Page:EB1911 - Volume 25.djvu/675

and let each side operate on $1/r$, then in virtue of (10), we have

$${\begin{aligned}&(rr')^{n}P_{n}\left({\frac {xx'+yy'+zz'}{rr'}}\right)=P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos({\overline {\phi -\phi '}}))\\&=P_{n}(\cos \theta )P_{n}(\cos \theta ')+2\sum {\frac {(n-m)!}{(n+m)!}}P_{n}^{m}(\cos \theta )P_{n}^{m}(\cos \theta ')\cos m(\phi -\phi ')\end{aligned}}$$

which is known as the addition theorem for the function $P_{n}$.

It has incidentally been proved that

$${\begin{aligned}P_{n}^{m}(\cos \theta )={\frac {(n+m!)}{2^{m}(n-m)!}}&\sin ^{m}\theta {\Bigl \{}\cos ^{n-m}\theta \\&-{\frac {(n-m)(n-m-1)}{2.2m+2}}\cos ^{n-m-2}\theta \sin ^{2}\theta +{\Bigr \}}\end{aligned}}$$

which is an expression for $P_{n}^{m}(\cos \theta )$ alternative to (4).

10. Legendre's Coefficients.—The reciprocal of the distance of a point $(r,\theta ,\phi )$ from a point on the z axis distant r' from the origin is

$$(r^{2}-2rr'\mu +r'^{2})^{-{\frac {1}{2}}}$$

which satisfies Laplace's equation, $\mu$ denoting $\cos \theta$. Writing this expression in the forms

$${\frac {1}{r'}}\left\{1-2{\frac {r}{r'}}\mu +{\frac {r^{2}}{r'^{2}}}\right\}^{-{\frac {1}{2}}},{\frac {1}{r}}\left\{1-2{\frac {r'}{r}}\mu +{\frac {r'^{2}}{r^{2}}}\right\}^{-{\frac {1}{2}}}$$

it is seen that when $r<r'$, the expression can be expanded in a convergent series of powers of $r/r'$, and when $r'<r$ in a convergent series of powers of $r'/r$. We have, when $h^{2}(2\mu -h)^{2}<1$

$${\begin{aligned}(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=1+h(2\mu -h)&+{\frac {1.3}{2.4}}h^{2}(2\mu -h)^{2}+\dots \\&+{\frac {1.3.5\dots 2n-1}{2.4\dots 2n}}h^{n}(2\mu -h)^{n}\end{aligned}}$$

and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of $h^{n}$ is then found to be

$${\frac {1.3.5\dots 2n-1}{1.2.3\dots n}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.4.2n-1.2n-3}}\mu ^{n-4}\right\}$$

this is the expression we have already denoted by $P_{n}(\mu )$; thus

$$(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=P_{0}(\mu )+hP_{1}(\mu )+\dots +h^{n}P_{n}(\mu )+\dots$$

the function $P_{n}(\mu )$ may thus be defined as the coefficient of $h^{n}$ in this expansion, and from this point of view is called the Legendre's coefficient or Legendre's function of degree n, and is identical with the zonal harmonic. It may be shown that the expansion is valid for all real and complex values of h and $\mu$, such that mod. h is less than the smaller of the two numbers mod. $(\mu \pm {\sqrt {\mu ^{2}-1}})$. We now see that

$$(r^{2}-2rr'\mu +r'^{2})^{-{\frac {1}{2}}}$$

is expressible in the form

$$\sum _{0}^{\infty }{\frac {r^{n}}{r'^{n+1}}}P_{n}(\mu )$$

when $r<r'$, or

$$\sum _{0}^{\infty }{\frac {r'^{n}}{r^{n+1}}}P_{n}(\mu )$$

when $r'<r$; it follows that the two expressions $r^{n}P_{n}(\mu )$, $r^{-n-1}P_{n}(\mu )$ are solutions of Laplace's equation.

