Difference between revisions of "Rodrigues formula"
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| − | + | 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 | + | 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 | ||
| + | |||
| + | $$ | ||
| + | |||
| + | \frac{h ^ \prime ( x) }{h ( x) } | ||
| + | |||
| + | = \ | ||
| + | |||
| + | \frac{p _ {0} + p _ {1} x }{q _ {0} + q _ {1} x + q _ {2} x ^ {2} } | ||
| + | |||
| + | \equiv \ | ||
| − | + | \frac{A ( x) }{B ( x) } | |
| + | ,\ \ | ||
| + | x \in ( a , b ) , | ||
| + | $$ | ||
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: | ||
| − | + | $$ | |
| + | \lim\limits _ {x \downarrow a } h ( x) B ( x) = \ | ||
| + | \lim\limits _ {x \uparrow b } h ( x) B ( x) = 0 , | ||
| + | $$ | ||
| − | then the orthogonal polynomial | + | 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 | + | 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) |
Rodrigues formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rodrigues_formula&oldid=12580