The plots in old (i.e. pre-computer) math books are impressive. These plots took a lot of effort to produce, and so they weren’t created lightly. Consequently they tend to be aesthetically and mathematically interesting. A few weeks ago I recreated a plot from A&S using Mathematica, and today I’d like to do the same for the plot below from a different source [1].
Here is my approximate reproduction.
I’ll give the mathematical details below for those who are interested.
The plot shows “normalized associated Legendre functions of the first kind.” There are two families of Legendre polynomials, denoted P and Q; we’re interested in the former. “Associated” means polynomials that are derived the Legendre polynomials by taking derivatives. Normalized means the polynomials are multiplied by constants so that their squares integrate to 1 over [−1, 1].
Mathematica has a function LegendreP[n, m, x] that implements the associated Legendre polynomials Pnm(x). I didn’t see that Mathematica has a function for the normalized version of these functions, so I rolled by own.
f[n_, m_, x_] := (-1)^n LegendreP[n, m, x]
Sqrt[(2 n + 1) Factorial[n - m]/(2 Factorial[n + m])]
I added the alternating sign term up front after discovering that apparently the original plot used a different convention for defining Pnm than Mathematica uses.
I make my plot by stacking the plots created by
Plot[Table[f[n, n, x], {n, 1, 8}], {x, 0, 1}]
and
Plot[Table[f[n + 4, n, x], {n, 0, 4}], {x, 0, 1}]
The original plot shows P4(x). I used the fact that this equals P40(x) to simplify the code. I also left out the flat line plotting P0 because I thought that looked better.
[1] Tables Of Functions With Formulae And Curves by Fritz Emde, published in 1945. Available on Archive.org.