Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
820
questions
1
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1answer
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}\...
1
vote
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
...
0
votes
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 ...
0
votes
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)
...
0
votes
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:
$$
f(a, b, c) = \int_{0}^{+\pi} \...
3
votes
2answers
121 views
Plotting the inverse function of a complicated function
So I have a function
F[x_] = Assuming[{Element[x, Reals], -1 < x < 1},
Integrate[1/Sqrt[(x^2 - 1)^2 + alpha*x], x]]
I'm now interested in the ...
0
votes
0answers
50 views
How's analytic continuation done in Mathematica?
I have the following expression:
$$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) \...
0
votes
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 ...
0
votes
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 ...
1
vote
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
votes
1answer
110 views
Appell series F3 on Mathematica
I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as
$$F_3(a_1,a_2,b_1,b_2;c;x,y)=\sum_{m,n=0}^{\infty}\frac{(a_1)_m(a_2)_n(b_1)_m(b_2)_n}{(c)_{m+n}m!\,n!}x^...
1
vote
0answers
24 views
Computing Hypergeometric Funtion of Matrix Argument [duplicate]
I'm new to Mathematica and unsure of how to compute functions or set up definitions.
I'd like to do some computations with the $_1F_1$ hypergeometric function of matrix argument as in the Koev and ...
0
votes
1answer
76 views
How to get solutions for recursive relations using RSolve?
I have
$$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') - ...
0
votes
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
votes
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 ∞?
1
vote
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]
...
0
votes
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
votes
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
votes
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 ...
0
votes
0answers
25 views
Expanding the complex conjugate of a Mathieu function as a series
In my calculations, I found this equation:
...
0
votes
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}] , ...
3
votes
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 "...
0
votes
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]/...
7
votes
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
votes
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.
...
0
votes
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?
1
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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}\...
0
votes
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 ...
0
votes
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
vote
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
...
0
votes
2answers
110 views
2
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
1answer
69 views
Hypergeometric differential equation with integer parameters?
Naively, the hypergeometric differential equation has two independent solutions as follows:
...
