Question

please show every step with legible hand writing! thank you!

f(2) dz 4. Show that an analytic function f(z) has a primitive in D if and only if 0 for every closed pathy in D

Answer 1

Let for the analytic function fc 2) has a primitive FC2) such that f'(2) = f (2) Now, det for any closed path dind. Let d. be any path from ze to Zrind then (412) da Fe 2) da 21 (22 F'(2) dz F(22) - F(21) for a closed path dir D, 2 = 22 so fea) da = F(21) - fez)=0. e of fea) has primitive then f (2) d 2=0 for any closed path din conversly let Sofeada=0 for any closed come d and the anonlytic function f(a) fixing to in the value of the function F(2) = s f(?) ds is in dependent of the path of integration from zo to z inside do choose & Small enough so that the line segment a to (2th) lies in D and consider the difference quotient F(2th) - F (2) + s fends - f Read the ea) as FC2+1) - FC2) _ +62) star 1 (fe 8) - f 2)) a$ - 2th 20 Then 2th

whose the path from a to 2th is taken to be the straight line segment. By the continuty of fez), we have 14es) - fealle for the sufficiently small. Since the ling segment from 2 to ath has the length del. it follows from h that It (2th) - F (2) _fa) et Elhl= € p14es) - 2! <E SO The best the dista 16 a stm F (2th) – F(2) hao h = f(2) = f'(2) = f(2) & fehas the primitive &(22)