From: dredmond@math.siu.edu (Don Redmond)
Subject: Re: Orthogonality of Legendre polynomials
Date: Tue, 02 Mar 1999 18:42:15 -0600
Newsgroups: sci.math
In article <36DC6C7E.3BDE8985@hermes.cam.ac.uk>, Loh Yen Lee
wrote:
> Dear people out there,
> Could you show me how to derive the orthogonality and normalisation of
> the Legendre polynomials
> Pm(x) and Pn(x) on the interval [-1,1]?
>
> 1 delta(m,n)
> Integral P(m,x) P(n,x) dx = ------------
> x = -1 n + 1/2
>
> "Math. Methods for Physics and Engineering" (Riley,Hobson&Bence)
> outlines a proof starting from the Rodrigues formula (RF), which is
> satisfactory. "Introduction to Mathematical Physics" (C.W.Wong) gives a
> proof starting from the generating function (GF), which I found not
> convincing. ...
>
> What is the general philosophy behind these special functions (Legendre,
> Bessel, Laguerre, Hermite)? It would be nice if one could start from
> the differential equation (DE) and find the RF and GF, but in all the
> books it is done the other way: the functions are defined according to
> the RF or GF's, and shown to satisfy the DE. Is this the only practical
> way of approaching the problem?
>
> Thanks a lot.
> Yen Lee
THe orthogonality of the orthogonal polynomials can be obtained in a wide
variety of ways. It mostly depends on where you want to start. It can be
done from the DE, the RF and GF, as you noted and from the recurrence
relation. It is mostly a matter of taste. Rainville's book on special
functions does it from the recurrence relations. If memory serves,
Erdelyi's book (the Bateman Manuscript project) does it from the DE
Of course, you could always look into Szego's book on orthogonal
polynomials.
Don
==============================================================================
From: "Charles H. Giffen"
Subject: Re: Orthogonality of Legendre polynomials
Date: Wed, 03 Mar 1999 13:29:06 -0500
Newsgroups: sci.math
To: Loh Yen Lee
Loh Yen Lee wrote:
[as above --djr]
Up to a scaling factor P(m,x) = D^m[(1-x^2)^m] -- where
D denotes derivative. So, just use integration by parts
to get a formula relating
\int_{-1}^{1} D^j[(1-x^2)^m]*D^k[(1-x^2)^n] dx
and
\int_{-1}^{1} D^{j+1}[(1-x^2)^m]*D^{k-1}[(1-x^2)^n] dx
--then start with
\int_{-1}^{1} D^m[(1-x^2)^m]*D^n[(1-x^2)^n] dx
and iterate the previous procedure.
You will need the fact that if k < n, then
D^k[(1-x^2)^n] is a degree 2n-k polynomial which is
divisible by (1-x^2). And, of course you will need
that D^{2m}[(1-x^2)^m] is a constant, namely (2m)! .
Here, I am assuming that m >= n (which is okay, by
symmetry). This is how I have my students do it when and
if I decide to cover Legendre polynomials in calculus (not
this year, though).
--Chuck Giffen
==============================================================================
From: rdownes@aol.com (RDownes)
Subject: Re: Orthogonality of Legendre polynomials
Date: 3 Mar 1999 01:37:05 GMT
Newsgroups: sci.math
aother excellent source is the text Orthogonal Polynomials by Jackson. Very
readable!
cheers
Rob