 # Every polynomial p has the property that p n p n 1 1

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(Every polynomialPhas the property thatP(n)P(n+1)1 since if the leadingcoefficient iscknk,P(n)P(n+1)cnkc(n+1)k= (1-1n+1)k1.)
Exercise 15:Supposefis a non-vanishing continuous function on¯Dthat isholomorphic inD. Prove that if|f(z)|= 1 whenever|z|= 1,thenfis constant.
11Chapter 2.7, Page 67Problem 3:Morera’s theorem states that iffis continuous onC, andRTf(z)dz=0 for all trianglesT, thenfis holomorphic inC. Naturally, we may ask ifthe conclusion still holds if we replace triangles by other sets.(a) Suppose thatfis continuous onC, andZCf(z)dz= 0or every circleC. Prove thatfis holomorphic.(b) More generally, let Γ be any toy contour, andFthe collection of alltranslates and dilates of Γ. Show that iffis continuous onC, andZγf(z)dz= 0 for allγ∈ Fthenfis holomorphic.In particular, Morera’s theorem holds underthe weaker assumption thatRTf(z)dz= 0 for all equilateral triangles.Chapter 3.8, Page 103Exercise 1:Using Euler’s formulasinπz=eiπz-e-iπz2i,show that the complex zeros of sinπzare exactly at the integers, and thatthey are each of order 1.Calculate the residue of 1/sinπzatz=nZ.

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Term
Fall
Professor
SOMEGUY
Tags
Sin, Cos, dx, lim es
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