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Homework Help: Complex integrals - help please!

  1. Mar 4, 2010 #1
    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.

    1. The problem statement, all variables and given/known data
    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.

    3. 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:

    1. The problem statement, all variables and given/known data
    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.


    3. 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!
     
  2. jcsd
  3. Mar 4, 2010 #2
    1.differentiate w and transform variables...?
    2.evaluate the residue...?
     
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