# closed curve, trace

• Jan 27th 2010, 07:11 PM
poincare4223
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.
• Jan 27th 2010, 08:19 PM
Jose27
Quote:

Originally Posted by poincare4223
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| 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 $\int_{\gamma } \frac{1}{z-z_j} = 2\pi i W(\gamma ,z_j)$