The map f (z) on the complex plane near the point z is in nitesimally expanded and twisted by f (z). In particular, a small disk is shrunk exactly when |f (z)| < 1. Since f (z) = 1 z2 , this happens for |z| > 1. images and identify the geometric objects they trace out, with proof. 1 z a iy a2 + y2 , with real and imaginary parts u = a a2 + y2 , v = y a2 + y2 . These satisfy or, equivalently, u2 + v2 = a2 + y2 (a2 + y2)2 = 4a2 which is the equation of a circle with radius 1/2|a| and center (1/2a, 0); that is, the image of a vertical line is a circle with center on the x-axis, passing throught the origin. 1 (2) evaluate the following sums and integrals: (b) z dx x4 + 1.