Questions about using complex numbers in Mathematica. This includes basic arithmetic, functions of complex numbers, plotting complex functions, and dealing with branch cuts.
37
votes
4answers
2k views
How can I generate this “domain coloring” plot?
I found this plot on Wikipedia:
Domain coloring of $\sin(z)$ over $(-\pi,\pi)$ on $x$ and $y$ axes. Brightness indicates absolute magnitude, saturation represents imaginary and real magnitude.
...
26
votes
6answers
2k views
Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$)
Say I want to quickly calculate $\sqrt[3]{-8}$, to which the most obvious solution is $-2$.
When I input $\sqrt[3]{-8}$ or Power[-8, 3^-1], Mathematica gives the ...
17
votes
3answers
711 views
Visualizing a Complex Vector Field near Poles
I've been playing around with a visualization technique for complex functions where one views the function $f: \mathbb{C} \rightarrow \mathbb{C}$ as the vector field $f: \mathbb{R^2} \rightarrow ...
15
votes
2answers
212 views
Inconsistent results from equivalent integrals
Why is Mathematica returning different values for these two integrals:
I am just being introduced to complex integration, so it's possible that I have a misunderstanding of how this works, but in ...
10
votes
2answers
717 views
Plotting complex Sine
I've got another plotting problem. I want to plot Sin[z] where z is complex. So, I've tried the following:
...
9
votes
4answers
3k views
Plotting Complex Quantity Functions
Trying to plot with complex quantities seems not to work properly in what I want to accomplish. I would like to know if there is a general rule/way of plotting when you have complex counterparts in ...
9
votes
2answers
1k views
How to specify assumptions before evaluation?
If I request mathematica evaluate an integral for me, I'll often get a more general ConditionalExpression than I want. Example :
...
8
votes
3answers
331 views
How can I convert a complex number a+b I to the exponent form A Exp(I phi)?
When I have an expression such as:
(1/4 + I/4) ((1 - 2 I) x + Sqrt[3] y)
it is hard to get an intuition of the number.
So I want to convert it to the complex ...
8
votes
1answer
377 views
Is it possible to set a variable as a positive one in the whole notebook?
I'm having issues during integration due to the fact that Mathematica doesn't know if an undefined variable is positive or not (it gives me complexes which bothers me in the end).
For example I do ...
8
votes
2answers
158 views
What is the value Re[Sqrt[1+I*2*x]]?
When I try to evaluate Re[Sqrt[z]], for some values of Mathematica fails to evaluate it.
For example,
Re[Sqrt[2 + I*x]]`
...
8
votes
2answers
518 views
RootSearch for complex or multiple equations
First the background. I'm trying to solve for the roots of a rather messy complex equation. This is not the exact equation, but it's a decent (simpler) stand in:
...
8
votes
0answers
102 views
7
votes
3answers
1k views
Derivative of real functions including Re and Im
When deriving functions using Re, Im or Arg (and probably some other functions as well), ...
7
votes
3answers
118 views
Wrapper for inexact numeric complex numbers that maintains polar form
Related question:
How can I convert a complex number a+b I to the exponent form A Exp(I phi)?
Mathematica insists on displaying complex number in form a+I b when ...
7
votes
1answer
149 views
Testing for primality in quadratic rings?
Testing for primality in $\mathbb{Z}[\sqrt{-1}]$ in Mathematica is easy:
PrimeQ[n, GaussianIntegers -> True]
But how can I test for primality in, say, ...
7
votes
2answers
225 views
Moving the location of the branch cut in Mathematica
According to the documentation, Mathematica chooses the branch cut for $\log(z)$ to lie along the negative real axis. It it possible to change this so that it lies along the positive axis or elsewhere ...
7
votes
1answer
235 views
Symbolic Integration along contour: branch cut problem?
Context
Following this question on path integrals in the complex plane, having defined again a numerical and symbolic integrator along a path as
...
7
votes
0answers
278 views
Dual complex integral over implicit path using contour plot
Context
I am interested in doing double contour integral over paths which are defined implicitely.
For the sake of debugging, let's assume its
$$\oint_{\cal C}\oint_{\cal C} \frac{1}{u\, x} d u d x$$ ...
6
votes
1answer
310 views
Is Abs[z]^2 a bad way to calculate the square modulus of z?
For a numerical quantity z, Abs[z] returns the square root of the sum of the squares of the real and imaginary parts of ...
6
votes
1answer
112 views
How to reduce expressions with complex coefficients in the form of a+0.*I
I'm trying to get an eigenvalue equation in Mathematica, and the result is an expression with coefficients of the form a + 0. I. For example,
Is there any clever ...
6
votes
1answer
116 views
Compiling Error functions of complex values
According to List of compilable functions Erf and Erfc are compilable functions.
However, I want to make a compiled version of ...
5
votes
3answers
2k views
Is there a simple way to plot complex numbers satisfying a given criteria
I think this should be straightforward, but I cannot seem to find a good source on how to do it after searching around, so
I'm trying to sketch sets of complex numbers that meet a given for criteria. ...
5
votes
2answers
286 views
5
votes
1answer
259 views
How do I put an image on the complex plane?
I watched this video and became interested in transforming an image. But I have no good idea on how to embed an image in the complex plane using Mathematica.
I have a method that seems to work, but ...
5
votes
1answer
487 views
Complex number operations: telling Mathematica variables are real
I want to do Conjugate[a + b*I], but when I do that, the solution is
Conjugate[a] - I*Conjugate[b]; when for me, a and b are ...
