Hyper-elliptic integral

The special case of an Abelian integral

(1)

where is a rational function in variables , which are related by an algebraic equation of special type:

(2)

Here is a polynomial of degree without multiple roots. For one obtains elliptic integrals (cf. Elliptic integral), while the cases are sometimes denoted as ultra-elliptic.

Equation (2) corresponds to a two-sheeted compact Riemann surface of genus if is even, and of genus if is odd; thus, for hyper-elliptic integrals . The functions , , and hence also , are single-valued on . The integral (1), considered as a definite integral, is given on as a curvilinear integral of an analytic function taken along some rectifiable path and, in general, the value of the integral (1) is completely determined by a specification of the initial and final points of alone.

As in the general case of Abelian integrals, any hyper-elliptic integral can be expressed as a linear combination of elementary functions and canonical hyper-elliptic integrals of the first, second and third kinds, having their specific forms. Thus, a normal hyper-elliptic integral of the first kind is a linear combination of hyper-elliptic integrals of the first kind

where , , is the simplest basis of Abelian differentials (cf. Abelian differential) of the first kind for the case of a hyper-elliptic surface . Explicit expressions for Abelian differentials of the second and third kinds and for the corresponding hyper-elliptic integrals can also be readily computed [2]. Basically, the theory of hyper-elliptic integrals coincides with the general theory of Abelian integrals.

All rational functions of variables and satisfying equation (2) above form a hyper-elliptic field of algebraic functions, of genus . Any compact Riemann surface of genus or has an elliptic or hyper-elliptic field, respectively. However, if or higher, there exist compact Riemann surfaces of a complicated structure for which this assertion is no longer true.

References

[1]	G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt. 10
[2]	R. Nevanlinna, "Uniformisierung" , Springer (1953) pp. Chapt.5
[3]	K. Neumann, "Vorlesungen uber Riemanns Theorie der Abelschen Integrale" , Leipzig (1884)