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Questions tagged [special-functions]

Questions on the special mathematical functions implemented in Mathematica.

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1answer
42 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
67 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
72 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
34 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
73 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
75 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
74 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
65 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
18 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} *) ...
2
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1answer
359 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
27 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
36 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
120 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
87 views

Plotting a 3D piecewise function

I would like to plot the following function: ...
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0answers
62 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
44 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
115 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
31 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
91 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
104 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
45 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
113 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
62 views

Hypergeometric differential equation with integer parameters?

Naively, the hypergeometric differential equation has two independent solutions as follows: ...
2
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1answer
100 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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0answers
28 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
22 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
121 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 ...
2
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2answers
81 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
52 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
46 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 ...
0
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0answers
68 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 ...
0
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1answer
46 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
79 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
203 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 ...
3
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2answers
85 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: ...
4
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2answers
212 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 ...
3
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0answers
99 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 ...
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1answer
64 views

Partial differential equation heat/diffusion equation 3d

I'm trying to solve the heat/diffusion equation in 3d in spherical symmetry $\partial_t f=D\Delta f$. I wrote : ...
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0answers
51 views

double Integral in complex number field

The following expression is an expansion of Hypergeometric2F1 function from the above expression with the help of 9.113(in 'Table Of Integrals, Series And Products'). but I got different results.WHY? ...
3
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1answer
87 views

Definition of WignerD function?

On Wikipedia, elements of Wigner's D-matrix are defined as $$D_{m'm}^{j}(\alpha,\beta,\gamma)=\langle jm'|e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}|jm\rangle=e^{-im'\alpha}d_{m'm}^j (\beta)e^{-...
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2answers
153 views

Does Mathematica gives us a wrong result for the integral of a function including elliptic functions?

Calculus tells us that the differentiation of the integral of a function should be itself, but at least in one case Mathematica answers NO. I feel very confused. The new figure seems to bifurcate at ...
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2answers
264 views
1
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1answer
156 views

Plot a Mathieu stability chart

I'm working with the Mathieu equation as a part of my research and as part of the analysis I'd like to include a stability chart such as Fig. 1 in this paper. Unfortunately I'm stuck with Mathematica ...
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2answers
57 views

Expansion of hypergeometric function with symbolic parameters

I just tried in Mathematica 11.3 ...
0
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1answer
100 views

Solving the spherical harmonics PDE using DSolve

I am trying to solve the spherical harmonics PDE in Mathemtica. My code is: ...
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2answers
144 views

How to solve a Bessel differential equation with a boundary condition at infinity?

I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use: ...
0
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2answers
68 views

DSolve - Unable to obtain plot of solution - 2nd order ODE

I am trying to solve the equation below with DSolve. The equation is that of a wave, expected to fall off exponentially as r approaches infinity. The solution is a combination of Spherical Bessel ...
0
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1answer
27 views

Solving an equation involving a determinant (including spherical recursive functions)does not compute

I'm trying to solve this matrix to get a resulting function that depends on the variable Q (or if impossible, H3). When I try to to that I get two results: if I try to solve for Q, it doesn't show ...
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0answers
96 views

Bug in SumConvergence ver. 11.2.0.0

Bug in V10.0.1 and persisting through 11.3 Version 11.2.0.0 on MacBook Pro: ...
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0answers
46 views

HypExp and HPL packages for hypergeometric functions: Evaluating a function HPL[{minus,plus},x]?

I am currently using the HypExp and HPL packages, which are useful for expanding hypergeometric functions in series around integer or half-integer values, as in common in dimensional regularization ...