View source for Hyper-elliptic integral

The special case of an [[Abelian integral|Abelian integral]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482202.png" /> is a [[Rational function|rational function]] in variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482204.png" /> which are related by an algebraic equation of special type:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482205.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482206.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482207.png" /> without multiple roots. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482208.png" /> one obtains elliptic integrals (cf. [[Elliptic integral|Elliptic integral]]), while the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h0482209.png" /> are sometimes denoted as ultra-elliptic.

Equation (2) corresponds to a two-sheeted compact [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822010.png" /> of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822011.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822012.png" /> is even, and of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822013.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822014.png" /> is odd; thus, for hyper-elliptic integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822015.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822017.png" />, and hence also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822018.png" />, are single-valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822019.png" />. The integral (1), considered as a definite integral, is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822020.png" /> as a [[Curvilinear integral|curvilinear integral]] of an analytic function taken along some rectifiable path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822021.png" /> and, in general, the value of the integral (1) is completely determined by a specification of the initial and final points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822022.png" /> 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822025.png" />, is the simplest basis of Abelian differentials (cf. [[Abelian differential|Abelian differential]]) of the first kind for the case of a hyper-elliptic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822026.png" />. Explicit expressions for Abelian differentials of the second and third kinds and for the corresponding hyper-elliptic integrals can also be readily computed [[#References|[2]]]. Basically, the theory of hyper-elliptic integrals coincides with the general theory of Abelian integrals.

All rational functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822027.png" /> of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822029.png" /> satisfying equation (2) above form a hyper-elliptic field of algebraic functions, of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822030.png" />. Any compact Riemann surface of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822031.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822032.png" /> has an elliptic or hyper-elliptic field, respectively. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822033.png" /> or higher, there exist compact Riemann surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048220/h04822034.png" /> of a complicated structure for which this assertion is no longer true.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Springer,   "Introduction to Riemann surfaces" , Addison-Wesley  (1957)  pp. Chapt. 10</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Nevanlinna,   "Uniformisierung" , Springer  (1953)  pp. Chapt.5</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Neumann,   "Vorlesungen uber Riemanns Theorie der Abelschen Integrale" , Leipzig  (1884)</TD></TR></table>