Cauchy Integral Formula

Complex Analysis · Axiom Academy

LESSON Cauchy's Integral Formula An analytic function's value at any interior point is locked in by its values on the boundary alone — and one integral hands it to you. 1. One Point, From the Whole Boundary Let f be analytic on and inside a simple closed contour C , traversed once counterclockwise, and let a be any point strictly inside C . Cauchy's Integral Formula says a single contour integral recovers the value f(a) : the integral runs over the boundary C only yet it pins down the interior value f(a) exactly f is analytic on C and on the region it encloses, C is simple — it does not cross itself, C is positively oriented (counterclockwise), a lies strictly inside C (not on it). 2. Why It Works: Shrink the Contour Onto a Look at the integrand . Since f is analytic, g is analytic everywhere except the single point z = a , where the denominator vanishes. By the deformation principle, we may slide C inward onto any small circle C_r centered at a — sweeping only through territory where g is analytic — and the integral does not change. Any shape, as long as it encloses a once. Its exact path will turn out not to matter. g(z)=f(z)/(z-a) blows up only here. Everywhere else inside C it is analytic. A small circle |z-a|=r . We are free to make r as tiny as we like. No poles are crossed during the deformation, so the two integrals are equal.

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