Cauchy's Theorems Summary

Complex Analysis · Axiom Academy

Section 7 wrapped: how a single closed-loop integral pins down everything an analytic function does inside it. If f is analytic on a simply connected domain, every closed contour integral of f is zero — contour integrals become path-independent. The Cauchy Integral Formula recovers any interior value f(z_0) from f 's values on a surrounding contour: interior is determined by boundary. The same formula, differentiated, shows analytic functions have derivatives of all orders — each given by its own contour integral. These results force a deep rigidity: an analytic function is locked in by its values on any single curve (Liouville, Maximum Modulus, the Fundamental Theorem of Algebra all follow). If f is analytic everywhere on and inside a closed contour lying in a simply connected domain, the loop integral vanishes. Equivalently, the integral between two points is independent of the path . When to use: the integrand has no singularities inside the loop. Watch out for: a single pole inside breaks it — then the value need not be 0 . Core Concept Cauchy Integral Formula For f analytic on and inside and z_0 interior to , the value at z_0 is fixed by the values of f on the contour. The boundary knows the interior. When to use: the integrand is with f analytic and z_0 inside. Watch out for: if z_0 is outside , the integral is 0 . Core Concept Derivative Formula

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