Complex analysis

Complex analysis studies functions of complex variables, especially those differentiable in the complex sense. Its central results connect local differentiability with strong global structure.

Complex number: A number expressible as a real part plus an imaginary part, where the imaginary unit squares to −1. Complex-valued inputs and outputs are the basic objects of the subject.
Contour integral: An integral of a function along an oriented curve in the complex plane. Integrating along closed curves is the starting point for Cauchy’s integral theorem.
Argument principle: A theorem relating the winding of a function’s values around zero to its zeros and poles inside a contour. It counts roots without solving for them individually.
Real analysis: The study of real-valued and real-variable functions, limits, integrals, and related structures. Complex differentiability imposes constraints absent from ordinary real differentiability.
Complex function: A function whose domain, codomain, or both involve complex numbers. These are the objects whose continuity, derivatives, and integrals complex analysis studies.
Cauchy's integral theorem: A theorem stating that the integral of a holomorphic function around a suitable closed contour is zero. It explains why contour integrals depend on domain topology and enclosed singularities.
Rouché's theorem: A theorem comparing two holomorphic functions on a contour to show they have equal numbers of zeros inside. It provides a practical way to count zeros by comparison.
Quaternionic analysis: The study of functions of quaternion variables, where multiplication is noncommutative. It extends complex-variable ideas into a setting where familiar holomorphic theory changes substantially.
Complex differentiability: Differentiability of a complex function under a limit that is independent of the direction of approach. Its direction-independent limit is stricter than ordinary real differentiability.
Cauchy's integral formula: A formula expressing a holomorphic function’s interior values through its values on a surrounding contour. It yields derivative formulas and strong bounds from boundary data.

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