Questions about functions of complex variables.

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4
votes
2answers
505 views

Order of a pole

Is there a simple way to determine the order of the poles of a rational function? I have a difficult function where I need Mathematica to find the poles. It would be interesting to also know what the ...
5
votes
1answer
69 views

How to calculate residues using different branches of the logarithm

My goal is to be able to compute the residues of something like $z^{\alpha} R(z)$ where $0 < \alpha < 1$, $R(z)$ is rational, and the exponential can be chosen with any branch. I found a way to ...
3
votes
2answers
130 views

Analytic expression for a Complex Hilbert Transform in Mathematica

I need to solve the following integral equations for a problem I'm working on - $\frac{-i}{2 \pi}$ $\int_{-a}^{a} \mathrm{dt}\,\, \frac{e^{i k t}}{t + i \tau}$ and $\frac{-i}{2 \pi}$ ...
0
votes
0answers
57 views

Plotting Real and Imaginary parts of Riemann Surfaces

I'm working on a project where I need to take curves like, for example, $y^{2}=(x^{2}-1)(x^{2}-3)$ and plot their real and imaginary parts. More specifically, let $x$ just be real and then plot ...
1
vote
2answers
119 views

Plot contour integral plots

I am $\TeX$ writing notes on complex analysis, I need to use figures of contour paths to integrate on them, how can I plot them on mathematica, something like this adding also the $\gamma_R$ ...
6
votes
1answer
214 views

Find exponential generating function from the first few terms

The function FindGeneratingFunction computes the ordinary generating function of a sequence, given a sufficient number of initial terms. I have two questions in ...
0
votes
1answer
58 views

Conjugate of I \[Phi] [closed]

I have a bigger expression of which I have to take the conjugate. But this is in itself perplexing. When I write ComplexExpand[Conjugate[I\[Phi]]] I just get ...
5
votes
0answers
139 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 ...
5
votes
3answers
595 views

Complex conjugate

How do I complex conjugate a vector? ...
3
votes
3answers
328 views

$L = \lim\limits_{z \to 1} \frac{1-z}{1-z^\ast}$ evaluates incorrectly

I'm about 90% certain this is a Mathematica issue, not me making a silly mistake. I'm trying to evaluate $$L = \lim_{z \to 1} \frac{1-z}{1-z^\ast}.$$ Naively entering ...
1
vote
1answer
131 views

Complex Integration

There are some sequential functions: ...
0
votes
0answers
85 views

Approximating The Fourier Transform

Eq.(1)$$f(z)=\frac {k}{2\pi}\int_{C}F(\theta)e^{ikz\cos \theta} \,d\theta$$ for $$|z|\le l$$ and the side condition Eq.(2) $$0=\frac {k}{2\pi}\int_{C}F(\theta)e^{ikz\cos \theta} \,d\theta$$ for ...
7
votes
3answers
546 views

How do I plot the images of oriented curves under complex transformation?

I'm working on a complex transformation sketch that I'd like to create via mathematica. I'm working with the function $$w=z^2\tag{0},$$ where its real and imaginary parts are $u=x^2-y^2$ and $v=2xy$, ...
0
votes
0answers
198 views

Differential equations with a complex variable

Is Mathematica able to handle ordinary differential equations where the variable itself is complex? I am looking for solutions of ODE systems of the form $$\left\{ \begin{align} i\frac{da_1}{dt} ...
2
votes
1answer
162 views

Check for holomorphy of a function

Given a (rather complicated) function H(z), what is the best approach to check symbolically whether it is holomorphic? What I tried is checking explicitly the ...
1
vote
1answer
633 views

Solving a complex-valued differential equation with NDSolve

I am trying to solve $dx/dt=\sqrt{1+(ix)^{1.8}}$ for initial condition $x[0] =-0.9877 + i 0.1563$, where $x$ is a complex variable. I would like to plot the imaginary part of the solution versus the ...
0
votes
0answers
160 views

A power series expansion

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 $n \in \mathbb{Z}$ and ...
0
votes
0answers
84 views

About how Mathematica understands the branchcuts of the complex logarithm [Part 3]

One understands that for the function $\log(z^2+a^2)$ Mathematica implicitly puts the branch cuts to be starting at $\pm ia$ and going up/down respectively on the imaginary axis. The phase of this ...
12
votes
1answer
996 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 ...
0
votes
1answer
204 views

