Question

Show that the following function is harmonic and

determine its conjugate function. Also determine its analytic function

f (z) =u+ iv

. Given

that

u= 2x(1-y)​

Answer 1

Step-by-step explanation:

Given imaginary part v=ex(xsiny+ycosy)v=ex(xsiny+ycosy)

∴vx=∂V∂x=ex(siny)+(xsiny+ycosy)ex∴v2x=∂2V∂x2=ex(siny)+ex(siny)+(xsiny+ycosy)exv2x=ex(−xsiny+y(−cosy)−siny−siny)vy=∂V∂y=ex(−xsiny+y(−csoy)−siny−siny)vy=∂V∂y=ex(−xsiny−ycosy)−2siny)v2