The Deformation Principle

Complex Analysis · Axiom Academy

EXAMPLE The Deformation Principle Deform a messy contour into a tidy circle, then finish with Cauchy's integral formula. Let C be the ellipse traced counterclockwise that encloses the point z = 2 . Evaluate the contour integral We never need the ellipse's exact shape — the Deformation Principle lets us swap it for a circle. The integrand is analytic everywhere except the pole at z = 2. In the shaded ring between C and the small circle C ᵣ it is analytic, so the two integrals are equal. Nice work. You evaluated a contour integral without ever computing a single parametrization — the geometry did the heavy lifting. Deformation Principle: a contour can slide to any other contour through a region where the integrand is analytic, and the integral doesn't change. The messy ellipse and a tidy circle give the same value. Cauchy Integral Formula: once the integral has the shape with f analytic inside and a enclosed, it equals — no integration required. Result: with f(z) = z^2 + 1 and a = 2 , the integral is . This pairing — deform to a circle, then read off the value with Cauchy's formula — is exactly the move behind residue calculus.

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