# Complex Analysis

Complex analysis is a branch of pure mathematics that grew out of arithmetic and algebra. Once complex numbers were accepted, mathematicians extended real analysis and calculus to this new domain. Complex analysis continues to be a field of study in pure mathematics. It is also used in more applied fields such as physics and engineering. To study complex analysis you need a solid background in multivariable calculus and a facility with logic and proofs.

The invention of complex numbers dates at the latest from the 16th century AD when Girolamo Cardano investigated and publicized the algebraic solutions to the cubic and quartic equations. The solution formulas involved expressions such as , even when the solutions were real. These expressions were not considered to be numbers, and caused a great deal of hand-wringing and controversy at the time.

Eventually mathematicians put the complex numbers on a solid theoretical footing. A new symbol, i, was invented to represent the principal solution to the equation x2 = -1, and new numbers of the form a + b·i were added to the objects of arithmetic. The familiar laws of real arithmetic were assumed to hold, and the complex numbers were born.

It was natural to progress to studies of complex-valued functions f : , f : , etc., to ask what continuity, differentiability, and integrability meant in this context, to explore sequences and series of complex numbers and functions, and generally to incorporate the complex number system into existing mathematics. Some of the familiar theory retains its old form, some is more subtle, and some of complex analysis is richer, more beautiful, and unexpected.

One of the most famous unsolved problems in mathematics, the Riemann Hypothesis, is in the field of complex analysis. It is one of the seven Millenium Problems and carries a prize of \$1 million for its solution. Surprisingly, the problem links the study of complex functions to the prime numbers.