Question

v=log (x^2+y^2)=x-2y is harmonic

Answer 1

Answer:

Show that u=12log(x2+y2) is harmonic and find its harmonic conjugate function.

I did the first part to show that ∂2u∂x2+∂2u∂y2=0

Now, to find v:

∂u∂x=xx2+y2=∂v∂y

I get v=tan−1(yx)

∂u∂y=yx2+y2=−∂v∂x

Here I get v=−tan−1(xy)