#### You've reached the end of your free preview.

Want to read both pages?

**Unformatted text preview: **z ) dz =W ( γ , a 1 ) +... + W( γ , a n) Math 185 Problem Set # 6 Karla Betel Palos Castellanos ID: 25612565 October 27, 2019 1 2 π i γ P ′ ( z ) P ( z ) dz = 1 2 πi γ n j =1 1 z − a j dz =n j =1 γ 1 aj dz = n j =1W ( γ , a j ) 2. Let f ( z ) = (z −z ) m h ( z ) where h is analytic on an open set U, andh ( z) ̸ = 0 for all z∈ U. Let γ be a closed path homologous to 0 in U, and such that zdoes not lie onγ . Prove that 1 2π i γ f ′( z )f ( z ) dz = W( γ , z ) m.Let’s begin by writing f′ ( z) f ( z ) = mz − z + h ′( z ) h( z )h ( z ) ̸= 0 =⇒ h′ h holomorphic on U. This means that by Cauchy’s, it’s integralalong γis 0 . Hence, 1 2 π i γ f ′ (z ) f( z ) dz = 1 2π iγ (m ( z− z ) + 0) dz = W ( γ , z ) m...

View
Full Document

- Fall '07
- Lim
- Derivative, dz, 2πi C ζ, z0 does