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
Let \(K\subset\CC\) be compact. Choose a closed square \(Q\) whose interior contains \(K\), put \(\delta=\operatorname{dist}(K,\partial Q)>0\), and let \(L\) be the perimeter of \(Q\).
1. \(f_n\to g\) uniformly on \(\partial Q\).
\(\partial Q\) is the union of four line segments, and \(f_n\to g\) uniformly on each.
2. \(\sup_{z\in K}\abs{f_n(z)-f_m(z)}\le{L\over2\pi\delta}\sup_{\xi\in\partial Q}\abs{f_n(\xi)-f_m(\xi)}\).
For \(z\in K\), the Cauchy integral formula applied to the entire function \(f_n-f_m\) gives \[ f_n(z)-f_m(z)={1\over2\pi i}\oint_{\partial Q}{f_n(\xi)-f_m(\xi)\over\xi-z}\dxi ,\] and \(\abs{\xi-z}\ge\delta\) on \(\partial Q\).
3. \(f_n\to g\) uniformly on \(K\).
By step 1 the right side of step 2 tends to \(0\) as \(n,m\to\infty\), so \((f_n)\) is uniformly Cauchy on \(K\). It converges uniformly on \(K\), and the limit is the pointwise limit \(g\).
4. \(g\) is entire.
By step 3, \(g\) is a uniform limit of continuous functions on every compact set, hence continuous. For every closed triangle \(T\), \(\int_{\partial T} g=\lim_n\int_{\partial T} f_n=0\) by uniform convergence on \(\partial T\) and Cauchy’s theorem. By Morera’s theorem, \(g\) is entire.
5. Q.E.D.
Steps 4 and 3 are the two claims.