Series expansion of a complex function

How do I expand a function $f(z)$ in a particular region? For example, how would I expand $f(z)=(z^2-3z+2)^{-1}$ in the region $0<|z-1|<1.$? I believe this can be done by the binomial theorem. ...
3
votes
1answer
88 views

How to apply RootLocusPlot corectly

I have to analyze the complex regions of variable c which is define obtained with InverseLaplace help in the form ...
0
votes
1answer
252 views

How to apply a conformal map to a jpg (or other image)

I would like to apply a conformal map to a jpg. For example, the map $z\mapsto z^2$ to this image: http://thumbs3.ebaystatic.com/d/l225/m/mGyeriV7l-YArkeBd9oMUng.jpg Any help on how to do this?
3
votes
1answer
177 views

How to do this complex integration on the real line?

$m, r$ are parameters in the following integral: Integrate[z Exp[I z r]/Sqrt[z^2 + m^2], {z, -∞, ∞}] How to do this integration directly? The result should be ...
5
votes
1answer
191 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}] ...
1
vote
0answers
70 views

Need to take infinite sum of residues, is there a way to choose the order of operations for ReleaseHold?

I'm computing an infinite sum of residues. I want to do something like this: ...
3
votes
2answers
180 views

Complex limit gives wrong answer

I'm evaluating $$\lim_{z\to e^{i \pi/2 n}} \frac{z-e^{i \pi/2 n}}{z^{2 n}+1}$$ with Mathematica 9.0.1: ...
0
votes
1answer
239 views

How to draw the plot of a function in the complex plane? [duplicate]

I'm new to Mathematica, and I was wondering how to plot $x^n$ in the complex plane. Is there a dedicated function for this purpose?
2
votes
2answers
260 views

Limit of a complex function

Can someone show me how to find the limit of a complex function? Example: z1 = 3 + 4*I some_function[z] = z * z1 Set-up: ...
6
votes
2answers
2k views

Paths integrals in the complex plane

I can't find how to calculate path integrals of complex functions in the complex plane. For example: $$\oint_{\mid z \mid =2}\frac{1-e^z+z}{z^3 (z-1)^2}dz$$
2
votes
2answers
163 views

Limit of integral gives incorrect output

I'm trying to evaluate $\lim_{e\to 0} \, \frac{i}{e}\int_{\pi }^0 \frac{1-\exp (i e \exp (i \theta ))}{\exp (i \theta )} \, d\theta$ with Mathematica 9.0.1.0 on OS X. However, I get "Undefined" for ...
19
votes
2answers
6k views

How to calculate contour integrals with Mathematica?

How to calculate the integral of $\frac{1}{\sqrt{4 z^2 + 4 z + 3}}$ over the unit circle counterclockwise for each branch of the integrand?
16
votes
0answers
2k views

How to visualize Riemann surfaces?

In WolframAlpha we can easily visualize Riemann surfaces of arbitrary functions, can we plot the Riemann surface of an arbitrary function using Mathematica and ...
5
votes
3answers
292 views
3
votes
3answers
219 views

Solving $(-1-z)^n = z$, for $z$ in the unit circle

I want to solve the following equation: $$(-1-z)^n = z,\quad |z|<1,\quad n>1\in \mathbb{N}$$ where $z$ is a complex number. However Solve or ...
4
votes
1answer
477 views

Multi-dimensional integral in the complex plane with poles and essential singularity

I've passed the last week searching a way to numerically integrate this multi-dimensional integral in the complex plane at the poles and avoiding the singularity at z=0: $$ \oint_{C}\oint_{C\ auound\ ...
8
votes
1answer
576 views

Numerical contour integrations in the complex plane - contour deformation gives different answer for analytic kernel

I am trying to do a contour integration in Mathematica numerically. In particular, I'm checking the identity: $$ H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m ...
1
vote
3answers
209 views

finding an argument of a complex number

What is the simplest way to find the argument of the following function? ((1 - E^((I π (1 - α))/(β - α)) z)/(1 - E^(-((I π (1 - α))/(β - α))) z)) as I tried the ...
2
votes
1answer
2k views

Laurent series expansion

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