Notes on Whittaker & Watson, Chapter XV, part 2

Whittaker & Watson, A Course of Modern Analysis. Chapter XV, Legendre Functions.

15.3 Legendre functions of the second kind.

In solving Legendre’s differential equation (§15.2), we instead consider the branch cut connecting $t=-1,1$​ and the contour $D$​​​​ which is an ellipse around the cut. Thus, we have the solution on the $z$​-plane cut on real axis from 1 to $-\infty$​ $$ Q_n(z)=\frac{1}{4i\sin n\pi}\int_D\frac{(t^2-1)^n}{2^n(z-t)^{n+1}}dt $$ for $n\notin\mathbb{Z}$​​ and under the condition that $\Re(n+1)>0$​​. In the limit when $D$​​ shrinks to the real interval $[-1,1]$​​​, we have the real integral $$ Q_n(z)=\frac{1}{2^{n+1}}\int_{-1}^1(1-t^2)^n(t-z)^{-n-1}dt $$ which defines $Q_n$ for non-negative integer values of $n$.

The function $Q_n$​​ is the Legendre function of degree $n$​​ of the second kind, and along with $P_n$​​​​, they are the two fundamental solutions to Legendre’s differential equation.

15.31 Expansion of $Q_n(z)$​​​ as a power-series in $z^{-1}$.

Using the real integral of $Q_n(z)$ for $\Re(n+1)>0$​, for $|z|>1$ on the cut plane we expand $(t-z)^{-n-1}$​ $$ \begin{align*}Q_n(z) &=\frac{1}{2^{n+1}z^{n+1}}\int_{-1}^1(1-t^2)^n\left(1+\sum_{r=1}^\infty\left(\frac tz\right)^r\frac{(n+1)\cdots(n+r)}{r!}\right)dt\\ & = \frac{1}{2^{n}z^{n+1}}\int_{-1}^1(1-t^2)^n+\sum_{s=1}^\infty\frac{(n+1)\cdots(n+r)}{r!z^{2s}}\int_{-1}^1(1-t^2)^nt^rdt & r\text{ odd terms vanish}\\ & = \frac{\pi^\frac12\Gamma(n+1)}{2^{n+1}\Gamma(n+\frac32)}\frac1{z^{n+1}}F\left(\frac12n+\frac12,\frac12n+1;n+\frac32;z^{-2}\right) & \text{Euler int. 1st kind} \end{align*} $$ which is also valid for other values of $n$​.

15.32 The recurrence-formulae for $Q_n(z)$. $$ \begin{gather*} Q'_{n+1}(z)-zQ'_n(z)=(n+1)Q_n(z),\\ (n+1)Q_{n+1}(z)-(2n+1)zQ_n(z)+nQ_{n-1}(z)=0,\\ zQ'_n(z)-Q'_{n-1}(z)=nQ_n(z),\\ Q'_{n+1}(z)-Q'_{n-1}(z)=(2n+1)Q_n(z)\\ (z^2-1)Q'_n(z) = nzQ_n(z)-nQ_{n-1}(z) \end{gather*} $$ 15.33 The Laplacian integral for Legendre functions of the second kind.

Make the change of variable $t=\frac{e^\theta(z+1)^\frac12-(z-1)^\frac12}{e^\theta(z+1)^\frac12+(z-1)^\frac12}$​ in the real integral definition of $Q_n(z)$ for $\Re(n+1)>0$, we have $$ Q_n(z)=\int_0^\infty\left(z+(z^2-1)^\frac12\cosh\theta\right)^{-n-1}d\theta. $$ 15.34 Neumann’s formula for $Q_n(z)$ when $n$ is an integer.

By the expansion $\frac1{z-y}=\sum_{m=0}^\infty\frac{y^m}{z^{m+1}}$, we have $$ \frac12\int_{-1}^{1} P_n(z)\frac1{z-y}dy=\frac12\sum_{m=0}^\infty z^{-n-2m-1}\frac{2^{n+1}(n+2m)!(n+m)!}{m!(2n+2m+1)!}=Q_n(z) $$ which is established for $|z|>1$ and is valid through analytic continuation for all $z\in\mathbb{C}$ except for the real interval $[-1,1]$.

15.4 Heine’s development of $(t — z)^{-1}$ as a series of Legendre polynomials in $z$.

Using the recurrence formulae that $(n+1)Q_{n+1}(t)-(2n+1)tQ_n(t)+nQ_{n-1}(t)=0$​ and $(n+1)P_{n+1}(z)-(2n+1)zP_n(z)+nP_{n-1}(z)=0$​​​, we have, by induction $$ \frac{1}{t-z}=\sum_{m=0}^n(2m+1)P_m(z)Q_m(t)+\frac{n+1}{t-z}\Big(P_{n+1}(z)Q_n(t)-P_n(z)Q_{n+1}(t)\Big) $$ The term $P_{n+1}(z)Q_n(t)-P_n(z)Q_{n+1}(t)$​​ can be estimated by Laplace’s integral and tends to 0 when $n\to\infty$. Thus, we have $$ \frac1{t-z}=\sum_{n=0}^{\infty}(2n+1)P_n(z)Q_n(t) $$ when $z$​ is inside the ellipse passing $t$ with foci at $\pm1$​. The convergence is uniform with respect to $t$.

15.41 Neumann’s expansion of an arbitrary function in a series of Legendre polynomials.

Using Heine’s expansion of $\frac1{t-z}$​ in Cauchy’s integral, we have $$ \begin{align*} f(z)=\frac1{2\pi i}\int_C\frac{f(t)}{t-z}dt=\frac1{2\pi i}\sum_{n=0}^\infty\int_C(2n+1)P_n(z)Q_n(t)f(t)dt. \end{align*} $$ Thus in the expasion $f(z)=\sum_{n=0}^{\infty}a_nP_n(z)$​​​ we have the coefficients (cf. §15.211) $$ a_n=\frac{2n+1}{2\pi i}\int_Cf(t)Q_n(t)dt. $$