Thread: Integration in a complex plane

Hello,

I have a problem:

Let gamma: z=e^(it) (0<= t <= pi). Show that

|integral (e^z / z) dz | <= pi e^2

I thought to use the Estimation theorem:

But at the end of |f(gamma(t))| i found e^(2cost)/2

And at the end of |(gamma'(t)| i founf 2.

But i don't understand how can i integrate e^(2cost)

Thank you

Richard

Use the ML inequality right? If on the contour and the length of the contour is L then:



On the unit circle , . Therefore .

Yes thank you very much.

Here z=2e^(it) so is true that |e^z|<=e^2 ??

So M=e^2 but why L should be pi??

I need help plz..

Originally Posted by rickgoz

Yes thank you very much.

Here z=2e^(it) so is true that |e^z|<=e^2 ??

So M=e^2 but why L should be pi??

If then but can you show that?

L is still the length of the contour which is but I believe in the original problem, since 2<e, then it was true for right?

Okay thank you very much. It's fine!

Last question: do you know how can I evaluate

|14 + z + z^2| because I don't know if I can use the triangle inequality here..

Thanks for your help

Richard

You can use the generalized version:



so