Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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_{...
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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 $A\cap[a,b]$), $[...
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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
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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 : ...
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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 ...
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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
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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
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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
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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 <...
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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
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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 ...
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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 ...
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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
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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}\...
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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
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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 ...
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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
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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 ...
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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
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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
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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
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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
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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
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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 ...
0
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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') - ...
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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
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
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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] ...
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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
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
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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 ...
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0answers
29 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
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
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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
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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
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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
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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
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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?
1
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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
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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
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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 ...
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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
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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
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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
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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
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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
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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
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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
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
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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 ...