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Math Help - closed curve, trace

  1. #1
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    closed curve, trace

    Let \gamma be a closed path in a domain D such that W(\gamma, \zeta)=0 for all \zeta \not \in D. Suppose that f(z) is analytic on D except possibly at a finite number of isolated singularities z_1, \ldots, z_m \in D \char`\\ \Gamma. Show that \int_{\gamma} f(z)dz = 2 \pi i \sum W(\gamma, \zeta) \text{Res}[f, z_k].


    Hint

    Consider the Laurent decomposition at each z_k.

    I do not see how to get the Laurent decomposition of the z_k to do this problem. By the way, the above notation means the winding number. It looks like perhaps the residue theorem would come into play here somehow too. I just do not see how to proceed now. Any suggestions would be very nice. Thank you.
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  2. #2
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    Quote Originally Posted by poincare4223 View Post
    Let \gamma be a closed path in a domain D such that W(\gamma, \zeta)=0 for all \zeta \not \in D. Suppose that f(z) is analytic on D except possibly at a finite number of isolated singularities z_1, \ldots, z_m \in D \char`\\ \Gamma. Show that \int_{\gamma} f(z)dz = 2 \pi i \sum W(\gamma, \zeta) \text{Res}[f, z_k].


    Hint

    Consider the Laurent decomposition at each z_k.

    I do not see how to get the Laurent decomposition of the z_k to do this problem. By the way, the above notation means the winding number. It looks like perhaps the residue theorem would come into play here somehow too. I just do not see how to proceed now. Any suggestions would be very nice. Thank you.
    For each z_j consider the set D_j= \{ z\in \mathbb{C} : 0<|z-z_j|<r_j \} where r_j is such that D_j\cap D_i =\emptyset if i\neq j. f has a Laurent expansion in each of these (shrink them if necessary) so f= \sum_{n=0}^{\infty } a_n(z-z_j)^n+ \sum_{n=1}^{\infty } \frac{b_n}{(z-z_j)^n}. Now define S_j(z)=\sum_{n=1}^{\infty } \frac{b_n}{(z-z_j)^n} then using a dual to Abel's lemma we get that S_j is uniformly convergent in sets of the form \{ z\in \mathbb{C} : |z-z_j|\geq \epsilon \} for all \epsilon >0. Now define g(z)=f(z)-\sum_{k=1}^{m} S_j(z) then it's clear that g is holomorphic in D (the z_i's are removable singularities) then \int_{\gamma } g=0 (because of Cauchy's theorem and the fact that \gamma is homotopic to a point in D) from which \int_{\gamma } f = \sum_{k=1}^{m} \int_{ \gamma } S_j. Now to conclude just calculate \int_{ \gamma } S_j where you need to remember this http://www.mathhelpforum.com/math-he...tml#post446504 and that \int_{\gamma } \frac{1}{z-z_j} = 2\pi i W(\gamma ,z_j)
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