I'm reading the post Bounded harmonic function is constant .I've some querry related to fiorerb's
answer.
I'm not getting how to prove,"if $u$ is harmonic then $f$ is analytic.
Related to this problem,i've found a theorem in Brown and Churchill's Complex analysis .Which is like
"A function $f(z)=u(x,y)+iv(x,y)$ is analytic in a domain $D$ iff $v$ is a harmonic conjugate of $u$"
from this theorem,if $v$ is harmonic conjugate of $u$ only then $f$ is analytic.But, it is nowhere given that $v $ is a harmonic conjugate of $u$.So, how can $f$ be analytic?