5
votes
2answers
167 views
How to eliminate the zero real part of a purely imaginary number?
In Mathematica 9, a purely imaginary number, e.g. 0.9 I, will display as 0. + 0.9i in the output form. How can I eliminate the ...
4
votes
3answers
487 views
How can I plot the complex graph of $x^x$ in Mathematica? [duplicate]
Possible Duplicate:
How to plot imaginary part of a function
Plotting Complex Quantity Functions
For example, this Wolfram Alpha query shows this graph:
But it does not show the code for ...
4
votes
6answers
344 views
Image of first quadrant under $f(z)=(z+i)/(z-i)$
I'm able to plot the region where Im[z] > 0 and Re[z] > 0:
...
4
votes
4answers
1k views
Plotting an Argand Diagram
I have the function:
$F(\omega) = \frac{5\; - \;i\;\omega}{5^2\; +\; \omega^2}$
When $\omega$ has the values :
$\{ -7, -2,\; 0,\; 2,\; 7\}$
How would I plot the Argand diagram in Mathematica? Or ...
4
votes
2answers
141 views
Convergence and value of a complex power series
I've done a little math and I got the following power series expansion of $\log z$ about $z_0=-2+i$.
$$\log z=\log(-2+i)+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n(-2+i)^n}[z-(-2+i)]^n$$
I've shown that ...
4
votes
1answer
364 views
Does the Im function work with symbolic arguments?
Does the Im function work with symbolic arguments?
...
4
votes
2answers
160 views
What can I do to eliminate the error FindFit::nrjnum:?
I am testing the "Power Law with finite-time singularity" hypothesis for world population growth for a project.
The data I'm using (same behaviour should also be exhibited by the stock market, thats ...
3
votes
4answers
870 views
Solving complex equations
I feel like I am missing a basic, but key point when using Mathematica's Solve or Reduce.
...
3
votes
2answers
182 views
presenting a real number as real instead of imaginary
I have an equation which results in an answer of the form
$\frac{i a}{\sqrt{c-d}}$
is there any way to get Mathematica to present it in it's real form? like
$\frac{a}{\sqrt{d-c}}$
I know that I ...
3
votes
2answers
443 views
Linear equation with complex numbers
I have to solve an equation of the type $$a z+b \overline{z}=c$$ with $a,b,c\in\mathbb{C}$.
My approach is to set $F(z)=a z+b \overline{z}-c$ transform $z$ to $x+i y$ and then get a real linear ...
3
votes
2answers
308 views
Is there a way to solve the Apollonius Circle problem in Mathematica?
Assuming x, a, and b are complex numbers, is there a way to reduce the equation Abs[x - a] == k Abs[x - b] to something like ...
3
votes
3answers
125 views
How to extract phase angle from sinusoid
I'm doing some electric circuit calcualtions and I'm trying to get the phasor representation of some arbitrary function of Sin or Cos. Could be complex like:
...
3
votes
3answers
169 views
Show does not combine the plots
i've the following problem:
because it's not possible to plot complex numbers (or is it?) i created my own "function":
...
3
votes
1answer
330 views
Symbolic integration in the complex plane
Context
While answering this question, I defined (symbolic and numerical) path integrations as follows
...
3
votes
1answer
305 views
Bifurcation diagrams for multiple equation systems
I am interested in constructing a bifurcation diagram for some of my parameters (especially for β and γ) in the dynamical system given in the code below. I want to see how parameter changes affect the ...
3
votes
1answer
105 views
Exporting/Importing a Table of complex numbers
I'm generating a long table of list of the form:
PN={{1,2,1+i},{3.5,2.6,2}...},{...},...
Using:
Export["PN.dat", PN, "Table"]
...
2
votes
5answers
411 views
Why is this Mandelbrot set's implementation infeasible: takes a massive amount of time to do? [closed]
The Mandelbrot set is defined by complex numbers such as $z=z^2+c$ where $z_0=0$ for the initial point and $c\in\mathbb C$. The numbers grow very fast in the iteration.
...
2
votes
2answers
304 views
Eigensystem, Eigenvalue doesn't output nonreal eigenvalues
Basically I have a matrix and when I used either Eigenvalue or Eigensystem, it doesn't output nonreal eigenvalues, instead it ...
2
votes
1answer
254 views
Finding residues of multi-dimensional complex functions
Say I have a function $f$ of $n$ complex variables, $\{ z_i \}_{i=1}^{i=Nc}$. And then I want to contour integrate the expression such that for each $z_i$ its an integration on an unit circle about ...
2
votes
1answer
61 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 ...
2
votes
0answers
78 views
How to make the imaginary part of a +0. I zero globally?
Values like a +0. I are really annoying. Answers from
How to reduce expressions with complex coefficients in the form of a+0.*I？
Is there a way to globally set when to treat a very small number as ...
2
votes
0answers
284 views
Simple contour integral with a parameter gone wrong
I run into the following problem, I tried to evaluate a very simple integral:
Assuming[ a > 0 , Integrate[Sin[a*s]/(s - I)^2, {s, -Infinity, Infinity}]]
...
1
vote
3answers
89 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 ...
1
vote
1answer
92 views
why there is a small imaginary part [closed]
I encountered a problem. I have a eigenvector eigvsI[1]
...
1
vote
1answer
339 views
Forcing FindRoot to return only real solutions
FindRoot documentation reports that if the Equation and the initial point are reals, the solutions are searched in the real domain.
However, in the following case I ...
