orthopolynom (version 1.0-6.1)

ultraspherical.weight: Weight function for the ultraspherical polynomial

Description

This function returns the value of the weight function for the order \(k\) ultraspherical polynomial, \(C_k^{\left( \alpha \right)} \left( x \right)\).

Usage

ultraspherical.weight(x,alpha)

Value

The value of the weight function

Arguments

x

the function argument which can be a vector

alpha

polynomial parameter

Author

Frederick Novomestky

Details

The function takes on non-zero values in the interval \( \left( -1,1 \right) \). The formula used to compute the weight function is as follows.

\(w\left( x \right) = \left( {1 - x^2 } \right)^{\alpha - 0.5} \)

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.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

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

Examples

###
### compute the ultraspherical weight function for arguments between -2 and 2
### polynomial parameter is 1.0
###
x <- seq( -2, 2, .01 )
y <- ultraspherical.weight( x, 1 )
plot( x, y )