Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
1
vote
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 ...
1
vote
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
3
votes
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 ...
1
vote
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 ...
0
votes
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....
-1
votes
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^{...
2
votes
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:
...
0
votes
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
...
0
votes
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
votes
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}\...
0
votes
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 ...
0
votes
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 ...
1
vote
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
...
0
votes
2answers
87 views
2
votes
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 ...
2
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
votes
1answer
62 views
Hypergeometric differential equation with integer parameters?
Naively, the hypergeometric differential equation has two independent solutions as follows:
...
2
votes
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 $x$? ...
0
votes
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]
...
0
votes
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
...
1
vote
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
votes
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 ...
0
votes
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 $n=1,2,\ldots$, $z>0$, $0<a<1$, ...
0
votes
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
votes
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
votes
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 ...
0
votes
1answer
79 views
Finding the symbolic inverse of a function
Is there a way of inverting this function to obtain $r(\rho)$?
...
3
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 :
...
0
votes
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
votes
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^{-...
6
votes
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 ...
0
votes
2answers
264 views
Sum a certain hypergeometric-function-based expression pertaining to an integration problem
I would like to sum over the index $h$ from 3 to $\infty$, the expression
...
1
vote
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 ...
1
vote
2answers
57 views
0
votes
1answer
100 views
Solving the spherical harmonics PDE using DSolve
I am trying to solve the spherical harmonics PDE in Mathemtica. My code is:
...
2
votes
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
votes
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
votes
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 ...
8
votes
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:
...
2
votes
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 ...
