Show that the following functions are harmonic and determine the conjugate functions. 1. u = 2x...

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Show that the following functions are harmonic and determine the conjugate functions. 1. u = 2x (1- y) Ans. v = -y+ 2 y + C 2. u = 2x-x+3 xy Ans. v = 2 y- 3 xy + y' + C_ 3. u = log (x + y?) Ans. y = – tan -1+C Determine the analytic function, whose real part is 4. 2x (1- y) Ans. iz+2 z+ C 5. log V +y (Madras 1997S, Madurai 1996S) Ans. log z *º 6. -y +5x+y- +y 7. cos x cosh y 9. -3 xy + 3 -3 y² + 2 x + 1 Ans. ? + (5 – i) z -+C 8. 3xły + 2x - y - 2 y Ans. 2 ? - i ?+C Ans. cos z + c 10. e-* (x sin y - y cos y). (A.M.I.E.T.E., Winter 1997, Punjab 1997) Ans. z+32+2z+C 11. (a) e (x cos 2 y- y sin 2 y) Ans. i (ze+C) Ans. ze+ iC Scanned with CamScanner Functions of a Complex Variable 471 (h) e (x cos y +y sin y) and f (0) = i. (A.M.J.E.TE Winter 1997 Maneglare (997 Ans. ze+i Show that the following functions are harmonic and determine the conjugate functions. 1. u = 2x (1- y) Ans. v = -y+ 2 y + C 2. u = 2x-x+3 xy Ans. v = 2 y- 3 xy + y' + C_ 3. u = log (x + y?) Ans. y = – tan -1+C Determine the analytic function, whose real part is 4. 2x (1- y) Ans. iz+2 z+ C 5. log V +y (Madras 1997S, Madurai 1996S) Ans. log z *º 6. -y +5x+y- +y 7. cos x cosh y 9. -3 xy + 3 -3 y² + 2 x + 1 Ans. ? + (5 – i) z -+C 8. 3xły + 2x - y - 2 y Ans. 2 ? - i ?+C Ans. cos z + c 10. e-* (x sin y - y cos y). (A.M.I.E.T.E., Winter 1997, Punjab 1997) Ans. z+32+2z+C 11. (a) e (x cos 2 y- y sin 2 y) Ans. i (ze+C) Ans. ze+ iC Scanned with CamScanner Functions of a Complex Variable 471 (h) e (x cos y +y sin y) and f (0) = i. (A.M.J.E.TE Winter 1997 Maneglare (997 Ans. ze+i