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Zeeree
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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.
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]
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]