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

[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)?

 

[Erikve]

I think that I have placed this topic on the wrong forum. Can anybody tell me how to delete my post or can it be closed by a forum-moderatort?

 

[tacman]

To calculate the index of a line in a complex integral, you will need to use the formula for the index of a curve, which is Ind(a) = 1/(2*pi*i) * integral(dz/(z-a)). In this case, the curve is defined as j1 * j2, where j1 is a half circle and j2 is a line from (-r,0) to (r,0). To calculate the index of the line, you will need to substitute the equation for j2 into the formula for the index of a curve. This will give you:

Ind(a) = 1/(2*pi*i) * integral(dz/(z-a)) = 1/(2*pi*i) * integral(dz/((r+z)*exp(i*t)-a))

You can then use the substitution u = (r+z)*exp(i*t) to simplify the integral. This will give you:

Ind(a) = 1/(2*pi*i) * integral(du/(u-a))

You can then use the Cauchy integral formula to evaluate this integral and calculate the index of the line. The Cauchy integral formula states that for a function f(z) with a simple pole at z0, the integral of f(z) over a closed curve C that encloses z0 is equal to 2*pi*i * Res(f,z0), where Res(f,z0) is the residue of f at z0. In this case, our function is 1/(u-a), which has a simple pole at a. So, we can use the Cauchy integral formula to evaluate the integral and calculate the index of the line. The result will be 1, indicating that the line crosses the curve once, from (-r,0) to (r,0).

In summary, to calculate the index of a line in a complex integral, you will need to use the formula for the index of a curve and substitute the equation for the line into it. Then, use the Cauchy integral formula to evaluate the integral and determine the index. I hope this helps. Let me know if you have any further questions.