Cauchy Formula for Derivatives | Complex Analysis, Part 21

In this episode of the Complex Analysis series, Prof. Happy Strawberry from the F.I.T. Department of Mathematics extends the Cauchy Integral Formula to handle higher powers in the denominator. We start from the basic result: \oint_\gamma \frac{f(z)}{z-a}\,dz = 2\pi i\, f(a) and ask: What happens if the denominator is (z-a)^2, (z-a)^3, or higher? The answer leads to a powerful generalization: \oint_\gamma \frac{f(z)}{(z-a)^{k+1}},dz = \frac{2\pi i}{k!} f^{(k)}(a) This formula shows a deep connection: * Higher powers in the denominator → higher derivatives of f Examples \oint \frac{e^z}{(z-1)^2}\,dz → First derivative → result: 2\pi i\,e \oint \frac{e^z}{(z-1)^3}\,dz → Second derivative → result: \pi i\,e Key Insight Complex integration can compute derivatives instantly. Instead of long calculations, we evaluate a derivative at a single point. This is one of the most powerful tools in complex analysis and a foundation for more advanced techniques like residues.