1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex countour integrals

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data

    1) evaluate the contour integral

    c.int [ e^z / ((z+1)^2 * (z-4)) ] dz

    about the rectangle with verticies -3 + 4i, 3 + 4i, -3-4i, 3 - 4i,

    use partial fractions and the cauchy integral formula to evaluate the integral.

    2) Let C be the closed contour that runs counterclockwise around the circle | z - 2 + i | = 3

    compute c.int [z - 3z^2]dz

    (the z raised to the first power in this equation is barred. i.e must integrate the complex conjugate of the first z.)


    2. Relevant equations

    cauchy integral formula f(z_0) = 1/(2*pi*i) * int [ f(z) / ( z - z_o) ] dz

    3. The attempt at a solution


    1. First I expanded e^z / ((z+1)^2 * (z-4)) using partial fraction decomposiion and got

    e^z * ( (-1/25)/(z+1) + (-1/5)/(z+1)^2 + (1/25)/(z-4) )

    brake up the integral into 3 parts

    -1/25 * c.int [ e^z / (z + 1) ] dz - 1/5 * c.int [ e^z / (z + 1)^2 ]

    + 1/25 * c.int [ e^z / (z - 4) ]

    applying the cauchy integral formula I get:

    -1/25 *2 *pi * i * e^(-1) - 1/5 *2 *pi * i * e^(-1) + 0

    the last term becomes zero because 4 lies outside the rectangle.

    The only part about this that makes me uncomfortable is not using the vertecies of the rectangle at all in the computation of the curve. Did I need to parameterize the rectangle? Thanks

    2) for the integral c.int [z - 3z^2]dz i parametrized the circle | z - 2 + i | = 3 as

    z(t) = 2 + i + 3e^(it)
    z'(t) = i3e^(it)

    then evaluated c.int [ f(z(t))z'(t)dt ]

    could someone please verify my methods.

    Thanks!
     
    Last edited: May 5, 2009
  2. jcsd
  3. May 6, 2009 #2
    Mostly correct, good job!

    Notice that for problem 1, the integral c.int [ e^z / (z + 1)^2 ] requires the CIF for derivatives, not the CIF you stated, but I believe the result is the same (double check it). Definitely don't parametrize the rectangle. The answer would be the same for any simple closed contour that encloses -1 but not 4, and that's part of what this question is testing that you know.

    For problem 2, the center of the circle is 2-i, not 2+i (unless the typo is in the question instead of your solution). Also, you could make it somewhat easier by breaking the integral into (integral of z bar dz) - (integral of 3z^2 dz), where the second integral is very easy, thus simplifying the amount of integration that has to be calculated "by hand."
     
  4. May 6, 2009 #3
    quickly done using the residue theorem i get [itex]-\frac{12 \pi i}{25 e}[/itex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Complex countour integrals
  1. Countour integral (Replies: 1)

Loading...