Questions about using complex numbers in Mathematica. This includes basic arithmetic, functions of complex numbers, plotting complex functions, and dealing with branch cuts.

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0
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1answer
90 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 ...
0
votes
1answer
111 views

Plotting the image of a curve under a complex polynomial using Mathematica

I would be able to plot the image of a curve (say the circle with radius $1$ centered at $1+i$) under a complex polynomial (say $p(z)=z^2$) using Mathematica. I know how to find the level curves of $...
5
votes
1answer
597 views

How to plot the solution of a Partial Differential Equation?

My attempt. I need to solve numerically the Complex Ginzburg-Laudau Equation (CGLE): $$ \frac{\partial A}{\partial t}=\epsilon A-(1+i\beta)|A|^2A+(1+i\alpha)\nabla^2A $$ I'm using a uniform initial ...
10
votes
1answer
297 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
0answers
101 views

Real part of a sum

I would like to take the real part of the following expression: Sum[(a[i] + I b[i]) r^i/(r^2 + z^2)^i, {i, 0, 2 n}] where all of ...
1
vote
1answer
618 views

How to plot the graph of the equation F[x,y]=0 if y is real number, x is complex? [closed]

I have an equation F(x,y;a,b,c,...)=0, where x and y are variables, a,b,c,... - parameters. y is a real number, x - complex. For every given y I need to solve this equation, i.e. to find Im(x) and Re(...
1
vote
2answers
177 views

Complex numbers from two arrays with Real and Imaginary parts

I have two arrays, containing the real and imaginary parts of a list of complex numbers. Re = {{Re_number1},{Re_number2},...} Im = {Im_number1},{Im_number2},...} ...
0
votes
1answer
314 views

How to get the modulus of following complex number

I want to get the modulus of following (1+(x+I*y)/2+(x+I*y)^2/12)/(1-(x+I*y)/2+(x+I*y)^2/12) I use ...
2
votes
1answer
212 views

Polar color coding for complex function plots?

Having browsed this Q on plotting complex functions and the Zeta page, I don't see anything as nice as this plot from Matilde Marcolli's slides "Geometry and physics of numbers". Looks like ...
8
votes
2answers
253 views

Strange behavior of Limit in Mathematica 9 and 10 (bug?)

fixed in 10.1 Consider a complex logarithm where the branch cut is defined along the negative axis. Then for $r$ and $\eta$ real and positive we can write $ \lim_{\eta \to 0} \log(-r+ i \eta) = ...
5
votes
3answers
1k views

Complex conjugate

How do I complex conjugate a vector? ...
2
votes
1answer
514 views

Working with derivative of conjugate of a complex number

I have a complex function, lets say $g(x)$. I want to take its and its conjugate's derivative. I need the solution of derivative which must be symbolically and computationally efficient. Lets take an ...
1
vote
1answer
77 views

Factor bivariate polynomial over the complex numbers

This is very much like Factoring polynomials to factors involving complex coefficients except that I'm concerned about bivariate polynomials, not univariate polynomials. Take for example the ...
1
vote
2answers
591 views

How to calculate the phase spectrum

I have the following frequency characteristics in the Fourier domain: $$H(\omega)=\frac{-\omega^2}{63170s^{-2}-\omega^2+355.1s^{-1}i\omega}$$ How do I find the phase spectrum from this? I should ...
0
votes
2answers
69 views

For the same function, why does /. give a complex result?

I define a very simple function like this, g[k_] := 1. + 2 Sum[1/(1 + (2 n)^2), {n, 1, k}] g[50] give the result is: 1.70279. But g[n] /. n -> 50 give aother result: ...
0
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1answer
115 views

Optimizing complex functions

I am trying to optimize a function containing complex numbers, however I keep running into errors. Consider the simple optimization problem: ...
0
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1answer
197 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: ...
9
votes
3answers
131 views

Pattern matching with Complex. Feature or a bug?

fixed in 10.1 (windows) I have run into some strange behavior while doing some pattern matching. First, this works as expected: ...
0
votes
1answer
167 views

Plotting Complex Numbers - Functions of Complex Numbers [duplicate]

So I have to generate a few different plots with z, where z is a complete number... z[x_, y_] := x + y*I F[z_] := (25*Pi*z*I)/(1 + 10*Pi*z*I) First, I need to ...
0
votes
1answer
189 views

Complex number plot with inequalities [duplicate]

I need to plot the function where z is a complex number: $$S = {z:|z + 0.15|<0.6} \wedge {z:Pi/4<=arg(z)<=Pi}$$ I can do this: ...
1
vote
2answers
279 views

Differentiation of an unknown function

I have to take the partial differentiation of an unknown function. For example, take the unknown function to be $g(x)$. Then it's derivative w.r.t $x$ is $g'(x)$. By default, Mathematica ...
4
votes
2answers
197 views

Trying to get Arg[1 + I a] -> ArcTan[a]

Can Mathematica evaluate Arg[1+ I a] when a is a positive real in order to get ArcTan[a]? ...
2
votes
1answer
247 views

Singularity on a simple Plot

While solving quartic equation ...
2
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0answers
86 views

Evaluating an integral

Here is the input: Integrate[E^(2 I*t)/(2*Pi*(E^(I*t) - z)), {t, 0, 2 Pi}] and here the output: ...
3
votes
2answers
241 views

NIntegrate and Integrate giving different results for ill-behaved function

I'm trying to integrate the following function with Mathematica 8: $$ I(a,b)= \int_0^1 \mathrm{d}x\int_0^1\mathrm{d}y \,\theta(1-x-y) \frac{1}{x a^2-y(1-y)b^2},$$ where $\theta$ is the Heaviside ...
1
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1answer
138 views

How to prevent tiny complex errors in Integrate?