The values of the first few Legendre's coefficients are

$$P_{0}(\mu )=1,P_{1}(\mu )=\mu ,P_{2}(\mu )={\frac {1}{2}}(3\mu ^{2}-1),P_{3}(\mu )={\frac {1}{2}}(5\mu ^{3}-3\mu )$$

$$P_{4}(\mu )={\frac {1}{8}}(35\mu ^{4}-30\mu ^{2}+3),P_{5}(\mu )={\frac {1}{8}}(63\mu ^{5}-70\mu ^{3}+15\mu )$$

$$P_{6}(\mu )={\frac {1}{16}}(231\mu ^{6}-315\mu ^{4}+105\mu ^{2}-5),P_{7}(\mu )={\frac {1}{16}}(429\mu ^{7}-693\mu ^{5}+315\mu ^{3}-35\mu ).$$

We find also

$P_{n}(1)=1,P_{n}(-1)=(-1)^{n}$

$P_{n}(0)=0$, or $(-1)^{{\frac {1}{2}}n}{\frac {1.3.5\dots n-1}{2.4\dots n}}$

according as n is odd or even; these values may be at once obtained from the expansion (13), by putting $\mu =1,0,-1$.

11. Additional Expressions for Legendre's Coefficients.—The expression (3) for $P_{n}(\mu )$ may be written in the form

$$P_{n}(\mu )={\frac {(2n)!}{2^{n}n!n!}}\mu ^{n}F\left(-{\frac {n}{2}},{\frac {1-n}{2}},{\frac {1}{2}}-n,{\frac {1}{\mu ^{2}}}\right)$$

with the usual notation for hypergeometric series. On writing this series in the reverse order

$$P_{n}(\mu )=(-1)^{{\frac {1}{2}}n}{\frac {n!}{2^{n}\left({\frac {1}{2}}n\right)!\left({\frac {1}{2}}n\right)!}}F\left(-{\frac {n}{2}},{\frac {n+1}{2}},{\frac {1}{2}},\mu ^{2}\right)$$

or

$$(-1)^{\frac {n-1}{2}}{\frac {n!}{{\frac {2^{n-1}n-1}{2}}!{\frac {n-1}{2}}!}}\mu F\left(-{\frac {n-1}{2}},{\frac {n}{2}}+1,{\frac {3}{2}},\mu ^{2}\right)$$

according as n is even or odd.

From the identity

$$(1-2h\cos \theta +h^{2})^{-{\frac {1}{2}}}=(1-he^{i\theta })^{-{\frac {1}{2}}}(1-he^{-i\theta })^{-{\frac {1}{2}}}$$

it can be shown that

$${\begin{aligned}P_{n}(\cos \theta )={\frac {1.3.5\dots 2n-1}{2.4.6\dots 2n}}&{\Bigl \{}\cos n\theta +{\frac {1.n}{1.2n-1}}\cos(n-2)\theta \\&+{\frac {1.3.n(n-1)}{1.2.(2n-1)(2n-3)}}\cos(n-4)\theta +{\Bigr \}}\end{aligned}}$$

By (13), or by the formula

$$P_{n}(\mu )={\frac {1}{2^{n}n!}}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}$$

which is known as Rodrigue's formula, we may prove that

$${\begin{aligned}P_{n}(\cos \theta )=1-{\frac {(n+1)n}{1^{2}}}\sin ^{2}{\frac {\theta }{2}}&+{\frac {(n+2)(n+1)n(n-1)}{1^{2}.2^{2}}}\sin ^{4}{\frac {\theta }{2}}\dots \\&=F\left(n+1,-n,1,\sin ^{2}{\frac {\theta }{2}}\right)\end{aligned}}$$

Also that

$${\begin{aligned}P_{n}(\cos \theta )=\cos ^{2n}{\frac {\theta }{2}}&\left\{1-{\frac {n^{2}}{1^{2}}}\tan ^{2}{\frac {\theta }{2}}+{\frac {n^{2}(n-1)^{2}}{1^{2}.2^{2}}}\tan ^{4}{\frac {\theta }{2}}-\dots \right\}\\&=\cos ^{2n}{\frac {\theta }{2}}F\left(-n,-n,1,-\tan ^{2}{\frac {\theta }{2}}\right).\end{aligned}}$$

By means of the identity

$$(1-2h\mu +h^{2})^{-{\frac {1}{2}}}=(1-h\mu )^{-1}\left\{1+{\frac {h^{2}(1-\mu ^{2})}{(1-h\mu )^{2}}}\right\}^{-{\frac {1}{2}}}$$

it may be shown that

$${\begin{aligned}P_{n}(\cos \theta )=\cos ^{n}\theta &\left\{1-{\frac {n(n-1)}{2^{2}}}\tan ^{2}\theta +{\frac {n(n-1)(n-2)(n-3)}{2^{2}.4^{2}}}\tan ^{4}\theta -\dots \right\}\\&=\cos ^{n}\theta F\left(-{\frac {1}{2}}n,{\frac {1}{2}}-{\frac {1}{2}}n,1,-\tan ^{2}\theta \right).\end{aligned}}$$

