Complex Analysis: Theory and Applications · Problem 6.5.18

Taylor and Laurent Series — Howell & Matthews

Mathematics — Complex Analysis · cauchy-integral-formula

Problem

Let ff be analytic in the simply connected domain DD and let z1z_1 and z2z_2 be two complex numbers that lie interior to the simple closed contour CC having positive orientation that lies in DD. Show that f(z2)−f(z1)z2−z1=12πi∫Cf(z)(z−z1)(z−z2) dz.\frac{f(z_2)-f(z_1)}{z_2-z_1} = \frac{1}{2\pi i}\int_C \frac{f(z)}{(z-z_1)(z-z_2)}\,dz. What happens when z2→z1z_2 \to z_1? Why?

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