Integrate is adding a tiny imaginary error to an easy result. Why? And how can I stop it? ...
4
votes
4answers
531 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 ...
0
votes
0answers
65 views

Why is PolyLog[] giving weird answers for ordinary values? [duplicate]

Possibly related to this question, but it seems slightly different: Strange behaviour of PolyLog Function Wikipedia says that for real s, z<1 should be real. So I was confused when MMa returned: ...
0
votes
1answer
72 views

Plot of the set complex numbers [duplicate]

I want to show this set (eigenvalues of my matrix): ...
1
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0answers
57 views

Complex input for Compile makes it slow

I asked a question here and then tried to use Compile for the solution to gain even more speed. Inside Compile there is a loop ...
2
votes
1answer
186 views

Complex Integration

There are some sequential functions: ...
0
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1answer
176 views

Matrix form of complex numbers [closed]

Complex numbers have matrix form. But some 2x2 matrices while differ, represent the same complex number. To compare complex numbers one has to transform the matrices to the form \begin{pmatrix} a &...
1
vote
1answer
314 views

Evaluating the real part of an expression

I want to get the Real part of this expression - shouldn't be too hard to evaluate. Why is Mathematica not evaluating but returning the same code? ...
5
votes
2answers
9k views

How to tell Mathematica that certain variables are real/imaginary, integer-valued, etc

I'm trying to expedite some quantum mechanical calculations by running them through Mathematica. Expectation values and stuff like that. When I say, for example, ...
0
votes
1answer
169 views

How do I declare that a variable represents a real number?

How do I tell Mathematica that a variable is a real number. I searched for this question here and in Google and nothing that was suggested seems to work for me. I must be missing something. My code is ...
4
votes
1answer
454 views

Problem on limit involving complex numbers

I have $${\frac{(6 k+1)^{k}}{(2 k+5)^{k}}}*(z-2 i)^k$$ and I need to find it's limit for $k$ approaching infinity. ...
0
votes
1answer
394 views

How to find all roots of a complex number [duplicate]

Finding all roots, and I know there are four f them, of this (1 - i)^(1/4) Not only real, but imaginary as well
1
vote
1answer
244 views

Rationalize complex numbers

What is the best way to rationalize complex numbers, if not both the real and imaginary part are actually rational? According to the documentation, Rationalize ...
0
votes
3answers
190 views

Plot Complex Function [duplicate]

I want to have a 3D plot for the following function: (1.6 - 0.005 I) E^(-0.5 p^2 + ( 2. I) p q + (6.5 - 11 q) q) p and q from -10 to 10 Is there any way?
2
votes
1answer
92 views

A question about expanding complex functions of real arguments

I have the following problem. There is such an expression as: P[x_,y_] := z[y] E^(I beta x) + Conjugate[z[y] E^(I beta x)]; The variables ...
7
votes
3answers
751 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 $v=2xy$, ...
39
votes
2answers
2k views

Möbius transformations revealed

Möbius Transformations Revealed is a short video that vividly illustrates the simplicity of Möbius transformations when viewed as rigid motions of the Riemann sphere. It was one of the winners in the ...
3
votes
1answer
403 views

Real and Imaginary parts of solutions to a complex linear ODE system

Consider a complex linear ODE system $x'=Ax$, where $$A=\left( \begin{matrix} 0&1\\ -2&-i \end{matrix}\right). $$ One can first find the eigenvalues and eigenvectors using ...
1
vote
0answers
384 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
2answers
446 views

Listplot imaginary part of complex numbers

I have the following list w={{0.01,99 +0.00001414 I},{0.15,6.6370108 +0.003144129 I},{0.25,3.9515722 +0.00854493297 I},{6,0.10041 +0.28132187 I}} and I want to <...
7
votes
4answers
446 views

Speed up Schwarz-Christoffel mapping

I would like to ask your advice how to speed up Contour and Parametric plots in the following example. Let as start by defining a function ...
7
votes
2answers
241 views

Why does N[Re@f] give complex result?

Consider this code: ...
4
votes
1answer
172 views

Optimizing a function containing a complex exponential

I've noticed the following significant different in performance for different formulations of the same function: ...
4
votes
2answers
251 views

Can't solve equation having complex coefficients

Why can't Mathematica solve $\quad\quad\frac{1-i}{\sqrt{2}}=e^{i \alpha } \tan \left(\frac{\beta }{2}\right)$ with the restrictions $\alpha \in [0, 2 \, \pi)$ and $\beta \in [0,\pi]$: ...
8
votes
3answers
206 views

Complex result for Real vectors in VectorAngle

I was expecting a real angle using VectorAngle when passing real valued vectors, but I obtained a complex angle: ...