Integrating legendre polynomials with weighting function

Click For Summary
SUMMARY

The discussion focuses on evaluating the integral of the product of Legendre polynomials, specifically the expression \(\int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) dt\) and its equivalence to \(\int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2} dx\). The user expresses difficulty in finding a straightforward analytic solution and mentions potential approaches, including Taylor series expansion and the use of triple integrals of Legendre polynomials as referenced in Arfken and Weber. The consensus is that while there are methods to tackle the problem, they are complex and may not yield a clean analytic result.

PREREQUISITES
  • Understanding of Legendre polynomials, specifically their properties and applications.
  • Familiarity with integral calculus, particularly techniques involving definite integrals.
  • Knowledge of Taylor series expansions and their applications in mathematical analysis.
  • Experience with advanced mathematical texts, such as Gradshteyn and Ryzhik, and Arfken and Weber.
NEXT STEPS
  • Research the properties and applications of Legendre polynomials in physics and engineering.
  • Study Taylor series expansions and their convergence in the context of polynomial functions.
  • Explore techniques for evaluating integrals involving products of orthogonal polynomials.
  • Examine triple integrals of Legendre polynomials as detailed in Arfken and Weber for potential insights.
USEFUL FOR

Students and researchers in mathematics and physics, particularly those dealing with problems in electromagnetism and integral calculus involving Legendre polynomials.

fantispug
Messages
101
Reaction score
0

Homework Statement


I recently came across this integral while doing a problem in electromagnetism (I'm not sure if there exists a nice analytic answer):

\int_{0}^{\pi}P_m(\cos(t))P_n(\cos(t)) \sin^2(t) = \int_{-1}^{1}P_m(x) P_n(x) \sqrt{1-x^2},

Homework Equations



P_m(x) is the m^th Legendre Polynomial.

The Attempt at a Solution



There are lots of close integrals in Gradshteyn and Ryzhik 7.1-7.2 but nothing close enough for me to use.

One way to evaluate it would be to expand the square root as a Taylor series, and then change basis to re-expand it as a series of Legendre polynomials, then use tricks involving triple integrals of Legendre polynomials (such as those in Arfken and Weber 12.9). However this is incredibly messy and I can't see how I could get an analytic expression from it.

Can anyone think of a nice way of approaching this integral? I can't use for example differentiation under the integral because I don't know how to integrate the product of Legendre functions on intervals other than [-1,1] (and [0,1]).
 
Physics news on Phys.org
Never mind...
 

Similar threads

Legendre Polynomial Integration
  • · Replies 1 ·
Replies
1
Views
2K
Legendre Polynomials & the Generating function
  • · Replies 1 ·
Replies
1
Views
3K
Coefficient Matching for different series
  • · Replies 5 ·
Replies
5
Views
2K
What Determines the Values of Legendre Polynomials at Zero?
  • · Replies 2 ·
Replies
2
Views
3K
Legendre polynomials and binomial series
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Expansion of 1/|x-x'| into Legendre Polynomials
  • · Replies 1 ·
Replies
1
Views
1K
Legendre Polynomials as an Orthogonal Basis
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How Does Gaussian-Legendre Quadrature Approximate Non-Polynomial Functions?
  • · Replies 2 ·
Replies
2
Views
2K
Orthonormal Set spanning the subspace (polynomials)
  • · Replies 3 ·
Replies
3
Views
2K
Expanding potential in Legendre polynomials (or spherical harmonics)
  • · Replies 1 ·
Replies
1
Views
2K