How Do I Solve These Complex Contour Integrals?

[drinkycrow]

Hello all. I've been browsing your forums for similar problems that might help me with these two integrals, but alas i am still stumped. please help me understand and solve these integrals! also please forgive my notation; the Latex controls aren't working for me.

Homework Statement

Determine the contour integral:

c.int(z_bar*(abs(z))^2)dz over the contour C which encloses the domain

abs(x^2-y^2) <=1 , 1<=xy<=2 , x>0. oriented in the clockwise direction.

z = x + iy. Hint: use the change of variable w=z^2

(here, z_bar is the complex conjugate of z, abs(z)^2 is (modulus of z)^2, and <= is less than or equal to.

The Attempt at a Solution

I think i understand the domain; it's in the first quadrant, each side part of a hyperbola, corners at (x,y) ~ (1.25,1.6), (1.6,1.25), (1.272,0.786), (0.786,1.272). so, f(z) here is not analytic, because the Cauchy-Riemann conditions aren't satisfied, right? Beyond this, I'm not sure how to start evaluating the integral; i really don't understand the hint in the first place. ie - if w=z^2, how do i express z_bar and (abs(z))^2? and then what good does this do me on the contour?

Second:

Homework Statement

Evaluate the integral

c.int(g(z)*exp(z)/(sin(z))dz , where g(z) = (z+4)/(z-4)

over the counter-clockwise rectangle C with corners at -2-i, -2+i, 2+i, 2-i.

The Attempt at a Solution

so this one isn't analytic inside the rectangle, right? because the domain includes z=0. so how do i even start?!

i've been staring at these equations and books for hours, so any help is much appreciated! i guess i might also mention that we haven't covered residues yet. thanks!

 

[xaos]

1.differentiate w and transform variables...?
2.evaluate the residue...?