bundles and a generalization of the theorem to algebraic curves. Examples The theorem will be illustrated by picking a point on the surface in question and regarding the sequence of numbers i.e., the dimension of the space of functions that are holomorphic everywhere except at where the function is allowed to have a pole of order at most . For , the functions are thus required to be entire, i.e., holomorphic on the whole surface . By Liouville's theorem, such a function is necessarily constant. Therefore, . In general, the sequence is an increasing sequence. Genus zero The Riemann sphere (also called complex projective line) is simply connected and hence its first singular homology is zero. In particular its genus is zero. The sphere can be covered by two copies of , with transition map being given by Therefore, the form on one copy of extends to a meromorphic form on the Riemann sphere: it has a double pole at infinity, since Thus, its divisor (where is the point at infinity). Therefore, the theorem says that the sequence reads 1, 2, 3, ... . This sequence can also be read off from the theory of partial fractions. Conversely if this sequence starts this way, then must be zero. Genus one The next case is a Riemann surface of genus , such as a torus , where is a two-dimensional lattice (a group isomorphic to ). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right. The standard complex coordinate on yields a one-form on that is everywhere holomorphic, i.e., has no poles at all. Therefore, , the divisor of is zero. On this surface, this sequence is 1, 1, 2, 3, 4, 5 ... ; and this characterises the case . Indeed, for , , as was mentioned above. For with , the degree of is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of elliptic functions. Genus two and beyond For , the sequence mentioned above is 1, 1, ?, 2, 3, ... . It is shown from this that the ? term of degree 2 is either 1 or 2, depending on the point. It can be proven that in any genus 2 curve there are exactly six points whose sequences are 1,