Complex Analysis/Meromorphic functions and the Riemann sphere

singularity Let be an open subset of , let and finally . Suppose that is unbounded in every neighbourhood of . Then (and only then) is called a singularity of . pole Let be an open subset of , let and finally . Suppose that is a singularity of , but there exists an such that the function is bounded in a neighbourhood of (whence by the Riemann removability theorem it may be holomorphically continued into the whole of ). Then (and only then) is called a pole of . order Let be an open subset of , let and finally . If is a pole of , the natural number from the definition of a pole is called the order of the pole . essential singularity An essential singularity is a singularity which is not a pole. meromorphic Let be an open subset of , let be discrete and let . We call a meromorphic function on if and only if at least one of the elements of is a pole of and all elements of are either poles or singularities of . existence and uniqueness of the Laurent expansion in a punctuated ball existence and uniqueness of the Laurent decomposition in a punctuated ball existence and uniqueness of the Laurent decomposition in an open set Marty's theorem A family of functions is normal if and only if for every sequence in , either the sequence or the sequence contains a subsequence that uniformly converges to a holomorphic function. Suppose first that is normal. Then there exists a constant such that . Let then be a sequence in . Then either is a bounded sequence, or there exists a subsequence of such that for a certain constant . Since , we have , we may infer from Montel's theorem that is normal, whence it contains a convergent subsequence. The opposite direction follows immediately from Montel's theorem and the symmetry of the spherical derivative. Exercises