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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5
votes
3answers
226 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: ...
1
vote
0answers
231 views

A power series expansion [on hold]

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 $n \in \mathbb{Z}$ ...
1
vote
2answers
85 views

Plot values of an $m\times n$ matrix on the complex plane with color varying along $m$

Let's say I have a complex valued matrix $\begin{pmatrix}1+I&2+2I&3+3I\\4+4I&5+5I&6+6I\end{pmatrix}$ represented by a list: ...
-1
votes
1answer
79 views

Image of the unit circle under a complex rational function

Let $$f(z)=\dfrac{z(z-a)}{(z-b)(z-c)(z-d)}$$ be a complex rational function with distinct non zero complex numbers $a,b,c$ and $d.$ I need to plot the image of the unit circle $S=\{z\in\mathbb{C} : ...
0
votes
3answers
55 views

how to take complex conjugate of a real variable

I have a function defined as follows: g[θ_] := 1/k (Exp[-I δ0] Sin[δ0] + 3 Exp[I δ1] Sin[δ1 Cos[θ] + 5/2 Exp[I δ2 (3 (Cos[θ])^2 - 1)) and I compute ...
0
votes
0answers
57 views

Test a complex number to see if it is a real number [closed]

I want to input a complex number and know if it is a real number. For example: I want to input Sqrt[2] and see Mathematica return True or if I input Sqrt[2]*I I want to see False.
-1
votes
1answer
63 views

Real and Imaginary matrix

I have a 2 by 2 matrix T: {{x, y}, {-y, x}} where x is the real part of some complex number and y is the imaginary part of something complex number. Let's say z1 = ...
0
votes
1answer
51 views

What makes imaginary parts show up?

If I define a function using alpha[y_, d_] := 2 ArcCos[1 - (2 y)/d] Then if $y$ and $d$ are equal, the result is $2\pi$ : ...
1
vote
1answer
59 views

How to find the pole of a equation where the residue exist?

I want to calculate the residue of an equation, such as: equ = 1/((x - 2)Sqrt[x - 1]) So first I need to find the pole of the equation. My method is find the ...
0
votes
3answers
71 views

Write complex number in terms of absolute value and phase

I have a long vector and some of the values (19 out of 64) are complex. I got them using the Mathematica Rationalize function, so the complex ones are written in the a+bi form. Is there a function I ...
6
votes
0answers
62 views

Generating OpenCL C code [closed]

I'm trying to make an interactive complex function plotter using OpenCl. So far I can only plot functions that I hardcode into the kernel by hand. I'd like to be able to use ...
1
vote
0answers
48 views

Finding Complex expression [closed]

How do you find a real part of a complex expression? Here is an example of what I am looking for complexExpand[Sqrt[x + I*b]] gives me ...
3
votes
1answer
106 views

Calling Integrate with many limits of integration

I accidentally discovered that you can put many numbers into the limits of integration of Integrate and Mathematica will still evaluate it without returning an ...
3
votes
2answers
177 views

How can I get Mathematica to simplify this i * sqrt expression

In[11]:= FullSimplify[I Sqrt[-1 + w^2], Element[w, Reals], Assumptions -> 0 <= w <= 1] Out[11]= I Sqrt[-1 + w^2] Is it possible to have Mathematica ...
2
votes
1answer
49 views

ComplexExpand and Arg function [closed]

Why does the command ComplexExpand[Arg[a]] gives Arg[a] instead of 0? Isn't it that ...
1
vote
1answer
47 views

Chop not working [closed]

I am running a function that has a large output in matrix format with small, complex parts added to the numbers- basically a rounding error. I am trying to use Chop to obtain use-able results, but ...
8
votes
2answers
806 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
0answers
55 views

Artifacts in the solutions of NSolve [closed]

I have (what I think are) some problems with the precision of NSolve. I'm trying to solve this cubic equation: $a f^3(x)-b f^2(x)-c f(x)-z=0$ for the function $f(x)$ and then plot the following ...
-1
votes
1answer
24 views

What does mean |sin(z)|<=1? [closed]

I already know that the magnitude of sin(z) its equal to sqrt(sin^2(x)+sinh^2(y))
14
votes
2answers
4k 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: ...
1
vote
1answer
41 views

How to find function existence border

I have a function like this f[x_, a_] = Log[-x - a]*(-x - a) + Sin[a]; this function does exist if x > a because of Log[]. a - parameters. if I plot a few graphs ...
0
votes
1answer
42 views

How can I plot an exponential Fourier series? [closed]

Iam triying to plot summatory of the next function: (e^(i*n*t*wo)*Sin[n*Pi/2])/(n*pi) I have no problem plotting a trigonometric fourier serie, so the only ...
1
vote
1answer
31 views

Distribute Conjugate over expression?

I have this output: Conjugate[Subscript[a, 0][x] + t^\[Beta] Subscript[a, 1][x]] I want to distribute Conjugate: ...
0
votes
3answers
90 views

ploting the graph of 1/z for a given value

I want to plot the graph of ${1}/{z}$ , for a specific single value of |z| (z - complex variable) . I would like something like ...
0
votes
0answers
51 views

Using DeleteCases to ignore term of Product

I'd like to calculate $\int_{-\infty}^{\infty}\mathrm{d}x/(1+x^6)$ through a variation of the residue formula, which is $\int_{-\infty}^{\infty}f(x)\mathrm{d}x=2\pi i\sum \text{Res }f$ for Residues in ...
68
votes
4answers
6k 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. ...
0
votes
0answers
36 views

How to specify in NDSolve that the sought function is holomorphic?

