Complex Analysis: Theory and Applications · Problem 6.2.3

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · contour-integrals

Problem

Consider the integral

\displaystyle\int_C z^2\,dz

, where

C

is the positively oriented upper semicircle of radius 1, centered at 0. (a) Give a Riemann sum approximation for the integral by selecting

n = 4

and the points

z_k = e^{i\frac{k\pi}{4}}

(

k = 0,\ldots 4

) and

c_k = e^{i\frac{(2k-1)\pi}{8}}

(

k = 1,\ldots 4

). (b) Compute the integral exactly by selecting a parametrization for

C

and applying Theorem 6.1.