Generalized Cauchy Formula

Complex Analysis · Axiom Academy

LESSON The Generalized Cauchy Formula One loop around a point recovers not just the function there — but every derivative it will ever have. Fix a point a inside a simple closed contour C , with f analytic on and inside it. Cauchy's formula recovers f(a) from f on C . The generalized formula says the same data on the same loop also gives f (n) (a) for every order n — just raise the denominator's power and multiply by n! . …delivers the n-th derivative at a Start from the base formula and differentiate with respect to a , slipping the derivative inside the integral. The only place a appears is the denominator, and one differentiation drops a power while pulling out a factor. Repeat n times: the powers and the factors accumulate into n! and (z − a) n+1 . The original Cauchy formula: f(a) = (1/2πi) ∮ f (z)/(z − a) dz. d/da [(z − a) −n ] = n·(z − a) −n−1 — power up by one, factor n pulled out. After n steps the pulled-out factors multiply to 1·2·⋯·n = n! . Each step adds one to the exponent, ending at (z − a) n+1 in the denominator. Assume the formula for n ; differentiate both sides in a . The right side gains exactly the factor and power needed to read as the formula for n + 1 — so it holds for all n ≥ 0. 3. Reading an Integral Off the Plane

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