Legendre functions

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Description

Legendre functions as per the Gnu Scientific Library reference manual section 7.24, and AMS-55, chapter 8. These functions are declared in header file gsl_sf_legendre.h

Usage

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legendre_P1(x, give=FALSE, strict=TRUE)
legendre_P2(x, give=FALSE, strict=TRUE)
legendre_P3(x, give=FALSE, strict=TRUE)
legendre_Pl(l, x, give=FALSE, strict=TRUE)
legendre_Pl_array(lmax, x, give=FALSE, strict=TRUE)
legendre_Q0(x, give=FALSE, strict=TRUE)
legendre_Q1(x, give=FALSE, strict=TRUE)
legendre_Ql(l, x, give=FALSE, strict=TRUE)
legendre_Plm(l, m, x, give=FALSE, strict=TRUE)
legendre_Plm_array(lmax, m, x, give=FALSE, strict=TRUE)
legendre_sphPlm(l, m, x, give=FALSE, strict=TRUE)
legendre_sphPlm_array(lmax, m, x, give=FALSE, strict=TRUE)
conicalP_half(lambda, x, give=FALSE, strict=TRUE)  
conicalP_mhalf(lambda, x, give=FALSE, strict=TRUE)  
conicalP_0(lambda, x, give=FALSE, strict=TRUE)  
conicalP_1(lambda, x, give=FALSE, strict=TRUE)  
conicalP_sph_reg(l, lambda, x, give=FALSE, strict=TRUE)  
conicalP_cyl_reg(m, lambda, x, give=FALSE, strict=TRUE)
legendre_H3d_0(lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d_1(lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d(l, lambda, eta, give=FALSE, strict=TRUE)
legendre_H3d_array(lmax, lambda, eta, give=FALSE, strict=TRUE)

Arguments

eta,lambda,x

input: real values

l,m,lmax

input: integer values

give

Boolean, with default FALSE meaning to return just the answers, and TRUE meaning to return a status vector as well

strict

Boolean, with TRUE meaning to return NaN if nonzero status is returned by the GSL function (FALSE means to return the value: use with caution)

Author(s)

Robin K. S. Hankin

References

http://www.gnu.org/software/gsl

Examples

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 theta <- seq(from=0,to=pi/2,len=100)
 plot(theta,legendre_P1(cos(theta)),type="l",ylim=c(-0.5,1), main="Figure 8.1, p338")
 abline(1,0)
 lines(theta,legendre_P2(cos(theta)),type="l")
 lines(theta,legendre_P3(cos(theta)),type="l")

x <- seq(from=0,to=1,len=600)
plot(x, legendre_Plm(3,1,x), type="l",lty=3,main="Figure 8.2, p338: note sign error")
lines(x,legendre_Plm(2,1,x), type="l",lty=2)
lines(x,legendre_Plm(1,1,x), type="l",lty=1)
abline(0,0)


plot(x,legendre_Ql(0,x),xlim=c(0,1), ylim=c(-1,1.5), type="l",lty=1,
main="Figure 8.4, p339")
lines(x,legendre_Ql(1,x),lty=2)
lines(x,legendre_Ql(2,x),lty=3)
lines(x,legendre_Ql(3,x),lty=4)
abline(0,0)

#table 8.1 of A&S:
t(legendre_Pl_array(10, seq(from=0,to=1,by=0.01))[1+c(2,3,9,10),])

#table 8.3:
f <- function(n){legendre_Ql(n, seq(from=0,to=1,by=0.01))}
sapply(c(0,1,2,3,9,10),f)