Complex Analysis

Introduction Complex functions are crucial in mathematical analysis, with applications in engineering and physics. Visualizing these functions helps to understand their behavior in the complex plane, revealing patterns and symmetries. This aids in both grasping theoretical concepts and exploring new properties. Domain Colring Domain coloring is a technique for visualizing complex functions by mapping the phase of complex numbers to colors. This creates a colorful representation of the complex plane, making it easier to observe function behaviors like zeros, poles, and overall structure. It’s a valuable tool in mathematics for enhancing understanding and analysis of complex functions. ...

5.5 Curvatures of Minimal Surface When a minimal surface $r(u,v) = (x(u,v),y(u,v),z(u,v))$ is represented by an isothermal coordinate system, we define $\phi_1(w), \phi_2(w), \phi_3(w), f(w)$ and a meromorphic function $g(w)$ as: $$ \phi_1(w) = \frac{\partial x}{\partial u} - i \frac{\partial x}{\partial v}, \quad \phi_2(w) = \frac{\partial y}{\partial u} - i \frac{\partial y}{\partial v}, \quad \phi_3(w) = \frac{\partial z}{\partial u} - i \frac{\partial z}{\partial v}, \quad f = \phi_1 - i\phi_2, \quad g = \frac{\phi_3}{\phi_1 - i\phi_2} $$ ...

Isothermal Coordinate System In the previous two sections, we have not been able to give a clear definition of minimal surfaces. In this section, we define minimal surfaces, compute their curvature, and find their Gauss map. Let us start with the definition. Definition of Minimal Surface We say a surface is a minimal surface if its mean curvature $H = 0$. Definition of Isothermal Coordinates Let $r = r(u, v)$ be a regular parametrized surface. We say that $r$ is isothermal if: ...

It is easy to see that the equation for a minimal surface is a partial differential equation. In this section, an attempt will be made to find its two famous special solutions: the catenoid and the helicoid. After that, we will give two other examples of minimal surfaces: Enneper’s surface and Scherk’s surface. Given the minimal surface equation: $$ f_{xx}(1 + f_y^2) - 2f_x f_y f_{xy} + f_{yy}(1 + f_x^2) = 0, $$ ...