How to Calculate the Index of a Line in a Complex Integral?

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SUMMARY

The discussion focuses on calculating the index of a curve defined by a complex integral involving a half-circle and a line segment. The curve is represented as j = j1 * j2, where j1 is defined as r*exp(i*t) for t in [0, pi], and j2 is the line segment from (-r, 0) to (r, 0). The index Ind(a) is calculated using the formula Ind(a) = 1/(2*pi*i) * integral(dz/(z-a)). The user successfully computes the index for the half-circle but seeks guidance on calculating the index for the line segment.

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Erikve
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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)?
 
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