Calculating Complex Integral for a Half Circle and Line - Step by Step Guide

[Erikve]

Hello,

I have a question about a complex integral. The question is about the index of a curve. This curve is defined as:

j = j1 * j2

with j1: r*exp(i*t) with t: [0,pi]
and j2: [-r,r]

This is quite simple: a half cirle followed by a line from (-r,0) through (0,0) to (r,0).
To calculate the index Ind(a) over the curve j:

Ind(a) = 1/(2*pi*i) * integral(dz/(z-a))





Now the problem: I used a curve (a+r*exp(it)) to solve the half circle, this give me 1/2. But how can I calculate the index of the line from (-r,0) to (r,0)?

 

[lippyka]

I cannot use the same formula as for the half circle, because it is not a closed loop.Any help would be greatly appreciated! Thank you.