Question

Complex Analysis:

. (a) Find a single function f(z) which has all of the following properties: f(z) is discontinuous at the origin z = 0, at z = 1, and at all points z with Arg(z) = 7/4, but f(z) is continuous at all other points of C; • f(z) has a simple zero at z = :i; and f(z) has a pole of order 3 at z = n. Justify that your function f(x) has each of the properties listed above. (b) Determine whether statement (*) below is true or false. If the statement is true, then prove it; if the statement is false, provide a counterexample that shows it is false. (*) Let h(2) be a function which has an isolated singularity at zo. If h(z) has an essential singularity at zo, then h'(z) has an essential singularity at zo.

Answer 1

z-i Let us take H2) = Z(z-17)3 sin (2-11/4) P(2) = (z-i) let P(z) 4 (2) H2) = where and 9 (2) = 2(2-1² sin ( Z - 19/4) 9 Now the sunetion 4(2) has a discontinuous PIZ=Zo gr P(20) to and A(Z) 20. at Now at Z.=0 P() = -izo 90 - 0 P(IT) = 5-izo 9 (IT) = 0 at z = IT Now sin Z=0> Z= N IT for nER. Sin (2-1/4) = Z-174 =mIT for NER. => z=2017 19/4 for NER <-> Ang (2) = 17/4 Arg (2) = 1/4 then P(2) - 201T+T-170 and 9(2) = Z (2-m) sun( 2-174) whene [: (204) = 0 Now for others values of $4(Z) = 0 clearly &(2) is conlon for others values of (other that Zzo, IT, and Zo such that Arg (20) = 14 Z

H2) is discontinuous at the origin zao, at 2215 and at all point Z with Arg(2) 20/4, but 82) y continuary at all other paéits of C. SO +2) has a simple o at Z= i power of (z-) is 1 in HZ) Now lim (z -IT)? He) Z- lim Z T z sin ZIT (ZNO - = M- T Sin 3 +0 IT sin (T-174 a denite no atf(z) has a pale of order 3 at Zz T.