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Math Help - contour integral-need help

  1. #1
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    Question contour integral-need help

    Having trouble solving this contour integral. Evaluate the integral e^5z/z^5 where c is the circle |z|=1 traversed in clockwise direction? Any suggestions where to start?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by durham2 View Post
    Having trouble solving this contour integral. Evaluate the integral e^5z/z^5 where c is the circle |z|=1 traversed in clockwise direction? Any suggestions where to start?
    Note that z=0 is a pole of order 5. We can apply the residue theorem to this by noting that

    \text{Res}\left(f,0\right)=\frac{1}{4!}\lim_{z\to 0}\frac{\,d^4}{\,dz^4}\left[z^5f\!\left(z\right)\right]=\frac{1}{4!}\lim_{z\to 0}\frac{\,d^4}{\,dz^4}e^{5z}=\frac{1}{4!}\lim_{z\t  o0}5^4e^{5z}=\frac{625}{24}

    Thus, \oint_{\left|z\right|=1}\frac{e^{5z}}{z^5}\,dz=2\p  i i\text{Res}\left(f,0\right)=\frac{625\pi i}{12}

    Since we're moving clockwise, it follows that the result will be negative.

    Does this make sense?
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  3. #3
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    Thumbs up

    Yes that makes since. I get confused on these integrals. That helps a lot thanks!!
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