Determine the analytic tunction, whose imaginary part is log (x? + y) +x- 2 y 12. v = (A.M.I.E.T.E., Summer 1997) Ans. 2 i log z- (2 i) z + C 13. v = sinhx cos y Ans. sin iz + C x- y 14. v =+Y Ans, ( + 15. v = - Ans. +C cos x+ sin x -e-y 2 cos x-2 cosh y 16. u- v = and = 0 (A.M.I.E.T.E., Winter 2000) 1- cot Ans. 17. v = sin 0 r- Ans. z + + c 18. If f (z) = u + iv is an analytic function of z = x+ iy and u-v = - cos x+ sin x cosh y - cos x find f(z) subject to the 3- condition that f(S (A.M.I.E.T.E. Summer 1999) Ans. f(z) = cot 19. Find an analytic function f (z) = u (r, 0) + iv (r, 0) such that V (r, 0) = r cos 2 0-r cos 0 + 2. Ans. i [2 -z+ 2] 20. Show that the function u = - y - 2ry - 2.x y-1 is harmonic. Find the conjugate harmonic function v and express u + iv as a function of z where z=x+ iy. (A.M.I.E.T.E., Summer 1997) Ans. (1+i)2+ (-2+ ) z 1 21. Construct an analytic function of the form f(z) = u+ iv, where v is v = tan' (y/x), x*0, y # 0. (A.M.I.E.T.E, Winter 1996, Tiruchirapalli, 1996S) Ans. log cz 22. Choose the correct answer: The value of m so that 2 x- + my may be harmonic is (c) 2 (b) 1 (d) 3 Ans. (b) (a) 0 23. If f (z) = u+iv is an analytic function of z in any domain, show that (A.M.I.E.T.E., Summer 2000) (a) %3D 2 sin 2x and f(z) = u + iv is an analytic function of z =x+ iy, find f (z) in terms 24. If u + v = 2y + e-2y 2 cos 2 0 (A.M.I.E.T.E., Winter 2002) Ans. (1 + i) co cot z+ u of z.