Lesson 7 · turning a region with holes into one connected boundary, so Cauchy’s theorem applies — f(z) = ez/z2 + 1/(z2+1), γ: |z| = 3
The z-plane 3 singularities, 3 crosscuts
width = —
∫ inward edge, all 3 corridors summed = —
∫ outward edge, all 3 corridors summed = —
net crosscut contribution (→ 0) = —
Why each γk must be a complete loop∮|z−z0|=r (z−z0)k dz = 2πi holds only for k = −1, and only when integrated over the entire 0→2π sweep — a partial arc gives a different, path-dependent value. However thin the corridor gets, each small circle is walked all the way around; only the corridor itself is walked there-and-back.∮(z−z0)−1dz = 2πi needs the full 0→2π sweep — a partial arc won’t do it. Each γk stays a complete loop; only the corridor is walked there-and-back.