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Math Help - Complex Integrals

  1. #1
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    Complex Integrals

    "Apply the Cauchy-Goursat theorem to show that \int_C f(z) \,dz=0 when the contour C is the circle |z| = 1, in either direction, and f(z) = \frac{1}{z^2+z+2}."

    After using quadratic formula, f(z) = \frac{1}{z^2+z+2} = \frac{1}{(z+1+i)(z+1-i)}.

    Tried breaking down to partial fractions:
    \frac{1}{(z+1+i)(z+1-i)} = \frac{A}{z+1+i} + \frac{B}{z+1-i} = \frac{A(z+1-i) + B(z+1+i)}{(z+1+i)(z+1-i)} = \frac{(A+B)z + (A+B) + (-A+B)i}{(z+1+i)(z+1-i)}

    Dead end since we're looking at A+B=0, A+B=1, and -A+B=0. Any suggestions?
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  2. #2
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    Where are your poles relative to your contour and the area enclosed by it?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    1) You don't have to break it down into partial fractions to see where the poles are!

    2) Why are you assuming that A+B and (B-A)i are the real and imaginary parts of the numerator? You can have A+B=0 and (B-A)i = 1.
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  4. #4
    Member Ruun's Avatar
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    You need to show that f(z) is holomorphic (Cauchy's integral theorem - Wikipedia, the free encyclopedia)
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