Page:EB1911 - Volume 25.djvu/676

on integrating the expression k times by parts, and remembering that $(\mu ^{2}-1)^{n}$ and its first $n-1$ derivatives all vanish when $\mu =\pm 1$, the theorem is established. This theorem derives additional importance from the fact that it may be shown that $AP_{n}(\mu )$ is the only rational integral function of degree n which has this property; from this arises the importance of the functions $P_{n}$ in the theory of quadratures.

The theorem which lies at the root of the applicability of the functions $P_{n}$ to potential problems is that if n and n' are unequal integers

$$\int _{-1}^{1}P_{n'}(\mu )P_{n}(\mu )d\mu =0,$$

which may be stated by saying that the integral of the product of two Legendre's coefficients of different degree taken over the whole of a spherical surface with its centre at the origin is zero; this is the fundamental harmonic property of the functions. It is immediately deducible from (18), for if $n'<n$, $P_{n'}(\mu )$ is a linear function of powers of $\mu$, whose indices are all less than n.

When $n'=n$ , the integral in (19) becomes $\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu$; to evaluate this we write it in the form

$${\frac {1}{2^{2n}n!n!}}\int _{-1}^{1}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}d\mu ;$$

on integrating n times by parts, this becomes

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

which on putting

$u={\frac {1}{2}}(1-\mu$, becomes $2{\frac {(2n)!}{n!n!}}\int _{-1}^{1}u^{n}(1-u)^{n}du,$

hence

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

14. Expansion of Functions in Series of Legendre's Coefficients.—If it be assumed that a function $f(\mu )$ given arbitrarily in the interval $\mu =-1$ to $+1$, can be represented by a series of Legendre's coefficients $a_{0}+a_{1}P_{1}(\mu )+a_{2}P_{2}(\mu )+\dots +a_{n}P_{n}(\mu )+\dots$ and it be assumed that the series converges in general uniformly within the interval, the coefficient a can be determined by using (19) and (20); we see that the theorem (19) plays the same part as the property $\int _{0}^{\pi }{\begin{matrix}\cos \\\sin \end{matrix}}n\theta {\begin{matrix}\cos \\\sin \end{matrix}}n'\theta d\theta =0$, $(n\neq n')$ does in the theory of the expansion of functions in series of circular functions. On multiplying the series by $P_{n}(\mu )$, we have

$$a_{n}\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu =\int _{-1}^{1}f(\mu )P_{n}(\mu )d\mu$$

hence

$$a_{n}={\frac {2n+1}{2}}\int _{-1}^{1}f(\mu )P_{n}(\mu )d\mu ,$$

hence the series by which $f(\mu )$ is in general represented in the interval is

$$\sum {\frac {2n+1}{2}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '.$$

The proof of the possibility of this representation, including the investigation of sufficient conditions as to the nature of the function $f(\mu )$, that the series may in general converge to the value of the function requires an investigation, for which we have not space, similar in character to the corresponding investigations for series of circular functions (see Fourier's Series). A complete investigation of this matter is given by Hobson, Proc. Lond. Math. Soc., 2nd series, vol. 6, p. 388, and vol. 7, p. 24. See also Dini's Serie di Fourier.

The expansion may be applied to the determination at an external and an internal point of the potential due to a distribution of matter of surface density $f(\mu )$ placed on a spherical surface $r=a$. If

$V_{1}=\sum A'_{n}{\frac {r^{n}}{a^{n+!}}}P_{n}(\mu ),V_{0}=\sum A_{n}{\frac {a^{n}}{r^{n+1}}}P_{n}(\mu ),$

we see that $V_{1}$, $V_{0}$ have the characteristic properties of potential functions for the spaces internal to, and external to, the spherical surface respectively; moreover, the condition that $V_{1}$ is continuous with $V_{0}$ at the surface $r=a$ is satisfied. The density of a surface distribution which produces these potentials is in accordance with a known theorem in the potential theory, given by

