orthopolynom (version 1.0-5)

legendre.inner.products: Inner products of Legendre polynomials

Description

This function returns a vector with \(n + 1\) elements containing the inner product of an order \(k\) Legendre polynomial, \(P_k \left( x \right)\), with itself (i.e. the norm squared) for orders \(k = 0,\;1,\; \ldots ,\;n \).

Usage

legendre.inner.products(n)

Arguments

n

integer value for the highest polynomial order

Value

A vector with \(n + 1\) elements

1

inner product of order 0 orthogonal polynomial

2

inner product of order 1 orthogonal polynomial

...

n+1

inner product of order \(n\) orthogonal polynomial

Details

The formula used compute the inner products is as follows.

\(h_n = \left\langle {P_n |P_n } \right\rangle = \frac{2} {{2\,n + 1}}\).

References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.

Courant, R., and D. Hilbert, 1989. Methods of Mathematical Physics, John Wiley, New York, NY.

Szego, G., 1939. Orthogonal Polynomials, 23, American Mathematical Society Colloquium Publications, Providence, RI.

See Also

spherical.inner.products

Examples

# NOT RUN {
###
### compute the inner product for the
###  Legendre polynomials of orders 0 to 1
###
h <- legendre.inner.products( 10 )
print( h )
# }