Reimann Surface e^(z^2)

This 3D object is made for Math 401: Mathematics Through 3D Printing at George Mason University under course instructor Dr. Evelyn Sander.

For this assignment, we are exploring Riemann Surfaces by looking at complex valued function of (z) where it can be divided into real and imaginary parts on the complex plane. I choose f(z) = e^(z^2) which is known as a holomorphic function. Holomorphic function essentially means it is complex differentiable in some neighborhood of a point that belongs to the domain in complex C space. The function f(z) = e^(z^2) is the composition of e^z and z^2.

The Cauchy-Riemann holds for this situation where we know that z = x+iy. Therefore, with having z^2, we can derive z^2 = x^2-y^2+2ixy. Thus, e^(z^2) = e^(x^2-y^2)*(cos(2xy)+i sin(2xy). For my object, I am plotting the real part e^(x^2-y^2)*cos(2xy) and imaginary e^(x^2-y^2)*sin(2xy) by parameterizing the equation. Using ParametricPlot3D on Mathematica, I used the intervals x from -1 to 1 and similarly for y. As I increase the interval for x, the object is intensively increasing its oscillation vertically, where increasing the y interval, the functions spreads out to the right and left (horizontally). We can see that the function has similar features to hyperbolic object due to the (x^2-y^2) but in exponential form.

The real and imaginary objects are translated in y and z directions to make sure they lay on the base accordingly. The base is created using ParametricPlot3D and then with Show command, I am able to put everything into one object. This object was printed on Makerbot Printer using Metallic Purple PLA filament and breakaway supports with a raft where it took approximately 2h40m to complete. There were several issues during the print because initially, my object was created with very little thickness, causing the printing to have problem with preciseness and not able to have the functions to have enough point of contact to the base.

Overall, riemann surfaces is very important in mathematics, especially as it serves as the core of complex analysis. Using the concept, we are able to study the behaviors of functions, curves, and surfaces. Moreover, we can consider analytic functions and use them to observe how complex plane C and Euclidean planes are related when mapped.