Complex analysis is a branch of mathematics that deals with the study of functions of complex numbers. It involves the investigation of complex-valued functions and their properties, such as differentiability, integration, and convergence.
Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). They consist of a real part (a) and an imaginary part (bi).
Real analysis deals with the study of real-valued functions, while complex analysis deals with the study of complex-valued functions. Additionally, complex analysis involves the use of techniques such as the Cauchy-Riemann equations and contour integration, which are not used in real analysis.
Complex analysis has various applications in different fields, including engineering, physics, and economics. It is used to solve problems involving electric circuits, fluid dynamics, signal processing, and many others. It is also essential in the study of quantum mechanics and general relativity.
Some important theorems in complex analysis include the Cauchy integral theorem, Cauchy integral formula, and the residue theorem. These theorems are used to evaluate complex integrals and to calculate the values of functions at certain points. Other important theorems include the maximum modulus theorem, Rouche's theorem, and the fundamental theorem of algebra.