Morera theorem
Theorem. If \(f\) is continuous on a domain \(U\) and \(\int_{\partial T}f(z)\,\mathrm{d}z=0\) for every triangle \(T\subset U\), then \(f\) is holomorphic. The vanishing triangle integrals construct local primitives, whose derivatives recover \(f\).
Locally uniform convergence
Definition. A sequence \(f_n\) converges locally uniformly to \(f\) if convergence is uniform on every compact subset of the domain. This is the natural convergence for holomorphic functions because contour integrals live on compact curves.
Weierstrass convergence theorem
Theorem. If \(f_n\) are holomorphic on \(U\) and \(f_n\to f\) locally uniformly, then \(f\) is holomorphic and \(f_n^{(k)}\to f^{(k)}\) locally uniformly for every \(k\). Cauchy's integral formula permits the limit to pass through the integral.
Proof by Morera
Proof. Continuity follows from locally uniform convergence. For every triangle, \(\int f\,\mathrm{d}z=\lim_n\int f_n\,\mathrm{d}z=0\), since the convergence is uniform on the triangle boundary. Morera's theorem then gives holomorphicity.
Normal convergence of series
Corollary. If \(\sum f_n\) converges uniformly on compact subsets and each \(f_n\) is holomorphic, then the sum is holomorphic and may be differentiated term by term locally uniformly.
Why uniformity is needed
Remark. Pointwise limits of holomorphic functions need not be holomorphic. Local uniform control prevents singular behavior from forming invisibly on compact sets.
Morera's converse
Theorem (Morera). Let \(f\) be continuous on a domain \(\Omega\). If \[ \int_{\partial\Delta}f(z)\,\mathrm{d}z=0 \] for every triangle \(\Delta\) whose closure lies in \(\Omega\), then \(f\) is holomorphic.
Proof. On each disc \(D\Subset\Omega\), fix \(z_0\) and define \(F(z)=\int_{[z_0,z]}f\,\mathrm{d}z\). Triangle-integral vanishing shows this agrees with integration along any polygonal path in the disc. The difference quotient along a segment converges to \(f(z)\), so \(F'=f\). Hence \(f\), as the derivative of a holomorphic function, is holomorphic; alternatively Cauchy's formula for \(F\) shows \(F'\) is holomorphic.
Weierstrass convergence theorem
Theorem. If holomorphic \(f_n\) converge locally uniformly on \(\Omega\) to \(f\), then \(f\) is holomorphic and, for every \(k\ge0\), \[ f_n^{(k)}\to f^{(k)} \] locally uniformly.
Proof. Local uniform convergence gives continuity of \(f\). Integrals over each triangle pass to the limit, so Morera gives holomorphy. For a compact \(K\Subset\Omega\), choose contours at positive distance around \(K\). Cauchy's derivative formula yields \[ f_n^{(k)}(z)-f^{(k)}(z) =\frac{k!}{2\pi i}\int_\Gamma \frac{f_n(\zeta)-f(\zeta)}{(\zeta-z)^{k+1}}\,\mathrm{d}\zeta, \] and the uniform bound tends to zero on \(K\).
Normally convergent series
Corollary. If \(\sum_n f_n\) is a series of holomorphic functions and for every compact \(K\Subset\Omega\), \[ \sum_n\sup_K|f_n|<\infty, \] then the series is holomorphic and may be differentiated termwise locally uniformly.
This “normal convergence” is stronger than pointwise absolute convergence and is the natural hypothesis for infinite products, Mittag--Leffler series, and parameter-dependent expansions.
Holomorphic parameter integrals
Proposition. Suppose \(f(z,t)\) is holomorphic in \(z\) for every \(t\), measurable in \(t\), and for each compact \(K\Subset\Omega\) is dominated by an integrable \(g_K(t)\). Then \[ F(z)=\int f(z,t)\,\nu(\mathrm{d}t) \] is holomorphic, and if derivatives admit corresponding domination, \(F'(z)=\int\partial_z f(z,t)\,\nu(\mathrm{d}t)\).
Proof. Dominated convergence gives continuity. Fubini and Cauchy's theorem make every triangle integral of \(F\) zero. Morera proves holomorphy. The derivative formula follows from Cauchy's formula or dominated difference quotients.