Complex Surfaces 3D Object

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Michael Tritle MATH 401 3D Printing October 29, 2019 The complex numbers are a generalization of the real numbers. Complex numbers can be written in the form x + yi where x,y are elements of the set of real numbers, and i is an imaginary number, which is sqrt(-1). Essentially, we view complex numbers as a combination of real and imaginary numbers, in that complex numbers have 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. It's standard that the x - axis represents the real part, and the y - axis represents the imaginary component. So, the complex number 2 + 3i would be analogous to the point (2, 3) on the usual Cartesian Plane. If we were are to a consider a function of a single real variable, we see that any input from the functions domain maps to a single real valued output. Functions of a complex variable are the same in that any complex input from the functions domain maps to a complex valued output. To represent a complex number geometrically, we need to 2 dimensions. 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, we need 4 dimensions to graph the relationship between its domain and range. This becomes a challenge since through usual graphing conventions, we visualize at most 3 dimensions. 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) 1) 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 let 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)).