Evaluating Complex Integrals: From Pi to i & Around Unit Circle & Square

[kathrynag]

Homework Statement

integral(e^(iz)dz) with bounds from pi to i.
I need to evaluate using : along the straight line joining the limits, along segments of the coordinate axes joining the limits, and estimate using integral <= L*M and compare the estimates with the values.

I am totally confused on how to do this. Do I convert z to x+iy and do the bounds change?



I have another integral with z conjugate about the unit circle. I think I have an idea on this one on converting to x-iy and seeing that this will equal 2ipi.



I am also asked to evaluate this same integral taken in the positive sense about the square with x=+-1, y=+-1. I know this involves 4 segments, but my problem is finding this 4 segments. I know the answer is 8i, but how to get there?

 

[Count Iblis]

You can parametrize the path from pi to i, e.g by taking:

z(t) = pi + (i-pi)t

The starting point is then at t = 0 and the end point at t = 1.

integral f(z)dz along the path then means:

f(z(t))d(z(t)) from t = 0 to t = 1.

We have dz = (i-pi) dt

and

f(z(t)) = exp[i pi - (i pi + 1)t] = -exp[-(i pi + 1)t]

 

[kathrynag]

That makes a lot of sense actually.