math185_homework_6_sol.pdf - Math 185 Problem Set 6 Karla Betel Palos Castellanos ID 25612565 this implies that the nth derivative of f will never

math185_homework_6_sol.pdf - Math 185 Problem Set 6 Karla...

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Math 185 Problem Set # 6 Karla Betel Palos Castellanos ID: 25612565 October 27, 2019 this implies that the nth derivative of f will never satisfy | f ( n ) | > n ! n n . 6. A more general form of Lemma 3 reads as follows: Let the function ϕ ( z, t ) be continuous as a function of both variables when z lies in a region and α t β . Suppose further that ϕ ( z, t ) is analytic as a function of z belongs to for any fixed t . Then F ( z ) = β α ϕ ( z, t ) dt is analytic in z and (26) F ( z ) = β α ∂ϕ ( z, t ) dz dt. To prove this represent ϕ ( z, t ) as a Cauchy integral ϕ ( z, t ) = 1 2 π i C ϕ ( ζ , t ) ζ z d ζ . Fill in the necessary details to obtain F ( z ) = C ( 1 2 π i β α ϕ ( ζ , t ) dt )) d ζ ζ z and use Lemma 3 to prove (26) Lang: 1. (a) Show that the association f −→ f /f (where f is holomorphic) sends products to sums. Let f = a 1 , a 2 , ..., a n then f = n i =1 j . = i a j −→ f f = n i =1 1 a i (b) If P ( z ) = ( z a 1 ) .... ( z a n ) where a 1 , ..., a n are the roots, what is P /P ? Using the outcome from part (a) P ( z ) P ( z ) = n i =1 1 z a i (c) Let γ be a closed path such that none of the roots of P lie on γ . Show that 1 2 π i γ ( P P )(
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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...
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