PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
Chebyshev equation (Definition)

Chebyshev's equation is the second order linear differential equation

$\displaystyle (1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + p^2y = 0$
where $ p$ is a real constant.

There are two independent solutions which are given as series by:

$\displaystyle y_1(x) = 1 - \tfrac{p^2}{2!}x^2 + \tfrac{(p-2)p^2(p+2)}{4!}x^4 - \tfrac{(p-4)(p-2)p^2(p+2)(p+4)}{6!}x^6 + \dotsb $
and
$\displaystyle y_2(x) = x - \tfrac{(p-1)(p+1)}{3!}x^3 + \tfrac{(p-3)(p-1)(p+1)(p+3)}{5!}x^5 - \dotsb $

In each case, the coefficients are given by the recursion

$\displaystyle a_{n+2} = \frac{(n-p)(n+p)}{(n+1)(n+2)} a_n $
with $ y_1$ arising from the choice $ a_0 = 1$, $ a_1 = 0$, and $ y_2$ arising from the choice $ a_0 = 0$, $ a_1 = 1$.

The series converge for $ \vert x\vert < 1$; this is easy to see from the ratio test and the recursion formula above.

When $ p$ is a non-negative integer, one of these series will terminate, giving a polynomial solution. If $ p \ge 0$ is even, then the series for $ y_1$ terminates at $ x^p$. If $ p$ is odd, then the series for $ y_2$ terminates at $ x^p$.

These polynomials are, up to multiplication by a constant, the Chebyshev polynomials. These are the only polynomial solutions of the Chebyshev equation.

(In fact, polynomial solutions are also obtained when $ p$ is a negative integer, but these are not new solutions, since the Chebyshev equation is invariant under the substitution of $ p$ by $ -p$.)



"Chebyshev equation" is owned by mclase.
(view preamble)

View style:

Other names:  Chebyshev differential equation
Log in to rate this entry.
(view current ratings)

Cross-references: invariant, negative, Chebyshev polynomials, multiplication, odd, even, polynomial, integer, ratio test, easy to see, converge, coefficients, series, solutions, independent, real, linear differential equation, second order
There is 1 reference to this entry.

This is version 3 of Chebyshev equation, born on 2002-11-21, modified 2002-11-21.
Object id is 3616, canonical name is ChebyshevEquation.
Accessed 7698 times total.

Classification:
AMS MSC: 34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)