Complex analysis MOC

Cauchy-Riemann equations

The Cauchy-Riemann equations are conditions imposed on the form of a complex-valued analytic (holomorphic) function. which naturally arise from the properties of complex arithmetic and Complex function decomposition. Namely, if a function is differentiable, then

which may be written in the equivalent complex form

As an immediate consequence of this, any holomorphic function will satisfy Laplace's equation when considered as a vector field . It also follows that integrals over complex numbers are path-independent, i.e. as a vector field holomorphic functions are irrotational. This simplifies Complex integration.

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