Questions tagged [special-functions]
Questions on the special mathematical functions implemented in Mathematica.
0
votes
0answers
3 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
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 ...
1
vote
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. ...
2
votes
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 ...
0
votes
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.
0
votes
0answers
25 views
Expanding the complex conjugate of a Mathieu function as a series
In my calculations, I found this equation:
...
0
votes
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}] , ...
3
votes
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 "...
0
votes
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]/...
0
votes
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 ...
7
votes
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} \,...
2
votes
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.
...
0
votes
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?
1
vote
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 ...
0
votes
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
3
votes
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 ...
1
vote
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 ...
0
votes
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....
-1
votes
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^{...
2
votes
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:
...
0
votes
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
...
0
votes
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} *)
...
3
votes
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}\...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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
...
0
votes
2answers
100 views
2
votes
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 ...
2
votes
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:
...
2
votes
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
votes
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 ...
2
votes
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 ...
1
vote
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
votes
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^{\...
4
votes
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
votes
1answer
66 views
Hypergeometric differential equation with integer parameters?
Naively, the hypergeometric differential equation has two independent solutions as follows:
...
2
votes
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 $x$? ...
0
votes
0answers
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]
...
0
votes
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
...
1
vote
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 ...
2
votes
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 ...
0
votes
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 $n=1,2,\ldots$, $z>0$, $0<a<1$, ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
81 views
Finding the symbolic inverse of a function
Is there a way of inverting this function to obtain $r(\rho)$?
...
3
votes
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 ...
3
votes
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:
...
4
votes
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
...
3
votes
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 ...
