Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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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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21 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 ...
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0answers
55 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. ...
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2answers
52 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
23 views

Relation between two Fox-H function with positive and negative argument [migrated]

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.
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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: ...
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1answer
70 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
96 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
49 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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0answers
38 views

Adding a Cosine function to alter the shape of the initial cosine function

I would like to be able to add a raised cosine to a "standerd" raised cosine. With a timedelay which I could manipulate (so the shape of the standerd cosine function is altered, meaning the function ...
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1answer
229 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} \,...
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1answer
50 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
28 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
45 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 ...
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0answers
86 views

Is it possible to find the Rössler attractor using only a set of Lyapunov exponents?

Is it possible to use a set of Lyapunov exponents to determine the orbits of the Rössler system? If so, could how would I go about plotting them? EG. Known Unknown
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1answer
85 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
55 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
83 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....
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1answer
98 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^{...
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1answer
86 views

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

I need to plot an interactive Cornu function like so: ...
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1answer
88 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 ...
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1answer
28 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} *) ...
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1answer
377 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
37 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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40 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 ...
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2answers
121 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
100 views

Plotting a 3D piecewise function

I would like to plot the following function: ...
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0answers
71 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 ...
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0answers
49 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: ...
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1answer
118 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
32 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 ...
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2answers
94 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 ...
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2answers
111 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
48 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^{\...
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0answers
114 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
66 views

Hypergeometric differential equation with integer parameters?

Naively, the hypergeometric differential equation has two independent solutions as follows: ...
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1answer
102 views

Power series representation of MeijerG function, $G_{m,n}^{p,q}(x)$ [closed]

I've been experimenting with Mathematica and I keep getting the following (where $G_{m,n}^{p,q}(x)$ is the MeijerG function): Is it possible to express those $f_{i}(x)$ as a power series in $x$? ...
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29 views

Generating DifferenceRoot Equation

I like to find a difference root equation I use the following methods use for mathematica but I have not get any result Clear[p] ...
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0answers
23 views

some errors the DiferenceRootReduce

I try to calculate DifferenceRootReduce[( Sqrt[\[Pi]] Gamma[3/2 + k] HypergeometricPFQ[{3/2 + k, -n}, {2 + k}, p])/ Gamma[2 + k], k] but any result ...
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1answer
125 views

How can I use the Stirling's approximation to approximate a factorial?

I'd like to exploit Stirling's approximation during the symbolic manipulation of an expression. Essentially, I want replace Factorial[n] with ...
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2answers
83 views

Derivative of integrated noise Gaussian likelihood

In a Bayesian problem with Gaussian likelihood with mean $\mu$ and a uniform prior on the standard deviation $\sigma$, it is possible to derive the marginal posterior (where $\sigma$ has been ...
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0answers
60 views

Summation involving 2F2 hypergeometric function

Trying to simplify the following sum: $$ \sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!}, $$ where $n=1,2,\ldots$, $z>0$, $0<a<1$, ...
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1answer
49 views

Complex Infinity of Hypergeometric+Gamma function

I am solving some integral which gives an hypergeometric function+gamma function . The point is that my values of n (see the code below) are integers, so ...
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0answers
96 views

Plotting the parameters of Mathieu equation for stability region

I am trying to plot stability regions of Mathieu equation with a and q parameters, I plot this and now i want that i fix the value of x between -5 and 5, and corresponding to each value of x,I get ...
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1answer
51 views

Using Mathematica to find series expansions for partial derivatives of the generalized Riemann zeta function

I am trying to use Mathematica to find a suitable series expansion for the expression $$ \zeta ^{(1,0)}\left(-1,1-\frac{i}{2}\right) - \zeta^{(1,0)}\left(-1,1+\frac{i}{2}\right),$$ which ...
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1answer
81 views

Finding the symbolic inverse of a function

Is there a way of inverting this function to obtain $r(\rho)$? ...
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5answers
208 views

Calculating the Dottie number using an infinite series

The Dottie number is the solution to the equation $\cos(x) = x$ It is approximately equal to $0.739085133215160641655312.$ This number can be expressed analytically in the following form (see this ...
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2answers
87 views

Legendre polynomials that evaluated with huge difference

I'm dealing with Legendre polynomials, involving the first kind, second kind, and the associated ones. However, I found this: ...
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2answers
215 views

Graph of Chebyshev's first polynomials, almost like the wikipedia graph

I want to graph the first polynomials of Chebyshev almost like the graph of Wikipedia: I have tried it this way ...
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0answers
114 views

How can I do a faster integration?

I have this part of my code, which takes forever to run. Does anybody know how to make it faster? Using NIntegrate I face error: "NIntegrate::eincr: The global error of the strategy GlobalAdaptive ...