I am reading Donald Sarason's "Notes on Complex Function Theory".
I have two questions about the following (taken from page $88$ of
the book):
-
Why did we had to use $g$ ? We already had $f$ which was claimed
to be holomorphic, so it seems that$-\frac{\partial u}{\partial y}$
is the harmonic conjugate of $u$ -
I'm guessing I am wrong in my statement that $-\frac{\partial u}{\partial y}$
is the harmonic conjugate of $u$, what is the harmonic conjugate
?
Best Answer
Because we want to prove the theorem stated above. The existence of $g$ with $\operatorname{Re}g=u$ is a part of the theorem.
This is incorrect, as Potato and anon already pointed out.
Since $\operatorname{Re}g=u$, the function $\operatorname{Im}g$ is a harmonic conjugate of $u$. (Not the harmonic conjugate, since we can add any constant and get other conjugate functions.)