Intro to complex analysis problems.

[nontradstuden]

1) How do you integrate 1/ [z^2] over the unit circle?

After you integrate, do you put it in polar form or do you replace z with x + iy then solve it?

I keep getting zero. It should exist since z=o is undefined, right?

2) How do you integrate x dz over gamma, when gamma is the straight line path from 0 to 1+i?

I have looked at some examples that put the line in y=mx +b form, then integrate it w/ respect to y, but my professor gave us this equation: gamma(t)= (1-t)P + tQ. I don't know where it comes from, so that's confusing me. The fact that it's the integral of x dz confuses me, too.

Thanks.

 

[lanedance]

1) you could either use the residue theorem, which will give the value for any closed contour containing the origin, or you could paramterise and perform the integration directly. Parameterising in terms of polar coordinates and using theta to integrate would be best

2) how about finding x(z) along the line gamma

 

[jackmell]

1) How do you integrate 1/ [z^2] over the unit circle?

After you integrate, do you put it in polar form or do you replace z with x + iy then solve it?

I keep getting zero. It should exist since z=o is undefined, right?

2) How do you integrate x dz over gamma, when gamma is the straight line path from 0 to 1+i?

I have looked at some examples that put the line in y=mx +b form, then integrate it w/ respect to y, but my professor gave us this equation: gamma(t)= (1-t)P + tQ. I don't know where it comes from, so that's confusing me. The fact that it's the integral of x dz confuses me, too.

Thanks.

Yes you can integrate it directly, find the antiderivative, evaluate it at it's endpoints (1 to 1 right?), zero. Bingo-bango as a consequence of Cauchy's Theorem. Or you can parameterize the contour, letting $z=re^{it}$, convert it to an integral in t, integrate it from 0 to 2pi, again, zero. Later when you learn the Residue Theorem, you can apply it and immdeiately conclude it's zero since the Laurent series for the intgrand has no residual term (no 1/z term) so again, it's zero.

The second one, you have got to get straight with parameterizing contours in the complex plane. Get good at it cus' you'll need it throughout the course. Just let z(t)=x(t)+iy(t). Not hard for a straigh-line path from the origin to the point 1+i right, just let x(t)=t, y(t)=t, then dz=dt+idt=(1+i)dt. Alright, plug that into the integrand and integrate from 0 to 1.

 

[nontradstuden]

Thank you both, very much!

This class confuses me because the professor never goes over examples in class. He only speaks in general [definitions, theorems, gamma, f/F etc] , so I'm always sooo lost when I actually have to apply it to different kinds of problems on a test/quiz. The book is hard for me to understand too, so it's no help either.

This is the book:

http://www.math.uiuc.edu/~jpda/jpd-complex-geometry-book-5-refs-bip.pdf

I understand a very small amount of it... lol

Thanks, again!