👉 The theory of Complex Analysis, or simply "ts" for short, is a branch of mathematics that studies the properties and behavior of functions defined on the complex plane, which is the extended number line including both real and imaginary axes. It leverages powerful analytic tools from calculus, such as contour integration and residue theory, to solve problems in analysis, number theory, and physics. Key concepts include analytic functions, which are complex functions that are differentiable at every point in their domain, and singularities, points where a function fails to be analytic. The residue theorem, a fundamental result in ts, allows for the evaluation of complex integrals by summing the residues of poles (points where a function becomes infinite) enclosed by a closed contour. This theory has profound applications in solving differential equations, evaluating real integrals, and understanding the behavior of physical systems in fields like quantum mechanics and fluid dynamics.
