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Difference between revisions of "Bilinear differential"

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An analytic [[Differential on a Riemann surface|differential on a Riemann surface]], depending on two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162402.png" />, and having the form
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<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/b/b016/b016240/b0162403.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162405.png" /> are local uniformizing parameters in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162407.png" /> respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162408.png" /> is an analytic function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b0162409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016240/b01624010.png" />. Bilinear differentials are used to express many functionals on a [[Finite Riemann surface|finite Riemann surface]].
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An analytic [[Differential on a Riemann surface|differential on a Riemann surface]], depending on two points  $  P $
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and  $  Q $,
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and having the form
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$$
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f (z, \zeta )  dz  d \zeta ,
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$$
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where $  z $
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and $  \zeta $
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are local uniformizing parameters in a neighbourhood of $  P $
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and $  Q $
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respectively, and $  f(z, \zeta ) $
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is an analytic function of $  z $
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and $  \zeta $.  
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Bilinear differentials are used to express many functionals on a [[Finite Riemann surface|finite Riemann surface]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Schiffer,  D.C. Spencer,  "Functionals of finite Riemann surfaces" , Princeton Univ. Press  (1954)</TD></TR></table>

Latest revision as of 10:59, 29 May 2020


An analytic differential on a Riemann surface, depending on two points $ P $ and $ Q $, and having the form

$$ f (z, \zeta ) dz d \zeta , $$

where $ z $ and $ \zeta $ are local uniformizing parameters in a neighbourhood of $ P $ and $ Q $ respectively, and $ f(z, \zeta ) $ is an analytic function of $ z $ and $ \zeta $. Bilinear differentials are used to express many functionals on a finite Riemann surface.

References

[1] M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)
How to Cite This Entry:
Bilinear differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bilinear_differential&oldid=19114
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article