Parametrizing Circles

Complex Analysis · Axiom Academy

Build the standard contour step by step, then read off its derivative and length. Parametrize the circle of radius 2 centered at z_0 = 1 + i , traversed once counterclockwise. Then find its derivative , its speed , and its total length. Circular contours like this one are the workhorses of complex analysis — they appear in Cauchy's integral formula and residue calculations — so getting the parametrization, orientation, and speed right is essential. Nicely done. You built a circular contour from scratch and recovered its derivative, constant speed, and length — and along the way confirmed that the length of a circle of radius r is . The master formula: a circle of radius r centered at z_0 , counterclockwise, is for — scale e^ it by r , then translate by z_0 . Derivative and speed: , so is constant. The tangent ire^ it is rotated , which is why it always points counterclockwise. This problem: gives speed 2 and length . The same template, four common variations: Every one of these is "take a known parametrization and transform it." Master the circle and the rest of contour integration has a foundation to stand on.

This is the written version of the interactive lesson above. See the full Complex Analysis course.