Laplace's definite integral expression (6) may be transformed into the expression

$${\frac {1}{\pi }}\int _{0}^{\pi }{\frac {d\phi }{(\mu -{\sqrt {\mu ^{2}-1}}\cos \psi )^{n+1}}},$$

by means of the relation

$$(\mu +{\sqrt {\mu ^{2}-1}}\cos \phi )(\mu -{\sqrt {\mu ^{2}-1}}\cos \psi )=1.$$

Two definite integral expressions for $P_{n}(\mu )$ given by Dirichlet have been put by Mehler into the forms

$$P_{n}(\cos \theta )={\frac {2}{\pi }}\int _{0}^{\theta }{\frac {\cos \left(n+{\frac {1}{2}}\right)\phi }{\sqrt {2\cos \phi -2\cos \theta }}}d\phi ={\frac {2}{\pi }}\int _{\theta }^{\pi }{\frac {\sin \left(n+{\frac {1}{2}}\right)\phi }{\sqrt {2\cos \theta -2\cos \phi }}}d\phi .$$

When n is large, and $\theta$ is not nearly equal to 0 or to $\pi$, an approximate value of $P_{n}(\cos \theta )$ is $\{2/n\pi \sin \theta \}^{\frac {1}{2}}\sin \left\{(n+{\frac {1}{2}})\theta +{\frac {1}{4}}\pi \right\}$.

12. Relations between successive Legendre's Coefficients and their Derivatives.—If $(1-2h\mu +h^{2})^{-{\frac {1}{2}}}$ be denoted by u, we find

$$(1-2h\mu +h^{2}){\frac {\partial u}{\partial h}}+(h-\mu )u=0;$$

on substituting $\sum h^{n}P_{n}$ for u, and equating to zero the coefficient of $h^{n}$, we obtain the relation

$$nP_{n}-(2n-1)\mu P_{n-1}+(n-1)P_{n-2}=0.$$

From Laplace's definite integral, or otherwise, we find

$$(\mu ^{2}-1){\frac {dP_{n}}{d\mu }}=n(\mu P_{n}-P_{n-1})=-(n+1)(\mu P_{n}-P_{n+1}).$$

We may also show that

$$\mu {\frac {dP_{n}}{d\mu }}-{\frac {dP_{n-1}}{d\mu }}=nP_{n}$$

$$(n+1)P_{n}=-\mu {\frac {dP_{n}}{d\mu }}+{\frac {dP_{n+1}}{d\mu }}$$

$$(2n+1)P_{n}={\frac {dP_{n+1}}{d\mu }}-{\frac {dP_{n-1}}{d\mu }}$$

$$(2n+1)\mu {\frac {dP_{n}}{d\mu }}=(n+1){\frac {dP_{n-1}}{d\mu }}+n{\frac {dP_{n+1}}{d\mu }}$$

$$(2n+1)(\mu ^{2}-1){\frac {dP_{n}}{d\mu }}=n(n+1)(P_{n+1}-P_{n-1})$$

$${\frac {dP_{n}}{d\mu }}=(2n-1)P_{n-1}+(2n-5)P_{n-3}+(2n-9)P_{n-5}+\dots$$

the last term being $3P_{1}$ or $P_{0}$ according as n is even or odd.

13. Integral Properties of Legendre's Coefficients.—It may be shown that if $P_{n}(\mu )$ be multiplied by any one of the numbers 1, $\mu$, $\mu ^{2}$, ... $\mu ^{n-1}$ and the product be integrated between the limits 1, -1 with respect to $\mu$, the result is zero, thus

$$\int _{-1}^{1}\mu ^{k}P_{n}(\mu )d\mu =0,\alpha =0,1,2,\dots ,n-1.$$

To prove this theorem we have

$$\int _{-1}^{1}\mu ^{k}P_{n}(\mu )d\mu ={\frac {1}{2^{n}n!}}\int _{-1}^{1}\mu ^{k}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}d\mu ,$$