Complex Analysis

Complex analysis is a branch of mathematics that studies complex numbers and the functions of complex variables.

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 i^2 = -1. The real part of the complex number is a, and the imaginary part is b.

Functions of a Complex Variable

Functions of a complex variable are functions that take complex numbers as inputs and produce complex numbers as outputs. These functions can exhibit remarkably different behavior compared to functions of real variables.

Analytic Functions

A function is analytic if it is differentiable at every point in its domain. For functions of a complex variable, being differentiable at a point also implies that the function is differentiable in a neighborhood of that point, a property that does not hold for functions of a real variable.

Contour Integration

Contour integration is a method of evaluating certain integrals along paths in the complex plane.

Cauchy's Theorem and Residue Theorem

Cauchy's theorem is a fundamental theorem in complex analysis which states that if a function is analytic in a simply connected domain, then the integral of the function around any closed contour in that domain is zero. The residue theorem, which is a direct consequence of Cauchy's theorem, provides a method for calculating contour integrals by summing the residues (values related to singularities) of a function.

Conformal Mapping

Conformal mapping is a technique used in complex analysis and applied to many physical problems. A conformal map is a function that preserves angles locally, meaning it maps infinitesimal angles from the domain to the same angle in the range.

Complex analysis has many applications in physics, engineering, and particularly in quantum mechanics and electrodynamics, where complex numbers are fundamentally tied to the nature of the physical world.