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Integrating a Complex Function Over a Contour

  1. Dec 3, 2014 #1
    1. The problem statement, all variables and given/known data
    ##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx##

    2. Relevant equations


    3. The attempt at a solution
    Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the path taken). So, I can choose a simpler contour which passes through these two points, such as a line.

    Using the two points to find the slope, and parameterizing it, we get the straight-line contour

    ##z_1(\tau) = \tau + i(4 \tau + 3)##.

    Here is where I am having difficulty. How exactly do I find the interval between the two points? This is how I did it, but am I unsure if it is correct:

    ##z(\tau) = (1,1) \implies##

    ##(\tau, 4 \tau + 3) = (1,1)##

    which gives us the two equations

    ##\tau = 1## and ##4 \tau + 3 = 1##.

    However, I get a different ##\tau## value from each equation. What can account for this?
     
  2. jcsd
  3. Dec 3, 2014 #2
    I found the mistake. The straight-line contour should actually be ##z_1(t) = \tau + i(2 \tau \underbrace{-} 3)##
     
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