Question

find the imagnary curve part of the analytic fungction​

Answer 1

We know what v(x,y)v(x,y) looks like, but we need u(x,y)u(x,y). The Cauchy Riemann equations tell us that

∂u∂x=∂v∂y∂u∂y=−∂v∂x∂u∂x=∂v∂y∂u∂y=−∂v∂x

Can we use these to deduce what uu is? Sure! By the first equation, we have

u=∫∂v∂ydx+f(y)u=∫∂v∂ydx+f(y)

Sub this into the second equation to get

∂∂y[∫∂v∂ydx+f(y)]=−∂v∂x∂∂y[∫∂v∂ydx+f(y)]=−∂v∂x

Use this to find f(y)f(y).