Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
832
questions
1
vote
1answer
55 views
Polar plotting Hankel Function with a lot of terms
I am trying to plot a normalized polar plot for the following function with different values of $a$
$$\left\lvert \sum_{n=1}^\infty i^n (2n+1) \frac {P_n^1(cos(\theta))}{\sqrt{\frac{\pi k a}{2}}[-H_{...
0
votes
0answers
25 views
Correctly Evaluating the existence of Large Solutions
I'm trying to correctly evaluate
$$D(S\cap[a,b])=\lim_{n\to\infty}\frac{\left|S\cap{F_n\cap[a,b]}\right|}{\left|F_n\cap[a,b]\right|}$$
where $D$ is the density of $S\cap[a,b]$ (in $A\cap[a,b]$), $[...
0
votes
0answers
77 views
Complicated Integral output with Unfamiliar Regularized Hypergeometric Function
I need the solution for following integral and it has output in MATHEMATICA as:
...
3
votes
1answer
129 views
Mathematica command to convert $\pi^{2n}$ to $\zeta(2n)$
Is there a command on Mathematica that helps me to get the answer of some harmonic series in terms on $\zeta(2n)$ instead of $\pi^{2n}$? Let me give you an example:
The command :
...
0
votes
1answer
60 views
Domain specifications for InverseFunction
I have difficulty implementing on how to specify the domain that I want for ry which is the inverse function of rho. The necessary condition is that, $\textbf{ry}$ must remain $\textbf{positive}$ for ...
0
votes
0answers
53 views
Inactive integral
I am trying to solve inactive integral but output is coming same as input.
Any idea about how to solve inactive integral which is a function of 'r'. I am using version 12. The integration contains ...
3
votes
1answer
132 views
What does this superscript on HypergeometricPFQ mean?
I was messing around with some integrals and I got as output the following:
What does the superscript on the last term in that expression mean? I looked at the documentation for HypergeometricPFQ, ...
1
vote
1answer
54 views
Unit Step Function
I will like to plot the unit step function of common function
, For example f(t)=H(t-4)t or f(t)=H(t-4)t^2
I have search through the Help Menu but cant really get a lead to the plotting. Thank you
6
votes
1answer
207 views
The differences amongst f[x_], f[x__], and f[x___] [closed]
This is probably very elementary but I have not used the following and I cannot find anything online, and I was unable to find something here.
Can someone explain to me the differences amongst
<...
0
votes
0answers
37 views
How to solve a system of five second order differential equations with boundary conditions?
I want to solve system of five differential equations of second order with their respective boundary conditions. So, I create a function that depends on their solutions. Such as
...
1
vote
1answer
77 views
About some hypergeometrical formulas for roots of trinomial and quadrinomial
Please correct my interpretation of formulas by Pietro Majer from post.
For equation $a=x+x^p$
one root is $x=a\sum\limits_{k=0}^{\infty}\frac{(-a)^{(p-1)k}}{(p-1)k+1} {{pk}\choose{k}}$
Wolfram ...
0
votes
1answer
52 views
Asymptotic solution of a second-order ODE containing InverseFunction
Essentially, I have a second-order differential equation given by ode below. In order to solve it, I need to obtain an asymptotic solution where $g(x)$ must vanish at infinity which will be used after ...
0
votes
1answer
40 views
Simplifying sums and showing equality - limitations?
Is it possible to verify the following lhs,rhs involving the sums
are equal, with Mathematica?
I can verify it for individual values of $d$ variable:
...
1
vote
1answer
63 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
88 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
...
1
vote
2answers
91 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
46 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)
...
3
votes
2answers
125 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
52 views
How's analytic continuation done in Mathematica?
I have the following expression:
$$I_1=\frac{\sqrt{a} c^{1-n} \Gamma (n-1) \, _2F_1\left(1,n-1;\frac{1}{2};\frac{b^2}{a c}\right)-\sqrt{\pi } b c^{\frac{1}{2}-n} \Gamma
\left(n-\frac{1}{2}\right) \...
0
votes
1answer
88 views
Performing a contour integration in Mathematica for a contour starting at $1$ and ending at $-\infty$ while avoiding the origin?
I would like to compute the following integral $I(k)$ in Mathematica to check if the result equals something I 'feel' is correct but I have no experience with contour integrals in Mathematica.
I want ...
0
votes
1answer
70 views
Trying to compute erfcx(x)? [duplicate]
The function erfcx(x) = exp(x^2)erfc(x) is sometimes provided in numerical packages to avoid numerical underflow for large values of ...
1
vote
2answers
62 views
Explicit series notation for hypergeometric functions
Is there an automated way to express hypergeometric functions in series form using gamma functions, factorials, double factorials or rising factorials? For example using the formula (on the ...
4
votes
1answer
118 views
Appell series F3 on Mathematica
I recently encountered the Appell series F3, defined on Wikipedia for $|x|<1$, $|y|<1$ as
$$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
78 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
73 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
37 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
58 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
59 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
29 views
Expanding the complex conjugate of a Mathieu function as a series
In my calculations, I found this equation:
...
0
votes
1answer
117 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
123 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}$$
I computed first few terms and used mathematica "...
0
votes
1answer
56 views
Strange result of MatrixFunction
Let us consider the sum of the matrix series
m = 2; n = 3; Sum[MatrixPower[{{1, 2}, {3, 4}}, m + k*n]/(m + n*k)!, {k, 0, Infinity}]
{{-(32 E^(-(5/4) - Sqrt[33]/...
8
votes
1answer
234 views
How could this asymptotic expansion be obtained?
I must precise that I am a very limited user of Mathematica (I can only run it from time when going at university).
Working this problem, I found that
$$\sigma_n=(1)^n\frac{\pi}{2} \big( j_{0,n+1} \,...
2
votes
1answer
53 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
48 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
96 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
60 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
94 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
59 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
564 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
110 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
96 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
101 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
403 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
42 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 ...
