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Introduction to Complex analysis

The complex analysis is usually termed as the theory of functions of a complex variable. It is a category of mathematical analysis which investigates the functions of complex numbers. It is helpful in numerous branches of mathematics, including the applied mathematics and number theory; as similar as in physics, that includes thermodynamics, electrical engineering and hydrodynamics.

Murray R. Spiegel explained the complex analysis as "one of the most beautiful and also useful branches of Mathematics".

The complex analysis is specifically concerned with the analytic functions of complex variables (more generally, meromorphic functions). Since the separate real and imaginary sections of any analytic function should satisfy Laplace's equation, and complex analysis is broadly applicable to 2-dimensional problems in physics.

The complex analysis is specifically the theory of conformal mappings, which has too many physical applications and is also used via analytic number theory. In modern times, it has become too much popular through a new boost from complex dynamics and the pictures of fractals formed by iterating holomorphic functions. One more significant application of complex analysis is in the string theory that studies conformal invariants in quantum field theory.

**Complex functions**

A complex function is a function whose domain and range are the subsets of complex plane. More accurately, a complex function is one in which the dependent variable and the self-governing variable are both complex numbers. For any complex function, both the dependent variable and the self-governing variable might be separated into real and imaginary parts:

*z = x + iy* and

*w = f(z) = u(x, y) + iv(x, y)*

Where *x, y? R* and *u(x, y), v(x, y)* are real-valued functions.

**Holomorphic functions**

Complex functions that are defined on an open subset of the complex plane are differentiable are known as Holomorphic functions. The Complex differentiability has much stronger consequences than normal (i.e., real) differentiability. For illustration, holomorphic functions are infinitely differentiable, whereas some of the real differentiable functions are not infinitely differentiable. Most of the basic functions, involve trigonometric functions, exponential function and all the polynomial functions which are holomorphic.

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