A hyperelliptic integral is by de?nition an integral of the form dz ? , f(z) ? where? is a path in the complex plane C with coordinatez andf(z)= (z? a )···(z?a ) with pairwise different constantsa.Ifd= degf is1or2,an 1 d i explicit integration by elementary functions is well known from calculus. Ifd= 3 or 4, integration is possible using elliptic functions. If howeverd? 5, no explicit integration is known in general. dz ? The reason for this is the following: the differential?= is not single valued, f(z) consideredasafunctiononC.LetC denotethecompactRiemannsurfaceassociated ? to f. By de?nitionC is the double covering of the Riemann sphere P , rami?ed 1 at the pointsa,...,a together with? ifd is odd. Now? may be considered as a 1 d holomorphic differential onC. It is essentially the topological structure ofC which causes the problem. The more complicated it is, the more dif?cult it is to integrate ?. At the beginning of the 19th century the Norwegian mathematician Niels Henrik Abel (1802–1829) and the German mathematician Carl Gustav Jacob Jacobi (1804– 1851) found a way to attack this problem. In geometric terms their method can be described as follows. The idea is to try to integrate not? alone, but simultaneously the whole set of holomorphic differentials dz d? 1 i?1 ?=z ? for i= 1,...,g= i f(z) 2 onC. For this, ?x a pointp?C and consider the map 0