11 5 1 Show that the following functions are harmonic and

Last updated: 4/24/2023

11 5 1 Show that the following functions are harmonic and find its har monic conjugates a x y c sinhrsing b xy 3x y y d e cos 2ry f Solution a Set u r y x y 1 2y 2r Au 0 thus is the function u x y is harmonic Cauchy Riemann give us Thus we have that thus b Set e tan y r x 0 x x y v u 2y v x y 2xy g y vy 2r g y v 2r u x y xy 3x y y g y 0 g y C v x y 2xy C 6y y 6y thus is the function u r y is harmonic Cauchy Riemann give us u y 6ry u x 3x 3y 52 u x 3x 3y v x y 3ry 9 y t 6ry g y Vy U y 6zy Thus we have that g x y g y C 2 Au 0