15: Legendre Functions

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import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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15: Legendre Functions#

The Legendre functions \(P_{\lambda}(x)\) and \(Q_{\lambda}(x)\) are the generalization of the Legendre polynomials to non-integer \(\lambda\) and the related Sturm Liouville problem. The Legendre polynomials satisfy (see also 12. Orthogonal Polynomials)

  • Interval: \(x \in [-1, 1]\).

  • Weight: \(W(x) = 1\).

  • Normalization (leading coefficient is \(x^{l}\) with coefficient 1):

    \[\begin{gather*} \int_{-1}^{1} P_k(x)P_l(x)\d{x} = \frac{2}{2l+1}\delta_{kl}. \end{gather*}\]

  • Completeness:

    \[\begin{gather*} \sum_{l=0}^{\infty} \frac{2l+1}{2}P_{l}(x) P_{l}(y) = \delta(x-y). \end{gather*}\]

  • Generating function:

    \[\begin{gather*} \frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^{\infty}P_{l}(x)t^l. \end{gather*}\]

  • Recurrence Relation:

    \[\begin{align*} P_{0}(x) &= 1, \\ P_{1}(x) &= x, \\ (l+1)P_{l+1}(x) &= (2l+1)xP_{l}(x) - lP_{l-1}(x).\tag{15.18} \end{align*}\]

  • Rodrigues’ formula:

    \[\begin{gather*} P_{l}(x) = \frac{1}{2^l l!}\diff[l]{}{x}(x^2-1)^l.\tag{15.30} \end{gather*}\]

  • Schläfli Integral:

    \[\begin{gather*} P_{l}(z) = \frac{1}{2^{l}}\oint_C \frac{(w^2 -1)^l}{(w-z)^{l+1}}\frac{\d{w}}{2\pi \I} \end{gather*}\]

    where the contour is closed, encloses the pole \(w=z\), and avoids any branch cuts (see below).

  • The Legendre polynomials satisfy the following second-order homogeneous linear differential equation (see also 7. Ordinary Differential Equations (ODEs), 8. Sturm-Liouville Theory):

    \[\begin{gather*} (1-x^2)y'' - 2xy' +l(l+1)y = 0, \qquad y(x) = P_{l}(x) \end{gather*}\]

The Legendre functions \(P_{\lambda}(z)\) extend these to non-integer \(l\). This presents the following issues:

  • The functions are no-longer orthogonal.

  • The generating function relation should still hold, but now with appropriately defined fractional derivatives. Similarly with Rodrigues’ formula.

  • The recurrence relations likely hold, but need new based cases for all non-integer streams.

  • The Schläfli integral is promising, but we must now be careful of the contour since the non-integer power introduced branch cuts.

  • The differential equation remains valid.

and satisfy

\[\begin{gather*} -\nabla^2 \psi \end{gather*}\]

Associated Legendre Functions#

The associated functions \(P^{m}_{l}(x)\) add an extra piece to the differential equation:

\[\begin{gather*} (1-x^2)y'' - 2xy' + \left(l(l+1) - \frac{m^2}{1-x^2}\right)y = 0, \qquad y(x) = P^{m}_{l}(x). \end{gather*}\]

They can be found by differentiating:

\[\begin{gather*} P^{m}_{l}(x) = (-1)^{m}(1-x^2)^{m/2} \diff[m]{}{x}P_{l}(x), \qquad m \geq 0,\\ P^{-m}_{l}(x) = (-1)^{m}\frac{(l-m)!}{(l+m)!}P^{m}_{l}(x). \end{gather*}\]

When \(m\) is even, the set of polynomials \(\bigl\{P^{m}_{l}(x) \quad \mid\quad l\in \{m, m+1, \cdots\}\bigr\}\) forms a completed orthogonal basis. For odd \(m\), we have a similar basis, but the functions are no longer polynomials.

For each value of \(0 \leq m\leq l\), we have a new set of orthogonal polynomials:

  • Interval: \(x \in [-1, 1]\).

