Complex analysis

Augustin-Louis Cauchy 1901

Exponentials of complex number within unit circle-2

Complex-plot

Complex Analysis is a branch of mathematics that investigates functions of complex numbers. It is fundamental in many areas of mathematics, including algebraic geometry, number theory, and applied mathematics, as well as in physics, particularly in quantum mechanics and general relativity. Complex analysis, with its rich theory and deep connections to other areas of mathematics, is known for its beauty and utility.

Overview

Complex analysis focuses on the study of functions that map complex numbers to complex numbers. These functions, known as complex functions, have properties and behaviors that are profoundly different from those of real functions. The field is primarily concerned with analytic functions of a complex variable, meaning functions that are locally given by power series and are differentiable in the complex sense.

Key Concepts

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 with the property that i² = -1. The set of all complex numbers is denoted by ℂ.

Functions of a Complex Variable

Functions of a complex variable are expressions that take complex numbers as inputs and produce complex numbers as outputs. These functions can exhibit behaviors and properties not seen in real functions, such as conformal mapping and complex differentiation.

Analytic Functions

Analytic functions, also known as holomorphic functions, are functions that are differentiable at every point in their domain. The concept of differentiability in complex analysis is much stronger than in real analysis, leading to powerful results such as the Cauchy-Riemann equations and Liouville's theorem.

Integration and Cauchy's Theorem

Integration in complex analysis is performed over paths in the complex plane. Cauchy's theorem is a central result, stating that if a function is analytic and defined throughout a simple closed path and its interior, then the integral of the function over the path is zero. This theorem has numerous corollaries, including Cauchy's integral formula, which provides a means to evaluate integrals of analytic functions.

Singularities and Residues

Singularities are points at which a function does not behave analytically. They play a crucial role in the evaluation of complex integrals, particularly through the residue theorem, which allows the calculation of integrals around singular points using residues, which are coefficients from the Laurent series expansion of the function.

Applications

Complex analysis has wide-ranging applications in both pure and applied mathematics. In physics, it is used to solve problems in electrodynamics, fluid dynamics, and quantum field theory. In engineering, complex analysis is applied in signal processing and control theory. It also has applications in number theory, through the study of modular forms and L-functions.

See Also

• Real Analysis
• Differential Equations
• Fourier Series
• Riemann Surfaces