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A few more minor typos fixed
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jirilebl committed Jul 8, 2022
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Expand Up @@ -13536,7 +13536,7 @@ \subsection{Real and imaginary parts of holomorphic functions}
As in \exerciseref{exercise:Liouvilleharmonic},
the analogue of \myquote{bounded} for holomorphic functions is
\myquote{nonnegative}
for harmonic functions. Afterall, if $f$ is
for harmonic functions. After all, if $f$ is
a bounded holomorphic function, then $\log \sabs{f(z)+M}$ or $\Re f(z) + M$
is nonnegative for large enough $M$. Conversely if $\log \sabs{f(z)} \geq
0$, then $\frac{1}{f(z)}$ is bounded, and if $\Re f(z) \geq 0$, then
Expand Down Expand Up @@ -14190,7 +14190,7 @@ \subsection{The Dirichlet problem in a disc and the Poisson kernel}
Define $\Phi(t) = \int_0^t \varphi(s)\, ds$, show that $\Phi$
is increasing, continuous, but not differentiable
on a dense set in $[0,1]$. Use it to construct a
$\psi(t)$ that is $2\pi$-periodic, continuous and
$\psi(t)$ that is $2\pi$-periodic, continuous, and
not differentiable on a dense subset of $\R$.
\item
Find a continuous $u \colon \overline{\D} \to \R$
Expand Down Expand Up @@ -15633,8 +15633,8 @@ \subsection{Applications, Rad\'o's theorem}
for simply connected domains provided we know the Riemann map. However,
the kernel is difficult to compute in general, and it requires a very nice
boundary to be able to integrate. The Perron method works much more
generally provided you can construct enough subharmonic functions (which can
afterall be pieced together unlike harmonic functions).
generally provided you can construct enough subharmonic functions (which
can, after all, be pieced together unlike harmonic functions).

If a solution exists, it clearly must equal to the Perron solution.

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