Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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59 views

Inversion of a hypergeometric function

I have been trying to invert the hypergeometric function $$\rho(r)=\frac{2b}{1-q}\sqrt{1-\left(\frac br\right)^{1-q}}\,_2F_1\left(\frac{1}{2},1-\frac{1}{q-1};\frac{3}{2};1-\left(\frac br\right)^{1-q}\...
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2answers
83 views

Solving of Equation which contains Hypergeometric Function 2F1

I am trying to solve this equation where I need the solution of K in term of v ...
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2answers
81 views

What series does Mathematica use for Hypergeometric1F1?

I'm trying to get an analytical expression for Hypergeometric1F1[-a, 1/2, X] Provided a is an integer number. I tried adding ...
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0answers
45 views

What is the representation of the Harmonic Number being used by Mma in this result?

The Fourier Transform of the function F[x_] = (m/Sqrt[\[Lambda]])*Tanh[(Sqrt[x^2]*m)/Sqrt[2]] where all variables are real, and $m>0$ is given by (Mma 11.0) ...
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0answers
72 views

Integrating an expression containing the error function, perf [migrated]

I am trying to integrate a formula, but I can’t figure out how to do it. Its integrand involves a special function — erf, the error function. Here's the formula: $$ f(a, b, c) = \int_{0}^{+\pi} \...
3
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2answers
121 views

Plotting the inverse function of a complicated function

So I have a function F[x_] = Assuming[{Element[x, Reals], -1 < x < 1}, Integrate[1/Sqrt[(x^2 - 1)^2 + alpha*x], x]] I'm now interested in the ...
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0answers
50 views

How's analytic continuation done in Mathematica?

I have the following expression: $$I_1=\frac{\sqrt{a} c^{1-n} \Gamma (n-1) \, _2F_1\left(1,n-1;\frac{1}{2};\frac{b^2}{a c}\right)-\sqrt{\pi } b c^{\frac{1}{2}-n} \Gamma \left(n-\frac{1}{2}\right) \...
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1answer
87 views

Performing a contour integration in Mathematica for a contour starting at $1$ and ending at $-\infty$ while avoiding the origin?

I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica. I want ...
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1answer
67 views

Trying to compute erfcx(x)? [duplicate]

The function erfcx(x) = exp(x^2)erfc(x) is sometimes provided in numerical packages to avoid numerical underflow for large values of ...
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2answers
60 views

Explicit series notation for hypergeometric functions

Is there an automated way to express hypergeometric functions in series form using gamma functions, factorials, double factorials or rising factorials? For example using the formula (on the ...
3
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1answer
110 views

Appell series F3 on Mathematica

I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as $$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_1)_m(a_2)_n(b_1)_m(b_2)_n}{(c)_{m+n}m!\,n!}x^...
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0answers
24 views

Computing Hypergeometric Funtion of Matrix Argument [duplicate]

I'm new to Mathematica and unsure of how to compute functions or set up definitions. I'd like to do some computations with the $_1F_1$ hypergeometric function of matrix argument as in the Koev and ...
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1answer
76 views

How to get solutions for recursive relations using RSolve?

I have $$a(x,t+2) - a(x,t) = -\cos{\theta} [a(x+1,t+1) + a(x-1,t+1)]$$ Setting $t' = -t\cos(\theta)$ fetches under the continuum approximation $$2 \frac{\partial a(x,t')}{\partial t'} = a(x-1,t') - ...
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1answer
49 views

How to solve this equation by Solve?

