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votes
0answers
25 views

Expanding the complex conjugate of a Mathieu function as a series

In my calculations, I found this equation: ...
0
votes
2answers
80 views

Result with assumptions contradicts previous result

Without assuming anything on the argument of the complex number inside the Gamma function ...
1
vote
0answers
36 views

Inverting a series

How do I invert the following, \[Rho]=r + b0 Sum[Pochhammer[1/2, k]/(k! ((1 - q) k - 1)), {k, 0, \[Infinity]}] + b0^(1 - q)/(2 q) r^q + O[r^(2 q - 1)] to get $r(...
3
votes
2answers
191 views

Why can’t mathematica find this residue?

I am trying to find the residue of the function $$f(z)=(z+1)^2e^{3/z^2}$$ at $z=0$. I tried the following in Mathematica Residue[(z+1)^2*Exp[3/z^2],{z,0}] which ...
6
votes
2answers
200 views

Series expansion gives incorrect result

Bug introduced after 10.4 and persisting through 11.3.0 Mathematica 11.1.1.0 tells me that ...
3
votes
1answer
124 views

Generating function for residues of a complicated function

I have a rather complicated function involving 3F2 Hypergeometric functions (see below), which has infinitely many poles. I can extract the residues individually. But it would be great if I could ...
3
votes
2answers
124 views

Series default assumptions?

I have a question about the "Series" command. Specifically, the following input Series[Sqrt[x^2], {x, 0, 2}] gives me the following output ...
0
votes
1answer
218 views

Why is Series with Abs giving an imaginary unit as output? [closed]

I have a simple integral of an absolute value. However, the output is showing an imaginary unit. In a previous question I have been told the problem is the Series. ...
5
votes
2answers
127 views

Why is Mathematica factoring I out of real expressions, and how can I reverse it?

I am doing some complicated series expansions involving integrals and whatnot, but everything is real the whole time (and I specify that using Assumptions at every ...
0
votes
0answers
63 views

Real vs. complex series coefficients

I hope that this question is not a duplicate but I was unable to find it here (or somewhere else in the internet). For some reason, mathematica makes some expressions unnecessarily complicated. ...
0
votes
1answer
91 views

Inverting a series with both positive and negative powers

I have an asymptotic series expansion that I would like to invert of the form $ t_2 z + t_3 z^2 + t_4 z^3 + \frac{A_1}{z} + \frac{A_2}{z^2} + ... + \frac{A_9}{z^9} + \mathcal{O}(z^{-10}) = w$ i.e. I ...
0
votes
0answers
258 views

Analytic Continuation Algorithm

I am trying to analytically continue the following sum starting from $z = 0$. $\sum_{n = 0}^{m} z^n$ I want $z = 2$ to be within the defined region. I have the following code: ...
0
votes
1answer
186 views

Laurent series 0 < |z-3| < 3

I wanna check my laurent series exercises on Mathematica, but can't seem to find a command or program to achieve the result of such type of interval. $f(z)=\frac{1}{(z-3) z},\\1<|z-3|<3$ The ...
8
votes
0answers
368 views

Expansion for Modified Bessel Function Around Infinity

I'm somewhat new to Mathematica, and I don't understand why I'm getting inconsistent series expansions for the modified Bessel Function of first kind near $x=\infty$. First problem: I get different ...
1
vote
1answer
214 views

Problem with complex conjugation of series data in mathematica 10

ComplexExpand in Mathematica 10 doesn't work with series data: Taylor = Series[Exp[I x], {x, 0, 4}]; ComplexExpand[Taylor\[Conjugate]] gives 1+I x-x^2/2-(I x^...
1
vote
0answers
80 views

Finding consecutive residues of large expression?

I have given a large expression expr (has LeafCount of 2772, you can find it in this file or ...
3
votes
0answers
632 views

Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
10
votes
1answer
731 views

Contour Integration along a contour containing two branch points

I need to compute following contour integrations: $$f(u)=\oint_\alpha dz \sqrt{z^3+z+u} \qquad ; \qquad g(u)=\oint_\beta dz \sqrt{z^3+z+u}$$ In which $\alpha$ and $\beta$ are two contours in ...
1
vote
1answer
697 views

How to obtain Laurent series coefficients of an (almost) arbitrary function?

I have observed that Series in Mathematica assumes that the given function is smooth in the point around which one wants to perform series expansion. For instance: ...
1
vote
0answers
302 views

A power series expansion [closed]

Consider the function, $f(z) = z\, \tanh(\pi z) \log (z^2 + a^2)$ for some $a>0$. Now I am considering 3 different situations, $z = i(n+0.5) - i\epsilon + \delta - it$ for $n \in \mathbb{Z}$ ...
22
votes
1answer
3k views

How does Mathematica understand branchcuts of the complex logarithm?

Say I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. Then it seems to be me that Mathematica when given such a ...
5
votes
1answer
3k views

How to get rid of all complex numbers and functions?

For almost esthetical reasons, I always have been fascinated by infinite sums of series and I always wonder if theyhave a closed form. A couple of days ago, I found on MSE a question related to the ...
13
votes
2answers
520 views

How to compute the residue of $e^{z-\frac{1}{z}}$ at z=0?

I've tried the following but it didn't work: Residue[Exp[z - 1/z], {z, 0}] not even this: Residue[Exp[1/z], {z, 0}] ...
2
votes
1answer
292 views

from complex function to a series

How can I express this complex function as a series? Log[ (1 - E^((I Pi (1 - a))/(b - a)) z)/ (1 - E^(-((I Pi (1 - a))/(b - a))) z) ] Where ...
5
votes
1answer
3k views

Laurent series expansion

Can someone share how to find the Laurent series expansion of $$f(z)=\frac{1}{(z^2-1)(z^2-4)}$$ centered at $0$ on the annulus $1<|z|<2$?