Complex Analysis: Theory and Applications · Problem 6.2.7

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · contour-integrals-circles

Problem

Recall

C_r^+(a)

is the circle of radius

r

that is centered at

a

and oriented counter-clockwise. (a) Evaluate

\displaystyle\int_{C_4^+(0)} z\,dz

. (b) Evaluate

\displaystyle\int_{C_4^+(0)} \bar{z}\,dz

. (c) Evaluate

\displaystyle\int_{C_2^-(0)} \frac{1}{z}\,dz

. (The minus sign means clockwise orientation.) (d) Evaluate

\displaystyle\int_{C_2^-(0)} \frac{1}{\bar{z}}\,dz

. (e) Evaluate

\displaystyle\int_C (z+1)\,dz

, where

C

C_1^+(0)

in the first quadrant. (f) Evaluate

\displaystyle\int_C (x^2 - iy^2)\,dz

, where

C

is the upper half of

C_1^+(0)

. (g) Evaluate

\displaystyle\int_C |z-1|^2\,dz

, where

C

is the upper half of

C_1^+(0)