I have an equation to be solved. But Mathematica does not work for it. I hope the solution x can be expressed as a function of a and b ...
3
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1answer
72 views

Evaluating an integral combining a Bessel function with some other functions [closed]

How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
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1answer
36 views

Limit of an integral and hypergeometric function

I want to evaluate the following integral: Integrate[Sin[θ]^(D1 - Nc - 1)/(A Cos[θ] - I ϵ)^(N1 - Nc), {θ, 0, π}, Assumptions -> A > 0 && ϵ > 0] ...
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0answers
23 views

Problem with integrating special function under assumption

i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible. In fact, if ...
2
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0answers
57 views

Computing Definite Sums of Rational Functions

I am attempting to compute a rather complicated sum, $S_n$, that in the end satisfies the relation $(S_n + T_n) = (\frac{(n+1)^2 - 1}{n+1})m^3 + O(m^2)$. I should note that $T_n$ is also unknown. ...
2
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2answers
58 views

Limit yielding wrong result for Hypergeometric2F1 but not for Hypergeometric2F1Regularized

I have to deal with an expression with some $_2F_1$ and take some limits for some values of the parameters. Let's call this parameter $m$. The issue is that I get a different result whether I take the ...
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0answers
25 views

Expanding the complex conjugate of a Mathieu function as a series

In my calculations, I found this equation: ...
0
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1answer
96 views

Mittag-Leffler function [closed]

What is the command to plot the Mittag-Leffler function in 3D by using Mathematica program? I have tried the command Plot3D[mittagLefflerE[v,t],{t,0,5}] , ...
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1answer
122 views

Exponential generating function

I need to find the $m^\text{th}$ term for the following expression: $$ \left.\frac{\partial^m}{\partial t^m}e^{a t^2}\right|_{t=0}$$ I computed first few terms and used mathematica "...
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1answer
54 views

Strange result of MatrixFunction

Let us consider the sum of the matrix series m = 2; n = 3; Sum[MatrixPower[{{1, 2}, {3, 4}}, m + k*n]/(m + n*k)!, {k, 0, Infinity}] {{-(32 E^(-(5/4) - Sqrt[33]/...
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1answer
230 views

How could this asymptotic expansion be obtained?

I must precise that I am a very limited user of Mathematica (I can only run it from time when going at university). Working this problem, I found that $$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
2
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1answer
51 views

How do I make InverseBetaRegularized function behave the same in Mathematica 11.3 compared to 11.2?

In Mathematica 11.3: InverseBetaRegularized[0.001, 4501, 500] Never finishes calculating. ...
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1answer
29 views

Plot a function defined on terms on its value on an interval

Besicovitch-Ursell family of fractal functions uses the following auxiliar definition: $\phi(x) = 2x$ on $[0,1/2]$. $\phi(-x) = \phi(x+1)$ otherwise. Is there a way to plot this in mathematica?
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1answer
47 views

Closed form solutions to functions involving Erfc

Is there a better way to get a closed form solution in terms of sigma? I've already attempted a significant amount of simplification up to this point and am unsure ...
3
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1answer
91 views

How to convert solution from ParabolicCylinderD to Bessel functions?

I am trying to verify my hand solution to an ODE. The solution I got is in terms of Bessel functions. Maple gives same solution. Mathematica gives the solution in terms of ParabolicCylinderD which I ...
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0answers
58 views

Flaky pattern-matching for Mittag-Leffler sums?

This sum correctly gives the Mittag-Leffler function: Sum[z^k/Gamma[α*k + α], {k, 0, ∞}] MittagLefflerE[α, α, z] Simply factoring the argument of ...
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5answers
92 views

Sum Over Solutions to an Equation

Two Related Questions Is there any general built-in functionality for computing a sum over solutions to an equation? This is common in number theory. For example, computing sums of the following form....
1
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1answer
56 views

Getting values of EllipticK with arguments that are very near 1

I need to evaluate EllipticK[m] very close to 1. However, when I get too close to 1 the function defaults to the exact solution for 1 , which is ...
4
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2answers
527 views

How to prevent Mathematica rounding extremely small numbers to zero?

