Question

Please do 2 only

please do 2 only

Exercises (1) Compute for de and c ) da where is the ultime center at the origin and oriented once in the counterclockwise (2) Computer da, where I is the circle {: € C: 1:= 3) once in the counterclockwise direction (3) (Mean Value Property of Holomorphic Functions) Supposed w = f(e) is holomorphic on and inside the circle {: € C:- Prove that f(20) == f( 70 +re) de. (4) Under the hypotheses of the preceding exercise, prove that for 0.1.2..... flm) (20) = f(zo + rek)e-*no do. (5) Let w = f(2) be a holomorphic function on and inside the unit circle with center at the origin. Prove that f(x + iy) de dy = f(0). Jizisi (6) (Cauchy's Estimates) Let w = f(z) be a holomorphic function on and inside the circle Cr given by Cr = {2 € C:\z- 20 = R}. Prove that if If(x) <M, ZE CR then if(n) (20) P n =0,1,2,.... (7) (Liouville's Theorem) Prove that every bounded entire function must be a constant function. (8) Let S be an entire function such that there exists a positive number M for which Ref(z) SM, ZEC. Prove that f has to be a constant function.

Answer 1

700 IZ =3? 1 (3-i9%8+z)a 3.- 1 416) '689 = (3-1):e'a _ (e)3(3-1)) 7' 4)= (-93* i empre 2*1*(-00) (29)4 -uietuie t'li) = = 0 (16) eis T Gates J let da Ť T8-ijo Je sailed to) -- =) de =] (82= ela *'(8) = (2+ifier-fel*) (013+)) 7'2- - 4 ie 'nijev 8+)* Ples Folder th