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Math Help - complex integral

  1. #1
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    complex integral

    Evaluate  \int_{C_{z}} \frac{e^{z} \ dz}{1+e^{z}} where  C_{z} is that piece of the imaginary axis lying between  y = 0 and  y = 1 and traversed upwards.

    So let  w = e^z . Then  C_{w}: |w| = 1, \ \ 0 \leq \text{arg}(w) \leq 1 traversed counterclockwise. So  I = \int_{C_{w}} \frac{dw}{1+w} . Then we get  I = \text{Log}(1+e^{i})- \ln 2 .

    Is this correct?
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  2. #2
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    I would parametrise the contour by y(t)=it, 0\leq t\leq 1

    Then I=\int_0^1\frac{e^{it}}{1+e^{it}}i~dt
    =\int_0^1 i-\frac{i}{1+e^{it}}dt
    =\left[ it-ln(1+e^{it}) \right]_0^1
    =i-ln(1+e^i)+ln(2)

    Not sure if that's correct but it makes more sense to me.
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