I'm studying the Dirichlet problem in the Complex Analysis book from Stein and Shakarchi.
For the exercises I was a bit confused so I looked up examples and came across this this useful page. I understand how they find/write the solution for $u(x,y)$ in the examples and also that you have to show that $u$ is truly a solution. But I am confused on how they show that $u$ is harmonic.
I know that if $f$ is a analytic/holomorphic function then Re$(f)$ and Im$(f)$ are harmonic. In the examples they use the logarithm, but I don't understand why they use it, how they use it and how they find the function $\Phi$ where $u = \text{Re}(\Phi)$.
Lastly, I don't see how exactly the theory in the book connects to the examples (chapter 8 section 1.3 The Dirichlet problem in a strip). I understand lemma 1.3, and see how they use that,
(lemma 1.3: Let $V$ and $U$ be open sets in $\mathbb{C}$ and $F: V \to U$ a holomorphic function. If $u: U \to \mathbb{C}$ is a harmonic function then $u \circ F$ is harmonic on $V$.)
but I don't know how to solve the questions using the theory. Without seeing the examples from the link I would have not known how to find $u$ and how to prove that $u$ is harmonic.