How can I evaluate a contour integral over a complex function?

[trickae]

Contour Integral question

Homework Statement

http://img300.imageshack.us/img300/8536/20075202330586331530065ra7.jpg
I need to evaluate the following integral over the above contour - also could someone do it for integral (ez cos z) as well?

Thanks in advance

Homework Equations

The Attempt at a Solution

I tried the method of deformation, i tried using the form

Integral [F(z) dz] = integral [f(z(t)) . z'(t)]

neither method works - because i get stuck near the end.

my working
:
Integral [F(z) dz] = integral [f(z(t)) . z'(t)]

the parametrized form of the contour - i used the deformation principle

that

http://img525.imageshack.us/img525/5272/myworkingcomplexq3wy8.jpg


yet i doubt this is the method - any help?

 

[trickae]

can someone explain this?

we know that ez sinz is an entire function. i.e. it has no singularities.and the region D is bounded by the simple closed path.
so by Cauchy integral theorem, we have http://img95.imageshack.us/img95/8707/cramsterequation2007521ed3.png =[/URL] 0
i.e. http://img221.imageshack.us/img221/9658/cramsterequation2007521mb9.png =[/URL] 0.
similarly http://img138.imageshack.us/img138/1315/cramsterequation2007521gx5.png =0

 

[siddharth]

It seems you help in understanding Cauchy's integral theorem? If so,
http://mathworld.wolfram.com/CauchyIntegralTheorem.html or look up a text on complex analysis.

 

[HallsofIvy]

we know that ez sinz is an entire function. i.e. it has no singularities.and the region D is bounded by the simple closed path.
so by Cauchy integral theorem, we have http://img95.imageshack.us/img95/8707/cramsterequation2007521ed3.png =[/URL] 0
i.e. http://img221.imageshack.us/img221/9658/cramsterequation2007521mb9.png =[/URL] 0.
similarly http://img138.imageshack.us/img138/1315/cramsterequation2007521gx5.png =0

Cauchy's integral formula says that the integral of an entire function, over a closed contour, is 0. The contour in this problem is not closed.
It is, however, true that you can replace the given contour by any other contour having the same endpoints. You might use the quarter circle or two lines, from i down the imaginary axis to 0 and then up the real axis to $\pi$.

 

[trickae]

if the contour isn't closed then in fact it isn't 0 then is it? or does the above still hold? I was thinking it was integrable ...

 

[HallsofIvy]

If the contour is not closed then the integral is not necessarily 0 but might be. I'm not sure what you mean by "I was thinking it was integrable"- e^zsin(z) obviously is integrable! As I said, you can use any contour having the same endpoints- in particular the quarter-circle you mentioned or two line segments. Yet another way to do this is to "ignore" the contour: find an anti-derivative of e^zsin(z) and evaluate at i and $\pi$.

 

[trickae]

thanks i'll do that.