Harmonic Conjugates in Complex Analysis: Finding the Right Solution

[Genericcoder]

Find the harmonic conjugate of u. u = u(z) = ln(|z|) so u(z) = ln(sqrt(x^2 + y^2))

so basically I am trying to find now its harmonic conjugate I did all the math

I got two solutions though one is v(z) = arctan(y/x) + C if I solve Au/Ax = -Au/Ay & other is
v(z) = - arctan(x/y) + C if I solved Av/Ay = Au/ax
so I was wondering can I ave two solutions or wat ? or is one solution more right than other?

 

[Office_Shredder]

It might help to know that $\arctan(x) + \arctan(1/x)$ is locally a constant. Try taking the derivative!

 

[Genericcoder]

so I took derivative and found x = 0 so does that mean C = 0 but I don't know how to get from here ?

 

[Office_Shredder]

so I took derivative and found x = 0 so does that mean C = 0 but I don't know how to get from here ?

I don't understand what you are saying here.

 

[Genericcoder]

I tried first to see arctan(x) + arctan(x^-1) to see if its a local constant on wolf ram alpha I didn't see any I don't understand how can that help.