This construction is helpful in the study of holomorphic and meromorphic functions. The remarkable behavior of holomorphic functions near essential singularities is described by Picard's Theorem. The Weierstrass preparation theorem implies that rings of germs of holomorphic functions are Noetherian rings. The theorem follows from the fact that holomorphic functions are analytic. For a proof of this theorem, see analyticity of holomorphic functions. The opposite is true in the algebraic and holomorphic settings. This follows directly from the identity theorem for holomorphic functions. These properties are consequences of the holomorphic functional calculus applied to matrices. The Bergman spaces are another family of Hilbert spaces of holomorphic functions. Let formula 15 be holomorphic on a compact Riemann surface formula 16. The etymology of "holonomy" shares the first part with "holomorphic" ("holos"). It is known from Morera's theorem that the uniform limit of holomorphic functions is holomorphic. The definition of a holomorphic function generalizes to several complex variables in a straightforward way. All holomorphic functions are complex-analytic. In several ways, the harmonic functions are real analogues to holomorphic functions.