Multiple Poles (Split the Integral) | Complex Analysis, Part 22

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 multiple poles inside one contour. The key idea is simple but powerful: A large contour enclosing several poles can be split into smaller contours, each surrounding exactly one pole. This allows us to compute the integral as a sum of contributions. We apply this to the example: \oint \frac{e^z}{z(z-1)^3}\,dz where the contour contains two poles: * z = 0 (simple pole) * z = 1 (higher-order pole) We split the integral into two parts: 1. Around z=0 → use the basic Cauchy formula 2. Around z=1 → use the higher-order Cauchy formula After computing both contributions, we obtain: \pi i \,(e - 2) Key Insight Each pole contributes its own term The total integral is the sum of all contributions This idea is the bridge to the Residue Theorem, where this process becomes systematic and extremely powerful.