I have a function that, while the maths itself is unimportant, at certain values it results in a very large number multiplying a very small number. E.g. 10^450000 * 10^-449998. As you can see, this ...
-1
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1answer
103 views

Plot fractional trigonometric functions with the Mittag-Leffler function

Can anyone help please? Im trying to plot the solution $X$ of the system as in the paper attached - about fractional calculus which is $X= [E_{\nu}(2t^{\nu})][2 \cos_{\nu}(3t^{\nu})+4 \sin_{\nu}(3t^{...
2
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1answer
91 views

Plotting an osculating circle at the leading edge of a developing Cornu spiral

I need to plot an interactive Cornu function like so: ...
0
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1answer
92 views

Plotting with the Mittag-Leffler function [closed]

I'm trying to plot the solution of fractional differential equations as shown in the photos below, The solutions are in terms of the Mittage-Leffler function, so I evaluated ...
0
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1answer
32 views

Different result with HypergeometricPFQ

I have a following sequence: Table[Sum[Sum[(1 + j + k + n)!/((1 + j + k) j! k! ), {j, 0, n}], {k, 0, n}], {n, 1, 5}] (* {16, 542, 31488, 2646024, 292224000} *) ...
3
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1answer
392 views

How to write the shifted Chebyshev polynomials (the first kind) in mathematica?

Is it possible to write the shifted Chebyshev polynomials (the first kind) in Mathematica? The formula is: $$P_n(x)=\sum_{k=0}^{\left\lfloor\tfrac{n}{2}\right\rfloor}(-1)^k 2^{n-2k-1}\frac{n}{n-k}\...
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0answers
38 views

Inverse of Poly Log function? Asymptotic behavior of Poly Log function?

I am unable to answer important questions such as what is the inverse of PolyLog[3/2,z]? I mean can you express the solution to w = PolyLog[3/2,z] (solve for z in terms of w) in terms of functions ...
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0answers
42 views

How to solve this error in numerical integration?

I am trying to integrate a function numerically and I get an error that I do not understand. The error is : NIntegrate::slwcon : Numerical integration converging too slowly; suspect one of the ...
1
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2answers
124 views

How to get details about how Mathematica did a definite Integral

Assuming[{Element[S, Reals],S>0},Integrate[Exp[-I*S*w]/(w^2 + 1)^(3/2) ,{w, 0, Infinity}]] gets ...
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2answers
110 views

Plotting a 3D piecewise function

I would like to plot the following function: ...
2
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0answers
81 views

Asymptotes of parabolic cylinder differential equations with boundaries at infinity

For context, I'm studying the paper Coulomb blockade in superconducting quantum point contacts by Averin from 1998. Specifically, I am trying to find how he obtains equation 11 from equation 10, which ...
2
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0answers
56 views

Simplifying long expressions leads to memory allocation failure

I am currently trying to manipulate a number of expressions into a particular form that will allow me to determine their poles and zeros: ...
2
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1answer
123 views

How to Plot a transcendental function

Let me try to be a bit schematic because my own expression is a bit complicated and could be not useful for future questions. x=f[y*a[x]] My problem is how to <...
3
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0answers
33 views

Trouble with simplifying trigonometric / hyper-trigonometric functions

Why (correct) expressions like Assuming[p > 0, 2 ArcTan[Sinh[p]] == Pi - 2 ArcTan[Csch[p]] // FullSimplify] Are not correctly evaluated to: True? What is the ...
2
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2answers
99 views

Numerically stable replacement for generalised incomplete gamma function [closed]

I am looking to replace the generalised incomplete gamma function (which appears in a solution to a problem I've posted about here) with a numerically stable formula involving other functions. This is ...
1
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2answers
119 views

Evaluation of a hypergeometric function

I am working with functions like f[z_] = Hypergeometric2F1[4, 4, 8, z] Here is a plot of this function over the interval $z \in [0,1]$: ...
3
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1answer
53 views

Calculation among Gamma functions

I was calculating gamma functions in Mathematica while it does not give me an agreed answer. By definition, $\Gamma[\alpha]=\int_0^\infty t^{\alpha-1}e^{-t}dt$, $\Gamma[\alpha,z]=\int_z^\infty t^{\...
4
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0answers
117 views

Another example where using FullSimplify gives different result than Simplify

I believe this question is very similar to Result of Series[expression] is different when I simplify the expression, however, due to my lack of Mathematica experience, I am reluctant to call it a bug. ...
3
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1answer
69 views

Hypergeometric differential equation with integer parameters?

Naively, the hypergeometric differential equation has two independent solutions as follows: ...