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Fix erratum, minor fix
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jirilebl committed Jul 19, 2023
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6 changes: 3 additions & 3 deletions ca.tex
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Expand Up @@ -16278,7 +16278,7 @@ \subsection{Factorization of sine}
The $\pi$ out front can be guessed by thinking what would we get if we
differentiate $\sin(\pi z)$ and evaluate at $0$.
Differentiating the product (using product rule) would give you $\pi \cdot 1
+ 0=1$, as any time the derivative falls on some factor other that $z$, when
+ 0=\pi$, as any time the derivative falls on some factor other that $z$, when
you evaluate at $0$, you get $0$. We can do this formal computation on the
finite products as the product converges uniformly on compact subsets and
therefore so does the derivative. See also \exerciseref{exercise:proddiff}.
Expand All @@ -16293,11 +16293,11 @@ \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. 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
Because $f$ captures the zeros of $\sin(\pi z)$, we write
(as in the factorization theorem)
\begin{equation*}
\sin(\pi z)
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