Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that studies complex-valued functions of one or more complex variables. It is used in many branches of mathematics, including functional analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. It also has applications in engineering fields, such as nuclear, aerospace, mechanical, and electrical engineering. Complex analysis primarily studies holomorphic functions, which are differentiable functions of a complex variable. In contrast with a differentiable real function, a holomorphic function is always infinitely differentiable and equal to the sum of its Taylor series in some neighborhood of each point of its domain. This makes methods and results of complex analysis significantly different from those of real analysis. Complex functions also behave very differently under contour integration: the integral of a holomorphic function over a contour in the complex plane does not depend on the details of the contour, only on how it winds around the singularities of the function. Being able to move a given contour to a more suitable one often leads to major simplifications in proofs. The theory of several complex variables generalizes one-variable complex function theory to more than one complex dimension. While many of the techniques of a single complex variable are used and generalized in this setting, the theory of several complex variables makes use of additional techniques such as Banach algebras and sheaf theory. It is often concerned with questions of interest in algebraic geometry and symmetric spaces.