Solution To Contour Integration Problem

Below, the solution to the problem given at the end of the December 30 post:

Note C is  now  the right hand half of the circle  êz ê = 2  from z = -2 i to z = 2 i. 

Hence we want to find the integral of:

I = ò _C  z* dz

For which z = 2 exp (i q)  (- p/2 <   q  <  p/2)

Then:

I=  ò_p/2 ^-p/2  (2 exp(i q)) (2 exp(- i q))  dq  =

  4i   ò_p/2 ^-p/2   dq

= 4 i(p/2  - (-p/2)) = 4i   (p)  =  4 pi

Note that for such a point z on the circle êz ê  = 2 , 

It follows that zz* = 4 or z* = 4/z. So that the result  4 pi can also be written:

I = ò _C  dz/ z =  pi