Author: Andrew J Barbour
Date: 2025-04-13

# Kelvin curves

#### Andrew J Barbour

#### 2025-04-13

    library(kelvin)

    ## Loaded kelvin (2.0.3) -- Solutions to the Kelvin differential equation

Setup a function to plot Kelvin solutions:

    plot_kelvin_curves <- function(){
      # Kelvin functions (order 0 by default)
      num.pts <- 1e3
      curve(expr=Kei,
            from=0.001, to=5, n=num.pts,
            ylim=c(-7,7), xlim=c(0,5),
            yaxs="i",
            xaxs="i",
            main="Fundamental Kelvin functions",
            ylab="Ke(x) or Be(x)", lwd=2)
      curve(expr=Ker,from=0.001,to=5,n=num.pts,add=T,col='red', lwd=2)

      # complementary Kelvin functions (order 0 by default)
      curve(expr=Bei, from=0.001, to=5, n=num.pts, add=TRUE, lty=2, lwd=2)
      curve(expr=Ber, from=0.001, to=5 ,n=num.pts, add=TRUE, col='red', lty=2, lwd=2)
      legend(0.5, 5, c(expression(Kei[0]), expression(Ker[0])), col=c(1,2), lty=c(1,1), lwd=2)
      legend(2.8, -3.8, c(expression(Bei[0]), expression(Ber[0])), col=c(1,2), lty=c(2,2), lwd=2)

      xseq <- seq.int(0.001, 5, length.out=num.pts)

      Knu <- Keir(xseq, nSeq=6, return.list=FALSE)
      matplot(xseq, Re(Knu), type="l", xaxs="i", xlim=c(0,5), yaxs="i", ylim=c(-7,7),
              lty=1, lwd=2, main="Fundamental and higher order Kelvin functions (Ker)")
      legend(3.5, 7, 0:5, col=1:6, lty=1, lwd=2)

      Bnu <- Beir(xseq, nSeq=6, return.list=FALSE)
      matplot(xseq, Re(Bnu), type="l", xaxs="i", xlim=c(0,5), yaxs="i", ylim=c(-7,7),
              lty=1, lwd=2, main="Fundamental and higher order complimentary Kelvin functions (Ber)")
      legend(0.5, 7, 0:5, col=1:6, lty=1, lwd=2)
    }

    plot_kelvin_curves()

Compare these curves with ones given by Wolfram.