Namespaces
Variants
Actions

Difference between revisions of "Rodrigues formula"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
A formula relating the differential of the [[Normal|normal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825401.png" /> to a surface to the differential of the [[Radius vector|radius vector]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825402.png" /> of the surface in the [[Principal direction|principal direction]]:
+
<!--
 +
r0825401.png
 +
$#A+1 = 11 n = 0
 +
$#C+1 = 11 : ~/encyclopedia/old_files/data/R082/R.0802540 Rodrigues formula
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/r/r082/r082540/r0825403.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825405.png" /> are the principal curvatures.
+
A formula relating the differential of the [[Normal|normal]]  $  \mathbf n $
 +
to a surface to the differential of the [[Radius vector|radius vector]]  $  \mathbf r $
 +
of the surface in the [[Principal direction|principal direction]]:
 +
 
 +
$$
 +
d \mathbf n  = - k _ {1}  d \mathbf r \ \
 +
\textrm{ or } \ \
 +
d \mathbf n  = - k _ {2}  d \mathbf r ,
 +
$$
 +
 
 +
where  $  k _ {1} $
 +
and $  k _ {2} $
 +
are the principal curvatures.
  
 
The formula was obtained by O. Rodrigues (1815).
 
The formula was obtained by O. Rodrigues (1815).
Line 9: Line 29:
 
''A.B. Ivanov''
 
''A.B. Ivanov''
  
A representation of [[Orthogonal polynomials|orthogonal polynomials]] in terms of a weight function using differentiation. If a weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825406.png" /> satisfies a Pearson differential equation
+
A representation of [[Orthogonal polynomials|orthogonal polynomials]] in terms of a weight function using differentiation. If a weight function $  h ( x) $
 +
satisfies a Pearson differential equation
 +
 
 +
$$
 +
 
 +
\frac{h  ^  \prime  ( x) }{h ( x) }
 +
 
 +
= \
 +
 
 +
\frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x  ^ {2} }
 +
 
 +
\equiv \
  
<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/r/r082/r082540/r0825407.png" /></td> </tr></table>
+
\frac{A ( x) }{B ( x) }
 +
,\ \
 +
x \in ( a , b ) ,
 +
$$
  
 
and if, moreover, at the end points of the orthogonality interval the following conditions hold:
 
and if, moreover, at the end points of the orthogonality interval the following conditions hold:
  
<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/r/r082/r082540/r0825408.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \downarrow a }  h ( x) B ( x)  = \
 +
\lim\limits _ {x \uparrow b }  h ( x) B ( x)  = 0 ,
 +
$$
  
then the orthogonal polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r0825409.png" /> can be represented by a Rodrigues formula:
+
then the orthogonal polynomial $  P _ {n} ( x) $
 +
can be represented by a Rodrigues formula:
  
<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/r/r082/r082540/r08254010.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x)  = \
 +
c _ {n}
 +
\frac{[ h ( x) B  ^ {n} ( x) ]  ^ {(} n) }{h ( x) }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082540/r08254011.png" /> is a constant. Rodrigues' formula holds only for orthogonal polynomials and for polynomials obtained from the latter by linear transformations of the argument. Originally, this formula was established by O. Rodrigues [[#References|[1]]] for the [[Legendre polynomials|Legendre polynomials]].
+
where $  c _ {n} $
 +
is a constant. Rodrigues' formula holds only for orthogonal polynomials and for polynomials obtained from the latter by linear transformations of the argument. Originally, this formula was established by O. Rodrigues [[#References|[1]]] for the [[Legendre polynomials|Legendre polynomials]].
  
 
====References====
 
====References====

Revision as of 08:12, 6 June 2020


A formula relating the differential of the normal $ \mathbf n $ to a surface to the differential of the radius vector $ \mathbf r $ of the surface in the principal direction:

$$ d \mathbf n = - k _ {1} d \mathbf r \ \ \textrm{ or } \ \ d \mathbf n = - k _ {2} d \mathbf r , $$

where $ k _ {1} $ and $ k _ {2} $ are the principal curvatures.

The formula was obtained by O. Rodrigues (1815).

A.B. Ivanov

A representation of orthogonal polynomials in terms of a weight function using differentiation. If a weight function $ h ( x) $ satisfies a Pearson differential equation

$$ \frac{h ^ \prime ( x) }{h ( x) } = \ \frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x ^ {2} } \equiv \ \frac{A ( x) }{B ( x) } ,\ \ x \in ( a , b ) , $$

and if, moreover, at the end points of the orthogonality interval the following conditions hold:

$$ \lim\limits _ {x \downarrow a } h ( x) B ( x) = \ \lim\limits _ {x \uparrow b } h ( x) B ( x) = 0 , $$

then the orthogonal polynomial $ P _ {n} ( x) $ can be represented by a Rodrigues formula:

$$ P _ {n} ( x) = \ c _ {n} \frac{[ h ( x) B ^ {n} ( x) ] ^ {(} n) }{h ( x) } , $$

where $ c _ {n} $ is a constant. Rodrigues' formula holds only for orthogonal polynomials and for polynomials obtained from the latter by linear transformations of the argument. Originally, this formula was established by O. Rodrigues [1] for the Legendre polynomials.

References

[1] O. Rodrigues, "Mémoire sur l'attraction des spheroides" Correspondence sur l'Ecole Polytechnique , 3 (1816) pp. 361–385

P.K. Suetin

Comments

For part 1) see also [a1], [a2]. For part 2) see also [a3], [a4].

References

[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972)
[a2] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a3] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a4] T.S. Chihara, "An introduction to orthogonal polynomials" , Gordon & Breach (1978)
How to Cite This Entry:
Rodrigues formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rodrigues_formula&oldid=12580
This article was adapted from an original article by A.B. Ivanov, P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article