Complex analysis MOC

Complex integration

Given a complex function , its complex integral along a path is

However, if we require be analytic, the Cauchy-Riemann equations imply that this integral should be path-independent.

Cauchy's Integral Theorem If is an analytic complex function for a closed region , then

That is, is defined inside and on a closed path .

Note that if contains a singularity and is hence non-analytic, then Cauchy's Integral Theorem does not apply and the path integral may be non-zero.¹ If does have singularities but they lie outside , the theorem still holds.

Core integration techniques

• Cauchy's Integral Formula
• Cauchy's Residue Theorem

#state/tidy | #lang/en | #SemBr