# Tagged Questions

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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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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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, $\... 2answers 842 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: $$H_m^{(1)}(z) =\frac{i^{-m}}{\pi}\int_{-\pi/2 + i \infty}^{\pi/2 - i \infty} \exp[i m \... 3answers 237 views ### Wrapper for inexact numeric complex numbers that maintains polar form Related question: How can I convert a complex number a+b I to the exponent form A Exp(I phi)? Mathematica insists on displaying complex number in form a+I b when <... 4answers 434 views ### Speed up Schwarz-Christoffel mapping I would like to ask your advice how to speed up Contour and Parametric plots in the following example. Let as start by defining a function ... 3answers 740 views ### How do I plot the images of oriented curves under complex transformation? I'm working on a complex transformation sketch that I'd like to create via mathematica. I'm working with the function$$w=z^2\tag{0},$$where its real and imaginary parts are u=x^2-y^2 and v=2xy, ... 2answers 240 views ### Why does N[Re@f] give complex result? Consider this code: ... 2answers 235 views ### Confusion regarding the incomplete elliptic integral of the first kind I am trying to manipulate a conformal map from the half-plane to a square z \rightarrow w(z) defined by:$$ w(z) = \int \limits^{z} dx \frac{1}{\sqrt{(1-x^2)x}} = \sqrt{2} \; F\left(\sqrt{z+1},\... 1answer 273 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}] ...
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### 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 ...