  • Weight: \(W(x) = 1\).

  • Orthogonality and Normalization (leading coefficient is \(x^{n}\) with coefficient 1):

    \[\begin{gather*} \int_{-1}^{1} P^{m}_{k}(x)P^{m}_{l}(x)\d{x} = \frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{kl}. \end{gather*}\]

    Note that they also satisfy the following orthogonality for fixed \(l\):

    \[\begin{gather*} \int_{-1}^{1} \frac{P^{m}_{l}(x)P^{m}_{l}(x)}{1-x^2}\d{x} = \frac{(l+m)!}{m(l-m)!}\delta_{mn}. \end{gather*}\]

  • Parity:

    \[\begin{gather*} P^{m}_{l}(-x) = (-1)^{l-m}P^{m}_{l}(x). \end{gather*}\]

  • Recurrence Relation: (one of many!)

    \[\begin{align*} (l-m+1)P^{m}_{l+1}(x) &= (2l+1)xP^{m}_{l}(x) - (l+m)P^{m}_{l-1}(x).\tag{15.88} \end{align*}\]

  • Rodrigues’ formula:

    \[\begin{gather*} P^{m}_{l}(x) = \frac{(-1)^{m}}{2^l l!}(1-x^2)^{m/2}\diff[l+m]{}{x}(x^2-1)^l. \end{gather*}\]

  • Schläfli Integral:

    \[\begin{gather*} P^{m}_{l}(z) = \frac{(-1)^{m}(l+m)!}{2^{l}l!}(1-z^2)^{m/2} \oint_C \frac{(w^2 - 1)^{l}}{(w-z)^{l+m+1}}\frac{\d{w}}{2\pi \I} \end{gather*}\]

    where the contour is closed, encloses the pole \(w=z\), and avoids any branch cuts (see below). (MMF: I am not certain yet that the same branch trick works here if \(m\) is not integer since the numerator and denominator have different powers.)

Schläfli Integral Countour#

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plt.rcParams['mathtext.fontset'] = 'cm'
from matplotlib_inline.backend_inline import set_matplotlib_formats
set_matplotlib_formats('svg')

rng = np.random.default_rng(seed=2)
th = np.linspace(0, 2*np.pi, 500)
n = np.arange(-5, 6)
dr = (np.exp(1j * n[:, None] * th[None, :])
      * (rng.random(size=len(n)*2)-0.5).view(dtype=complex)[:, None]).sum(axis=0)
z = 1.5 + 1j
w0 = (z + 1)/2
r = 2*abs(z-w0)
C = ((r + 0.1*dr.real)*np.exp(1j*th) + w0)

eta = 1/np.linspace(1, 0, 200)[:-1]
br = (1+eta*z)/(z+eta)

fig, ax = plt.subplots(figsize=(4, 2))
ax.plot(C.real, C.imag, lw=2)
ax.plot([-1, 1, z.real], [0, 0, z.imag], 'ko', ms=10)
ax.plot(br.real, br.imag, '-k', lw=3)
ax.plot([-3, -1], [0, 0], '-k', lw=3)
ax.text(z.real-0.3, z.imag + 0.1, "$z$", size=15)
ax.grid(True)
ax.set(xlim=(-1.5, 2.5), aspect=1, xticks=[-1, 0, 1, 2], yticks=[-1, 0, 1],
    xlabel=r"$\mathrm{Re}(w)$", ylabel=r"$\mathrm{Im}(w)$");
_images/389d9f94a6608488875d2c6f7e9943d1f9b63ae98ced6bf0a03b8ddf6520f56f.svg

The Legendre function \(P_{l}(z)\) has the following Schläfli contour integral representation:

\[\begin{gather*} P_{l}(z) = \frac{1}{2^{l}}\oint_{C} \frac{(w^2-1)^l}{(w-z)^{l+1}}\frac{\d{w}}{2\pi \I} \end{gather*}\]

with the contour shown on the right after choosing the branch cuts as shown.

15.3: Multipole Expansion#

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