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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-3
votes
0answers
29 views

Get function creating variables inside [on hold]

I want to use Mathematica to create this function in pseudocode: ...
10
votes
1answer
374 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 ...
1
vote
1answer
56 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 ...
0
votes
1answer
52 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 ...
7
votes
1answer
144 views

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

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) = log (r) + i \pi \\ ...
4
votes
3answers
262 views

Complex conjugate

How do I complex conjugate a vector? ...
1
vote
1answer
30 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
133 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
51 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
votes
1answer
44 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
votes
1answer
41 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
2answers
78 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 ...
1
vote
0answers
31 views

Fourier transform of a real white noise on a 3D cubic lattice [migrated]

I'm facing the following problem: I have a cubic domain of side $L=2\pi$; this domain is divided in a cubic grid, each side is divided in $N$ points, where $N$ is an even integer number. the ...
8
votes
2answers
97 views

Pattern matching with Complex. Feature or a bug?

I have run into some strange behavior while doing some pattern matching. First, this works as expected: Exp[2 I u x] /. Exp[Complex[0, a_] u x] :> a (* 2 *) ...
0
votes
1answer
84 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 ...
4
votes
2answers
126 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]? ...
0
votes
1answer
45 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
1answer
137 views
2
votes
1answer
78 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
vote
1answer
89 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? ...
0
votes
0answers
46 views

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

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
44 views

Plot of the set complex numbers [duplicate]

I want to show this set (eigenvalues of my matrix): ...
1
vote
0answers
46 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 ...
0
votes
1answer
87 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
votes
1answer
87 views

Is it possible to have Logarithm with base 1 or 0? [closed]

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)?? Out-coming question is if you can define a logarithm with ...
1
vote
3answers
1k views

How to take conjugate of a function?

Naïvely this is what happens and it obviously is not helpful! ...
1
vote
1answer
59 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? ...
3
votes
1answer
108 views

Decomposing a complex numbers equation into its real and imaginary part

I have an equation such as: a*z^2 + b*(z - c)^2 == d where z is a complex variable and ...
1
vote
2answers
561 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, ...
8
votes
4answers
6k views

Factoring polynomials to factors involving complex coefficients

I've run into some problems using Factor on polynomials with complex coefficient factors. Reading the documentation it looks like it only factors over the ...
0
votes
1answer
90 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
232 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. ...
-2
votes
1answer
95 views

Why is it so complicated to simplify a complex conjugate? [closed]

My question is very simple. I would like to multiply a complex number by its conjugate. Unfortunately, this seems to be extremely complicated in Mathematica. Consider the very basic example: ...
0
votes
0answers
40 views

Simplification of a complex-valued expression with purely real components

(I'm including all the information here, as I'm not sure how much of it is necessary to solve the problem. For a tl;dr skip to the bottom) I've been working on a problem involving calculating ...
0
votes
1answer
127 views

How find solve the equation complex conjugate? [closed]

Tried solving it with every function I know but couldn't z*=z^2
0
votes
1answer
101 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
87 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
106 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
45 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 ...
0
votes
0answers
80 views

ComplexExpand does not treat everything as real number

ComplexExpand does not treat everything as real number?? Please read the following simple example ...
36
votes
6answers
5k 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 ...
4
votes
1answer
142 views

Optimizing a function containing a complex exponential

I've noticed the following significant different in performance for different formulations of the same function: ...
7
votes
3answers
469 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 $v=2xy$, ...
18
votes
1answer
856 views

Mobius transformations revealed

Mobius Transformations Revealed is a short video that vividly illustrates the simplicity of Mobius transformations when viewed as rigid motions of the Riemann sphere. It was one of the winners in the ...
2
votes
1answer
141 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 ...
2
votes
2answers
228 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 ...
0
votes
0answers
129 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} ...
3
votes
2answers
204 views

Contourplot of complex Roots

Suppose I have this equation: $$ z^2 + 3z + (x^2 + y^2) = 0 $$ I want the real and complex contour plot of $z(x,y)$. Analytically, the real/imaginary boundary is separated by condition $x^2 + y^2 ...
4
votes
2answers
661 views

Why does this integral have a complex component?

I wanted to find the probability of my normally-distributed random variable being at least 15, so I set up this integral: ...
4
votes
2answers
118 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 $\beta \in [0,\pi]$: ...