MATH 227 Final, Spring 2005

DEPARTMENT OF MATHEMATICS AMERICAN UNIVERSITY OF BEIRUT MATH 227, FINAL EXAMINATION, SPRING 2005 Answer the following questions: 1. (a) Define the residue of a function at a point zo and state the Cauchy residue theorem (4 pts.) (b) Use the Cauchy residue theorem to evaluate the integral (6 pts.) | _Lbogz dz (m=1,2,--+; GC: z(t) = 2e" 0 <t < 2r). co (2+1)™ 2. (a) Use the Cauchy residue theorem to evaluate the integral (5 pts.) [exe dz (m=1,2,---; C: z(t) =e",0 St < 2m). c (b) Use residues to show that (5 pts.) °° cos ax T | wea ae (> 9) 3. (a) State the single residue theorem regarding the value of a line L f(z) dz, where C is a positively-oriented closed contour and f is analytic in the integral plane except at finitely many points z1,--- , zn lying within C. (4 pts.) (b) Use the single residue theorem to evaluate the integral (6 pts.) pellz ; [ dz (C:2(t) = 2e",0<t < 2r). C l+23 4. (a) State Laurent’s theorem. (4 pts.) (b) Find the Laurent’s series of the function (6 pts.) z+1 fa) = z(2z — 1) for values (i) jz} > 1/2, and (ii) |z - 1/2| < 1/2. 5. (a) State the Cauchy integral formula. (4 pts.) (by Use the Cauchy integral formula to Compute (6 pts.) [eera (m= 0,41, £2,---; C:2(t) =e", 0<t < Qn). c