Problem Set 2: Complex Analysis (6) Let G be a region and G1 := { z : z-bar E G } where z-bar means the complex conjugate of z. Assume that f : G --------> C is analytic. Define g : G1-----------> C as g(z) = f(z)-bar where z E G1 (once again, here g(z)-bar means the conjugate of g(z) , one can see that why writing mathematical notation in FB is difficult !!!! ). Prove that g (z) is analytic on G1. (7) Let f be analytic. Then which one is correct ? (i) [ D_x ( |f(z)| ) ]^2 + [ D_y ( |f(z)| ) ]^2 = |f (z)|^2 (ii) [ D^2_x + D^2_y ] (|f(z)|)^2 = 4|f (z)|^2 (here D_x stands for the partial derivative with respect to x; D^2_x denotes second order partial derivative with respect to x. Similalr notation for D_y and D^2_y ) (8) Justify: any harmonic function satisfies the equation : D^2 (u)/(DzD(z-bar)) = 0 where D stands for partial derivative. (9) Justify: There cannot exist any analytic function f on an open set E (subset C )with real part x - 2y^2 . Link to previous set https://facebook/groups/325029324237962/permalink/758648774209346/
Posted on: Sat, 22 Nov 2014 05:12:35 +0000
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