Complex Surfaces 3D Object

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Michael Tritle MATH 401 3D Printing October 29, 2019 Complex numbers are a generalization of real numbers. They can be written in the form x + yi where x and y are elements of the set of real numbers, and i is an imaginary number which equals sqrt(-1). Essentially we view complex numbers as a combination of real and imaginary numbers, with complex numbers having both a real and an imaginary part. To represent real numbers visually we consider a number line. To represent complex numbers visually we use a plane where the x-axis represents the real part and the y-axis represents the imaginary component. Therefore the complex number 2 + 3i is analogous to the point (2,3) on the usual Cartesian Plane. If we consider a function of a single real variable, any input from the function's domain maps to a single real-valued output. Functions of a complex variable are similar in that any complex input from the function's domain maps to a complex-valued output. To represent a complex number geometrically, two dimensions are needed. We use a plane to represent the relationship between the domain and range of a function of a single real variable. For complex-numbered functions, four dimensions are required to graph the relationship between its domain and range, which becomes a challenge since we visualize at most three dimensions through usual graphing conventions. A way to visualize a complex function is to consider two options: (1) Let our (x,y) domain represent a complex variable; then we graph only the real component of the range of our complex-valued function. (2) Let our (x,y) domain represent a complex variable, then we graph only the imaginary component of the range of our complex-valued function. In the Mathematica notebook we consider letting the domain of our function represent a disk of radius 2 around the origin of the complex plane so as r -> 2 and 0 <= theta <= 2 pi, we see that our domain is x = r*cos(theta) y = r*sin(theta). Our complex function f is f(z) = sin(exp(z)^(2/3)) where z = x + iy, so f(z) = sin(exp(r*cos(theta) + i*r*sin(theta))^(2/3)).