Weierstrass' convergence theorem. The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives. This is an immediate consequence of Morera's theorem for the first statement. Cauchy's integral formula gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta.
weierstrass' convergence theorem the uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives this is an immediate consequence of morera's theorem for the first statement cauchy's integral formula gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta