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    Some simple applications of the Cauchy-Goursat theorem

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    Use the Cauchy theorem to show that the integral around the unit circle |z|=1, traversed in either direction, is zero for each of the following functions:

    1) f(z)=z exp(-z)
    2) f(z)=tan(z)
    3) f(z)=Log(z+2)

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    Solution Preview

    The Cauchy-Goursat theorem tells us that if is analytic in a simply connected domain D, then for every closed contour C in D. Since is a closed contour, we can apply the ...

    Solution Summary

    One version of Cauchy's theorem for complex functions states that the integral around a closed simple curve of a complex function is zero, provided the integrand is analytic in the region enclosed by the curve. The solution comprises half a page written in Word with Mathtype illustrating the application of this theorem in 3 concrete examples.