Complex
analysis is the study of
Complex numbers together with their derivatives, manipulation, and other properties.
Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems.Countour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the
function in regions of the complex plane near and between the limits of integration.
Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible.
The key result in complex analysis is the Cauchy integral theorem, which is the reason that single-variable complex analysis has so many nice results. A single example of the unexpected power of complex analysis is Picard''s great theorem, which states that an
analytic function assumes every complex number, with possibly one exception, infinitely often in any neighbourhood of an essential singularity.
A fundamental result of complex analysis is the Cauchy-Riemann equations which give the conditions a function must satisfy in order for a complex generalization of the derivative, the so-called complex derivative, to exist. When the complex
derivative is defined "everywhere," the function is said to be analytic.
More abstracts about the Complex Numbers analysis