Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. It involves the manipulation and analysis of functions that have complex numbers as their inputs and outputs.
An entire function is a complex-valued function that is defined and analytic on the entire complex plane. In other words, it does not have any singularities or poles. Examples of entire functions include polynomials and trigonometric functions.
Some key concepts in complex analysis include analyticity, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. These concepts are used to study the behavior of complex functions and their properties.
Complex analysis has various applications in science, particularly in physics and engineering. It is used to solve problems involving electric circuits, fluid dynamics, and quantum mechanics. It also plays a crucial role in the development of mathematical models and theories in these fields.
Some common techniques used in complex analysis include power series, Laurent series, and residues. These techniques are used to find the behavior of complex functions, calculate integrals, and determine the presence of singularities.