| ultraspherical.weight {orthopolynom} | R Documentation |
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)
Arguments
x |
the function argument which can be a vector |
alpha |
polynomial parameter |
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}
Value
The value of the weight function
Author(s)
Frederick Novomestky fnovomes@poly.edu
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 )