# Complex analysis

{{#invoke:Hatnote|hatnote}}Template:Main other Plot of the function f(x) = (x2 − 1)(x − 2 − i)2 / (x2 + 2 + 2i). The hue represents the function argument, while the brightness represents the magnitude.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics and thermodynamics and also in engineering fields such as; nuclear, aerospace, mechanical and electrical engineering.

Murray R. Spiegel described complex analysis as "one of the most beautiful as well as useful branches of Mathematics".

Complex analysis is particularly concerned with analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

## History

Complex analysis is one of the classical branches in mathematics with roots in the 19th century and just prior. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. 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 has become very popular through a new boost from complex dynamics and the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in string theory which studies conformal invariants in quantum field theory.

## Complex functions

A complex function is one in which the independent variable and the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane.

For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts:

$z=x+iy\,$ and
$w=f(z)=u(x,y)+iv(x,y)\,$ where $x,y\in \mathbb {R} \,$ and $u(x,y),v(x,y)\,$ are real-valued functions.

In other words, the components of the function f(z),

$u=u(x,y)\,$ and
$v=v(x,y),\,$ can be interpreted as real-valued functions of the two real variables, x and y.

The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponential functions, logarithmic functions, and trigonometric functions) into the complex domain.

## Holomorphic functions

{{#invoke:main|main}} Holomorphic functions are complex functions defined on an open subset of the complex plane that are differentiable. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, whereas some real differentiable functions are not. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic.