Complex Analysis
Introduction
Complex analysis is a branch of mathematics that studies functions of complex numbers. It is a fundamental area of mathematics with applications in many fields, including physics, engineering, and computer science. Complex analysis is known for its beauty and its surprising power.
Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i² = −1. The real part of the complex number is a, and the imaginary part is b.
Functions of a Complex Variable
A function of a complex variable is a function whose domain and range are subsets of the complex numbers. Such functions are generally expressed in terms of the complex variable z = x + iy, where x and y are real numbers.
Analytic Functions
An analytic function is a function that is locally given by a convergent power series. In complex analysis, every differentiable function is analytic, a property that does not hold for differentiable functions in real analysis.
Complex Integration
Complex integration is a fundamental concept in complex analysis. The most important result in this area is the Cauchy's integral theorem, which states that if a function is analytic in a simply connected domain, then its integral around any closed curve in that domain is zero.
Residue Theory
Residue theory is a powerful tool in complex analysis that is used to evaluate real integrals, among other things. The central concept is that of a residue, which is the value of a certain integral around a singularity of a function.
Applications
Complex analysis has many applications in various fields. In physics, it is used in quantum mechanics and the theory of relativity. In engineering, it is used in signal processing and control theory. In computer science, it is used in the analysis of algorithms.