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