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) |
Hyper-elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyper-elliptic_integral&oldid=13571