Cauchy Integral Formula | Complex Analysis, Part 20

In this episode of the Complex Analysis series, Prof. Happy Strawberry from the F.I.T. Department of Mathematics connects everything you’ve learned to one of the most powerful results in the subject: the Cauchy Integral Formula. We begin by computing the classic integral \oint_{|z|=1} \frac{1}{z}\,dz by direct parametrization using z = e^{it}. The result is: 2\pi i Now comes the key insight. We rewrite the integrand as \frac{1}{z} = \frac{1}{z - 0} which fits the general form of the Cauchy Integral Formula: \oint_\gamma \frac{f(z)}{z-a},dz = 2\pi i , f(a) This formula allows us to compute integrals instantly, without parametrization , as long as certain conditions are satisfied: * f is holomorphic inside the curve * The curve is closed * The point a lies inside the curve * No other singularities are present We then apply the formula to a more complex example: \oint_{|z|=1} \frac{z^2 + 3}{z}\,dz Instead of a long computation, we simply evaluate the function at z=0, giving: 6\pi i Key Takeaway The Cauchy Integral Formula turns difficult integrals into simple evaluations. Once the conditions are met, integration becomes substitution.