Skip to content

Commit

Permalink
Very minor fixes, up the version
Browse files Browse the repository at this point in the history
  • Loading branch information
jirilebl committed May 16, 2023
1 parent 47efda0 commit c14284a
Show file tree
Hide file tree
Showing 2 changed files with 9 additions and 9 deletions.
16 changes: 8 additions & 8 deletions ca.tex
Original file line number Diff line number Diff line change
Expand Up @@ -409,7 +409,7 @@
Ji{\v r}\'i Lebl\\[3ex]}
\today
\\
(version 1.3)
(version 1.4)
\end{minipage}}

%\addtolength{\textwidth}{\centeroffset}
Expand All @@ -427,7 +427,7 @@
\bigskip

\noindent
Copyright \copyright 2019--2022 Ji{\v r}\'i Lebl
Copyright \copyright 2019--2023 Ji{\v r}\'i Lebl

%PRINT
% not for the coil version
Expand Down Expand Up @@ -841,7 +841,7 @@ \subsection{The geometry and topology of the plane}
This version ought to be called
Cauchy inequality, but lamentably that name could refer to a different
inequality, one that we will call the Cauchy estimates.}, note: $\Re z
\bar{w}$ is the real dot product),\index{Cauchy--Schwarz}
\bar{w}$ is the $\R^2$ dot product),\index{Cauchy--Schwarz}
\item
$\sabs{z+w} \leq \sabs{z} + \sabs{w}$ \quad (Triangle inequality).%
\index{triangle inequality!complex numbers}
Expand Down Expand Up @@ -16106,7 +16106,7 @@ \subsection{In the plane}
with no limit points in $\C$ and $\{ m_k \}$ is a sequence of
natural numbers.
Then there exists an entire holomorphic $f \colon \C \to \C$ that
has zeros exactly at $c_k$, with orders given by $m_k$.
has zeros exactly at $c_k$, with orders given by~$m_k$.

More precisely, suppose $\{ a_n \}$ is the sequence of nonzero $\{ c_k \}$ with points
repeated according to the multiplicities $\{ m_k \}$ and $m$
Expand Down Expand Up @@ -16279,12 +16279,12 @@ \subsection{Factorization of sine}
\pi z \prod_{n=1}^\infty
\left(1-\frac{z^2}{n^2}\right) .
\end{equation*}
That's a rather nice factorization. Of course we still do not know if $f(z)$ is
That's a rather nice factorization. Of course, we still do not know if $f(z)$ is
$\sin(\pi z)$.
All we know is that the two have the same zeros,
and the derivative at 0 is $\pi$ as it should be.
As $f$ captures the zeros of $\sin(\pi z)$, we write
(as in the factorization theorem),
(as in the factorization theorem)
\begin{equation*}
\sin(\pi z)
=
Expand All @@ -16293,7 +16293,7 @@ \subsection{Factorization of sine}
\pi z
e^{g(z)}
\prod_{n=1}^\infty
\left(1-\frac{z^2}{n^2}\right)
\left(1-\frac{z^2}{n^2}\right) ,
\end{equation*}
for some entire function $g$. We need to show that $g \equiv 0$. By the
computation of the derivative above, $g(0) = 0$.
Expand Down Expand Up @@ -16465,7 +16465,7 @@ \subsection{The product theorem in any open set}
with no limit points in $U$, and $\{ m_k \}$ is a sequence of
natural numbers.
Then there exists a holomorphic $f \colon U \to \C$ with
zeros exactly at $c_k$, with orders given by $m_k$.
zeros exactly at $c_k$, with orders given by~$m_k$.
\end{thm}

\begin{proof}
Expand Down
2 changes: 1 addition & 1 deletion changes-draft.html
Original file line number Diff line number Diff line change
Expand Up @@ -3,7 +3,7 @@
<li>In the description of the typical application of Hurwitz,
be a bit more precise (besides fixing an erratum).
<li>In the proof of Proposition B.3.13, use j,k for the same thing
as in the definition B.3.12 and Proposition B.3.11
as in the Definition B.3.12 and Proposition B.3.11
for consistency. This required changing k to h',
which is more consistent with the naming of A' anyway.
Also use p for the fixed point rather than x for
Expand Down

0 comments on commit c14284a

Please sign in to comment.