3. (a) Given the function u x y e y x ( , ) = cos , find it's harmonic conjugate v(x, y). Determine the analytic function f (z) and its derivative, f '(z) . (b) If 3 3 ln ( ) + ? = z z f z , where ? 5 2 f (1) = ?i and any branch line lies on the negative real axis, evaluate ? f ( about the circle z)dz z ? 6 = 1. (Hint: First determine the specific branch where the function is to be evaluated.)

(a) Given the function u(x, y) = e\"r cos y , nd it's harmonic conjugate v(x, y) . Determine the analytic function f (z) and its derivative, f '(z) . 1n z ,where f (l) = i 2 7: and any branch line lies on the negative If 2 = 09) f ( ) m_ 3 5 real axis, evaluate {a f (z)dz about the circle lz 6| =1. (Hint: First determine the specic branch where the function is to be evaluated.)