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Homework Help: Integral of e^(-z) dz / (z - (pi*i)/2)

  1. Apr 20, 2007 #1
    1. The problem statement, all variables and given/known data

    Let C denote the positively oriented boundary of the square whos sides lie on the lines (x= + / - 2 and y = + / - 2)

    Evaluate int_c (e^(-z) dz) / (z - (pi*i/2))

    The answer is simply 2*pi


    2. Relevant equations

    This is a text book question after the topic of Cauchy Integrals but before residues


    3. The attempt at a solution

    I am not very cluely at the Contour integration as the teacher just gave us the text book notes and said do it... so I figure to use the Cauchy Integral

    f(z_0) = 1/(2*pi*i) int_c f(z) dz / z - z_0

    Where my belief is that z_0 is interior to the given contour (in this case it's the square)

    so I thought that the question looks like it will just fit inside the formula, so to speak, because (pi*i)/2 is interor to the contour (well if it's not then I have no idea what i'm talking about) so then i put

    f(z_0) = 1/(2*pi*i) int_c e^z dz/ z - ((Pi*i) / 2)

    so I sub in f(z_0) = e^(Pi*i) and get -1

    so -1 = 1/(2*pi*i) int_c e^z dz/ z - ((Pi*i) / 2)
    -2*pi*i = int_c e^z dz/ z - ((Pi*i) / 2)

    and thats where I stop :surprised because I dont know how to integrate e^z dz/ z - ((Pi*i) / 2)

    Any suggestions?
     
  2. jcsd
  3. Apr 20, 2007 #2

    Päällikkö

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    Homework Helper

    Cauchy's integral formula (I'm not gonna list the assumptions it requires, you should make sure that they are satisfied in this problem, though):
    [tex]f(\alpha) = \frac{1}{2\pi i}\int_C \frac{f(z)}{z-\alpha}dz[/tex]


    Substitute f(z) = e-z:

    [tex]\int_C \frac{e^{-z}}{z-\alpha}dz = 2\pi i e^{-\alpha}[/tex]

    Can you do the rest?
    As you can see, you don't really have to integrate anything.
     
    Last edited: Apr 20, 2007
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