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=n∈Z.