Complex Analysis

Complex analysis is the study of complex numbers together with their derivatives, manipulation and other properties. It is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. The key result in complex analysis is the Cauchy Integral Theorem, which is the reason that single- variable complex analysis has so many positive results. An example of the influence of complex analysis is Picard’s theorem, which states that an analytic function assumes every complex number, with one possible exception. A fundamental result of complex analysis is the Cauchy- Riemann equations, which gives the conditions that a function must satisfy in order form a complex generalization of the derivative. When the complex derivative is defined, the function is said to be analytic.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating functions of complex numbers. It is useful in branches of mathematics, including number theory and applied mathematics, as well as in physics, including hydrodynamics, thermodynamics and electrical engineering. Murray Spiegel describes complex analysis as “one of the most beautiful as well as useful branches of mathematics”.

Complex analysis is particularly concerned with the analytic functions of complex variables or more generally meromorphic elements. Complex analysis is widely applicable to two- dimensional problems in Physics, as the separable real and imaginary parts of any analytic function that must satisfy Laplace’s equation.

History

Complex analysis is considered as one of the classical branches with its roots in the 19^th century. Important names are Euler, Gauss, Riemann, Cauchy, Weierstrass and many more in the 20^th century. Traditionally, complex analysis in particular the theory of conformal mappings has many physical applications and is also used throughout analytic number theory. In modern times, it became very popular through a new boost of complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis today is in string theory which is a conformally invariant quantum field theory.

Complex functions

A complex analysis is a function in which the independent variable and the dependent variable are both complex numbers. For any complex function, both the independent and the dependent variable may be separated into real and imaginary parts.

Holomorphic functions

Holomorphic functions are complex functions defined on open subset of the complex plane which are differentiable. For instance holomorphic functions are infinitely differentiable, whereas some real differentiable functions are not. Complex differentiability has much stronger consequences than usual differentiability.

Questions:

• What is Complex Analysis?
• What are complex functions and holomorphic functions?