If I include the Cauchy-Riemann equations condition in complex form for the sought holomorphic function gf in NDSolve like this: ...
2
votes
2answers
87 views

Plot a line on the complex plane knowing one point and its orientation

I want to draw a line on the complex plane which begins at $0.4+0.1i$, and with an angle of $\frac{\pi}{4}$ with respect to the real axis. Is there a simple way to do this in Mathematica?
7
votes
1answer
258 views

Branch cuts of sqrt

I was trying to plot some complex functions with branch cuts on mathematica but I have two problems. 1) The function is with z-Sqrt[z-1]*Sqrt[z+1] with z complex. Now Mathematica says that the ...
3
votes
1answer
133 views

Why does Compile not make my function faster?

I need to speed up a large program that repeatedly calls the module below (I've not trimmed the code for this question to ensure there is enough calculation to show a speed improvement.) The obvious ...
0
votes
1answer
69 views

Simplify Re[z] in equation to more “natural” form [closed]

If you have a triangle in the complex plane with vertexes {a,b,c} (all complex numbers), the length of the altitude from A to the side BC can be computed as: ...
3
votes
1answer
337 views

How to simplify to form without imaginary unit

I have an integral that I compute with Mathematica and as a result I get a seemingly complex expression (i.e. the expression contains the imaginary unit, $i$, at some places). However, if I try to ...
0
votes
2answers
96 views

How to plot list of numbers in the complex plane? [closed]

This should have a trivial solution, but how do I plot a list of (complex) numbers in the complex plane? Or, put another way, why doesn't the code below work? I get the error "... is not a list of ...
7
votes
1answer
85 views

Why can't I replace I with HoldPattern@I?

Just try this sample: I /. HoldPattern@I -> a (* I *) Why does it fail to replace I with ...
2
votes
1answer
76 views

Can't obtain Real results using NSolve

I am trying to find the critical properties of the Carnahan-Starling equation of state, $$p=\rho R T\frac{1+b\rho /4+(b\rho/4)^2-(b\rho/4)^3}{(1-b\rho /4)^3}-a\rho^2$$ using this system of equations, ...
1
vote
2answers
88 views

Simplify complex square roots

I want to simplify the following expression as follows: $$\sqrt{\left(\gamma -\kappa _2\right){}^2-16 G_2^2}=i\sqrt{-\left(\gamma -\kappa _2\right){}^2+16 G_2^2}$$ I am trying ...
4
votes
1answer
84 views

Fitting real and imaginary parts of a function

I have an analytic form of a function G[w_] = -I*z0*w/(1 + I*z0*w*a) I know this function can be obtained (up to some overall normalization) from solving a ...
3
votes
1answer
55 views

Why doesn't a real variable's conjugate evaluate to itself?

I have this code (it's just a snippet to exemplify my issue): Assuming[Element[a, Reals],Conjugate[a]] Which results in the output: ...
2
votes
3answers
110 views

Contour plot of the imaginary portion of a function given the real part is positive

How do I plot the imaginary part of the function while its real part is positive? For example, for the below code I could have the imaginary part but I am not able to find the positive or negative ...
0
votes
2answers
89 views

How to plot in the complex plane? [duplicate]

I am new to Mathematica and I would like to ask how to plot in the complex plane in general. Also, as an example, how do you plot $e^{i\theta}$ in Mathematica? In physics the function $e^{i\theta}$ is ...
5
votes
3answers
165 views

How to tell Mathematica to get the real-only inverse function?

The problem I have is that Mathematica seems to play better with complex numbers so that the result of N[InverseFunction[Function[x,x^3]][-1]] is ...
1
vote
3answers
72 views

replacing complex numbers

I know: I'm going to make a poor showing, but really I can't understand this: a is an expressione whose FullForm is ...
1
vote
2answers
85 views

Imaginary part of complex number, nothing works?

I want to find the imaginary part of the complex number (a+i*b). say I have a function in the following form: ...
3
votes
1answer
81 views

Is there a simple way to get the Conjugate of a complex term, provided all information is available?

I've found a few threads addressing this, but they all seem to have pretty clunky answers, and it's hard to believe there's not a more elegant solution in Mathematica: Remove annoying Conjugate a ...
4
votes
3answers
427 views

Solving complex equations with variable exponents

How would I go about solving $(1+i)^n = (1+\sqrt{3}i)^m$ for integer $m$ and $n$? I have tried ...
1
vote
1answer
58 views
0
votes
1answer
105 views

Solve this equation explicitly for $\beta$

Solve this equation explicitly for $\beta$: $$T_{12}=\frac{x_{11}+x_{22}}{1+|\beta|^2}\beta$$ $x_{11}$ and $x_{22}$ are positive real numbers where $x_{11}>x_{22}$ and $T_{12}$ and $\beta$ are ...
0
votes
2answers
57 views

Unable to simplify the real part of a complex expression

Simplifying this expression shouldn't be hard. I tried adding assumptions asserting that all involved variables are real, but still the Re expression remains ...