Question

A)

1 Find the Laurent series for 22 +22 for 0 < 121 < 2

B)

C)

Answer 1

SWh에 A) let f(2)= 전 <시리<2 저2 02: 뭘시 2 나르 2 그글 자 [ 대를게 <월의 굴 다중 - ② - + - which is the required Laurent series for f(Z). 2근 9 Z 23 임근 33 - 굿 + 4 B Given that, ㅈ ge) (3) (3-2) 에 2-31>S 8 + (2+2] + 3E Now, A R+2)(3-25 42) A +2) (3-2) + e (3-2) + C (2+2) | putting 쯔3, -2,0 we get fr끗23, C(5)를 ml for 2, 5B19B22 5 For 2=0, 6A+BB +4e2 6A2 - 6 q 3 s 뻐 I 2

i, e glu = [(- 1 4 0 2 ) + (- 2 + 30 - ] g(u) un 7 1 13 or gez) 7 (7-3) which is the required Laurent series for g(2). (2-3)2+ Given, h (2) Klz-11 22 Z"(z-i) We here, B At Zrezis zat z. w ) stor C (1) A Z(Zi) A B Cz i) + c 2 = 1 putting 2=0, we get, Bei) = lan pruiting zzi, a -c=1 Cz-1 purting a=-1, Ail-22) + t2iB - C=1 0x90 ai Aai (i) +1=1 2Atat 131 A2-1. a i - hla) t Z 22 z- Yok a za vt I CIVIL et r1=0 here) - Then 1 i + uti (+12 (V+l-i)

* H(2)=-6(1+)7 (1+6)-(187)(*** [ Chok. 12 ana INI2V2 1ural = 10,21, (1) V2 21 i b6 + z- -} + - 4 63 + --} & {- {1 - 2 + 3 262 102 2 re W I- + (1 ir Le3 w (1-1)3 7 04- =t-hot von 2 1 6 3 - -) +i (to - 2 3 4 5 6 ----) -till- (1-2 2- (1-233+ -- tle - u3 li t h(z)= [-12) til +--] 2 (2-1) (2-4)3 + (2_102 – 2133+. cé-jy---- - 1 [l- (2-1) (1-2) Which is the required Laurent series + (24)? (- - for h (Z). This completes the problem.