Hyper-elliptic integral

The special case of an Abelian integral

$$ \tag{1 } \int\limits R ( z, w) dz, $$

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

$$ \tag{2 } w ^ {2} = P ( z). $$

Here $ P( z) $ is a polynomial of degree $ m \geq 5 $ without multiple roots. For $ m = 3, 4 $ one obtains elliptic integrals (cf. Elliptic integral), while the cases $ m = 5, 6 $ are sometimes denoted as ultra-elliptic.

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

$$ \int\limits \frac{z ^ {\nu - 1 } dz }{w} ,\ \ \nu = 1 \dots g, $$

where $ ( z ^ {\nu - 1 } / w) d z $, $ \nu = 1 \dots g $, is the simplest basis of Abelian differentials (cf. Abelian differential) of the first kind for the case of a hyper-elliptic surface $ F $. 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 $ R( z, w) $ of variables $ z $ and $ w $ satisfying equation (2) above form a hyper-elliptic field of algebraic functions, of genus $ g $. Any compact Riemann surface of genus $ g = 1 $ or $ g = 2 $ has an elliptic or hyper-elliptic field, respectively. However, if $ g = 3 $ or higher, there exist compact Riemann surfaces $ F $ 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)