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The special case of an [[Abelian integral|Abelian 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>
+
$$ \tag{1 }
 +
\int\limits R ( z, w) dz,
 +
$$
  
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:
+
where $  R $
 +
is a [[Rational function|rational function]] in variables $  z $,  
 +
$  w $
 +
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>
+
$$ \tag{2 }
 +
w  ^ {2}  = P ( z).
 +
$$
  
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.
+
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|Elliptic integral]]), while the cases $  m = 5, 6 $
 +
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.
+
Equation (2) corresponds to a two-sheeted compact [[Riemann surface|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|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
 
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>
+
$$
 +
\int\limits
 +
\frac{z ^ {\nu - 1 }  dz }{w}
 +
,\ \
 +
\nu = 1 \dots g,
 +
$$
  
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.
+
where $  ( z ^ {\nu - 1 } / w)  d z $,  
 +
$  \nu = 1 \dots g $,  
 +
is the simplest basis of Abelian differentials (cf. [[Abelian differential|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 [[#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.
+
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====
 
====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>
 
<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>

Latest revision as of 22:11, 5 June 2020


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)
How to Cite This Entry:
Hyper-elliptic integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyper-elliptic_integral&oldid=47286
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article