AMATH 501 Homework #7 Due: Monday, December 7, 2015 1. Find and classify all singularities of the following functions. Be sure to justify your answer. ez f (z) = 3 2 z (z + 1) (a) (b) 2. f (z) = z+1 z sin z Evaluate I = C (5z + 7) dz (z 2 + 2z 3) using the calculus of residues (i.e., the residue theorem). Here, C is the closed, counterclockwise circle of radius 4 centered at the origin, z = 4. 3. Evaluate the following integral using contour integration and the calculus of residues. Be sure to discuss the behaviour of your integral on each part of your contour in the complex plane. I = x dx . (x 2 2x + 2) 2 4. Evaluate 2 I = (cos3 + sin2 ) d 0 using the calculus of residues. (Hint: After you reformulate your problem as a complex integral, your integrand can easily be rewritten as a Laurent series. You can then nd the residue by inspection.) \f