Question

Q3: 5 marks (A) Expand f(z) (2-1)(2-3) in a Laurent series valid for (i) Iz - 11 < 2, and (ii) Iz - 31 < 2. 1.5 marks each part (B) Use Laurent series to find the residue of f(2)= e (x - 2)-2 at its pole z = 2. 2 marks

Answer 1

8 3(A) i) FOr 1211<2 le 1z-11c1 We have find laurent seriel enpan&ion to f the function t(2)=1219(2-3) (Z-1)2(2-1-2) (z4-2 (1-) 2(2)2 [1+)+E)E)-] 2 (24)2 tre)3= 16 4(24) 2 (24)2 Laerent series of The Jiven ,u the fun chion.

O(3) c) when 12-31<2 ce 12-3/</ have we to find Laurent Series enpansion of the funchon t)= Now t(2)= (Z+)"(2-3) (Z-3) (Z-3+2)2 4 (2-3) (1+ N-3 12 4(2-3) ++2)E3) 2! 42-3) 3 (z-3) - 1(2-3) 16 4(2-3) ů The Laurent Series enpanlion of The given fon Chion F2)= in r2-3) the Region 12-3/e2

0(3) (B) te)= ē(2-2) eZ+2-2 %3D (2-2)2 -z+2 (Z-2)2 (2-2)3 [1-(=-2)+ 2-2)= (z-2)? (z-2/3 21 3! 2. ē? (Z-2)3 (2-2)2 (2-2) (Res z=2) te)= coefficient f in the Z-2 Laurent Series enpansion of fe) Ans.