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)