Cauchy-Riemann Equations
Introduction
The complex analysis field of mathematics introduces a set of equations known as the Cauchy-Riemann equations. These equations are a significant part of complex analysis and are fundamental to the study of functions of a complex variable. They are named after Augustin-Louis Cauchy and Bernhard Riemann, two mathematicians who made significant contributions to the field of complex analysis.
Definition
The Cauchy-Riemann equations are a set of two partial differential equations that are used in the field of complex analysis to study functions of a complex variable. They are given by:
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
Here, u and v are real-valued functions of two real variables, x and y, and are usually considered to be the real and imaginary parts of a function of a complex variable z = x + iy.
Derivation
The derivation of the Cauchy-Riemann equations begins with the definition of a complex function. A function f(z) of a complex variable z = x + iy, where i is the imaginary unit, is said to be complex differentiable (or holomorphic) at a point z0 in its domain if the limit
f'(z0) = lim (z->z0) [f(z) - f(z0)] / (z - z0)
exists. This limit is a complex number and gives the derivative of f at z0.
Properties
The Cauchy-Riemann equations have several important properties that are used in complex analysis. These include:
1. If a function f(z) = u(x, y) + iv(x, y) satisfies the Cauchy-Riemann equations at a point z0 in its domain, then f is complex differentiable at z0.
2. If f is complex differentiable at every point in a region D, then f is said to be holomorphic in D. In this case, f also satisfies the Cauchy-Riemann equations at every point in D.
3. If f is holomorphic in a region D, then the derivatives of u and v exist and are continuous in D. This means that u and v are real differentiable in D.
Applications
The Cauchy-Riemann equations are used in various fields of mathematics and physics, including fluid dynamics, image processing, and quantum mechanics. They are also used in the proof of the fundamental theorem of algebra, which states that every non-constant polynomial has at least one complex root.