[Math] About the existence of harmonic conjugate

complex-analysisharmonic functions

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):

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  1. 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 $u$

  2. 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

Why did we had to use $g$

Because we want to prove the theorem stated above. The existence of $g$ with $\operatorname{Re}g=u$ is a part of the theorem.

it seems that $−\dfrac{\partial u}{\partial y}$ is the harmonic conjugate of $u$

This is incorrect, as Potato and anon already pointed out.

what is the harmonic conjugate of $u$?

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.)

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