Finding an upper bound for a contour integral (Complex)

[Zeeree]

C₁ 1. Homework Statement :
Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz
where the contour C = C₁ + C₂.

C₁ is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C₂ is the line segment from the origin to B.

Homework Equations

The Attempt at a Solution

I figured I have to find M1 and M2, and also L1 and L2 given the two different contours. Along C₁, I have found the length to be pi/6 by integrating a parameterized equation of the unit circle from pi/6 to pi/3. I have also found the length along C₂ to be 1 (no surprises here)

However, I've hit a bump when trying to find the upper bound for e^2z - z^2.
I figured I should use the triangle inequality i.e. |e^2z - z^2|
<= |e^2z| + |-z^2|
= |e^2z| + |z^2|
= |e^2z| + |z|^2
= |e^2z| + 1 (since the length of the line segment is one).

I'm kind of stuck when it comes to the e^2z. It's the same problem for finding the upper bound on C₂ as well.

I'm sorry for not using Latex. I tried formatting it but it doesn't come out as intended in the Preview, so I decided not to take the risk of fudging it up and making it look confusing. I'm quite sleepy, sorry.[/B]

 

[pasmith]

If $z = x + iy$ then $e^z = e^x (\cos y + i \sin y)$...

 

[Zeeree]

If $z = x + iy$ then $e^z = e^x (\cos y + i \sin y)$...

Thank you so much, I've got it :)