Suppose \(\theset{f_n}_{n\in \NN}\) is a sequence of entire functions where
\(f_n \to g\) pointwise for some \(g:\CC\to\CC\).
On every line segment in \(\CC\), \(f_n \to g\) uniformly.
Show that
\(g\) is entire, and
\(f_n\to g\) uniformly on every compact subset of \(\CC\).
Solution
Goal: Suppose \((f_n)\) is a sequence of entire functions with \(f_n \to g\) pointwise, and \(f_n \to g\) uniformly on every line segment in \(\CC\). Show that \(g\) is entire and \(f_n \to g\) uniformly on every compact subset of \(\CC\).
1. \(g\) is holomorphic on \(\CC\).
1.1. It suffices to show \(\oint_\gamma g = 0\) for every closed polygonal path \(\gamma\).
Morera’s theorem.
1.2. For a closed polygonal path \(\gamma\) (a finite union of line segments), \(\int_\gamma g = \lim_n \int_\gamma f_n\).
\(\gamma\) is a finite union of line segments; on each, \(f_n \to g\) uniformly by hypothesis, so the integrals converge.
1.3. \(\int_\gamma g = 0\).
each \(f_n\) is entire, so \(\oint_\gamma f_n = 0\) (Cauchy’s theorem); 2 gives the limit \(0\).
2. \(f_n \to g\) uniformly on compact sets.
2.1. Let \(K \subset \CC\) be compact; choose a bounded polygonal domain \(\Omega\) with \(K \subset \Omega\) and \(\dist(K, \bd\Omega) > 0\).
e.g. a large square (or union of squares) around \(K\), whose boundary is a finite union of line segments.
2.2. For \(z \in K\): \(f_n(z) - g(z) = \frac{1}{2\pi i}\oint_{\bd\Omega} \frac{f_n(\zeta) - g(\zeta)}{\zeta - z}\, d\zeta\).
Cauchy integral formula applied to the holomorphic functions \(f_n\) and \(g\) (entire by 1) on the domain \(\Omega\).
2.3. \(\sup_{z \in K}|f_n(z) - g(z)| \le \frac{\operatorname{length}(\bd\Omega)}{2\pi\, d} \sup_{\zeta \in \bd\Omega}|f_n(\zeta) - g(\zeta)|\) where \(d = \dist(K, \bd\Omega)\).
bound the integral in 2: \(|\zeta - z| \ge d\).
2.4. The right-hand side tends to \(0\).
\(\bd\Omega\) is a finite union of line segments, and \(f_n \to g\) uniformly on each, so \(\sup_{\bd\Omega}|f_n - g| \to 0\).
3. Q.E.D.
1 shows \(g\) is entire; 2 shows uniform convergence on compacta.