2k+1}$. Mathematica ...
2
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2answers
84 views

How to calculate Gamma functions values for Quaternions using Mathematica?

How can we calculate the values for the Gamma function using Quaternions on Mathematica? For example: ...
3
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2answers
88 views

Two identities involving contour integrals in the presence of a branch point where the integrand explodes, and the Kummer function

I need to understand how to establish two identities. The first is $$ \int_{C} z^{-1-q}(1-z)^{-1-\lambda } dz=\frac{2 \pi \Gamma (q+\lambda +1)}{\Gamma (\lambda +1) \Gamma (q+1)}, q\geq 0, \...
1
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3answers
139 views

Solving equations involving integrals

I need to find the value of $z$ for a particular value of $D_c$ (eg. $500$), but $z$ is inside an integral, and I'm not able to use Solve since the integral is ...
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0answers
61 views

Understanding Working Precision - two example equations

I am trying to understand how WorkingPrecision works, and in general how very small/large numbers are to be handled. Let me preface I probably do not have ...
1
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0answers
41 views

Change the PolyLog branch cut in Wolfram Mathematica

The problem is to change the branch cut of PolyLog[2,z] function from the default half line z=1+x, x>0 to the new one ...
1
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2answers
152 views

Using Convolve with functions that contain error functions, and iterative convolutions

I need to produce convolutions of convolutions to study some probability distributions. My starting distribution is the Rayleigh distribution, which I convolve with itself as: ...
0
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1answer
41 views

Substitute gives me different result

I'm working with Legendre polynomials (& associated ones). When I do the following calculation: ...
0
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0answers
36 views

Gamma and Pochhammer simplification

I want to perform some simplification on Gamma function and Pochhammer symbols. For example, ...
1
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0answers
36 views

Jacobi elliptic function JacobiSN[z,1/2] should be equal to JacobiSN[z , -1/2], But MATHEMATICA shows different values, Why?

Jacobi elliptic function JacobiSN[z,1/2] should be equal to JacobiSN[z , -1/2] by their properties(NIST Handbook of Mathematical functions, SN[z,k]=SN[z,-k]). But in MATHEMATICA they are giving ...
2
votes
2answers
73 views

Why is FindRoot throwing a “singular Jacobian” error when function has small values?

Running the code: FindRoot[PolyLog[2, -E^-(10 + Sqrt[1 + x^2])] == 10^-30, {x, 10}] Gives me the error: ...
0
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1answer
64 views

Integrating FindRoot

I have a certain function F[a_] := FindRoot[f[a,x],{x,0.5}] and I want to compute NIntegrate[F[t],{t,0,1}]. However, ...
2
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0answers
52 views

Replacing an intrinsic Mathematica function by some other function

I want to express the Hypergeometric2F1 series (which is an intrinsic function of Mathematica ) wherever it appears by its definition. Where ever this function appears with whatever the arguments I ...

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