The question asked me to find the harmonic conjugate of $u$ and then express $f(x,y)$ in terms of $z$. I found the conjugate. I also reasoned that it must be the expanded form of $z^2$. However, they give the answer and it includes a $c$. Why is this?
My answer: $$f(x,y)=x^2-y^2+2xyi$$ $$f(z)=z^2$$
Given answer: $$f(z)=z^2+c$$