$$\sigma ={\frac {1}{4\pi }}\left({\frac {\partial V_{1}}{\partial r}}-{\frac {\partial V_{0}}{\partial r}}\right)r=a,$$

hence

$\sigma ={\frac {1}{4\pi a^{2}}}\sum (2n+1)A_{n}P_{n}(\mu )$; on comparing this with the series (21),

we have $A_{n}=2\pi a^{2}\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '$,

hence

$${\begin{cases}V_{1}&=2\pi a\sum {\frac {r^{n}}{a^{n}}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '\\V_{0}&=2\pi a\sum {\frac {a^{n+1}}{r^{n+1}}}P_{n}(\mu )\int _{-1}^{1}f(\mu ')P_{n}(\mu ')d\mu '\\\end{cases}}$$

are the required expressions for the internal and external potentials due to the distribution of surface density $f(\mu )$.

15. Integral Properties of Spherical Harmonics.—The fundamental harmonic property of spherical harmonics, of which property (19) is a particular case, is that if $Y_{n}(x,y,z)$, $Z_{n'}(x,y,z)$ be two (ordinary) spherical harmonics, then,

$$\iint Y_{n}(x,y,z)Z_{n'}(x,y,z)dS=0,$$

when n and n' are unequal, the integration being taken for every element dS of a spherical surface, of which the origin is the centre. Since $\nabla ^{2}Y_{n}=0$, $\nabla ^{2}Z_{n'}=0$, we have

$$\iiint (Y_{n}\nabla ^{2}Z_{n'}-Z_{n'}\nabla ^{2}Y_{n})dxdydz=0,$$

the integration being taken through the volume of the sphere of radius r; this volume integral may be written

$${\begin{aligned}\iiint {\Bigl \{}{\frac {\partial }{\partial x}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial x}}-Z_{n'}{\frac {\partial Y_{n}}{\partial x}}\right)&+{\frac {\partial }{\partial y}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial y}}-Z_{n'}{\frac {\partial Y_{n}}{\partial y}}\right)\\&+{\frac {\partial }{\partial z}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial z}}-Z_{n'}{\frac {\partial Y_{n}}{\partial z}}\right){\Bigr \}}=0;\end{aligned}}$$

by a well-known theorem in the integral calculus, the volume integral may be replaced by a surface integral over the spherical surface; we thus obtain

$${\begin{aligned}\iint {\Bigl \{}{\frac {x}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial x}}-Z_{n'}{\frac {\partial Y_{n}}{\partial x}}\right)&+{\frac {y}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial y}}-Z_{n'}{\frac {\partial Y_{n}}{\partial y}}\right)\\&+{\frac {z}{r}}\left(Y_{n}{\frac {\partial Z_{n'}}{\partial z}}-Z_{n'}{\frac {\partial Y_{n}}{\partial z}}\right){\Bigr \}}dS=0;\end{aligned}}$$

on using Euler's theorem for homogeneous functions, this becomes

$${\frac {n'-n}{r}}\iint Y_{n}Z_{n'}dS=0,$$

whence the theorem (22), which is due to Laplace, is proved.

The integral over a spherical surface of the product of a spherical harmonic of degree n, and a zonal surface harmonic $P_{n}$ of the same degree, the pole of which is at $(x',y',z')$ is given by

$$\iint Y_{n}(x,y,z)P_{n}dS={\frac {4\pi }{2n+1}}a^{n+2}Y_{n}(x',y',z');$$

thus the value of the integral depends on the value of the spherical harmonic at the pole of the zonal harmonic. This theorem may also be written

$$\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\theta ,\phi )P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu d\phi ={\frac {4\pi }{2n+1}}V_{n}(\theta ',\phi ').$$

To prove the theorem, we observe that $V_{n}$ is of the form

$$a_{0}P_{n}(\mu )+\sum _{1}^{n}(a_{m}\cos m\phi +b_{m}\sin m\phi )P_{n}^{m}(\mu );$$

to determine $a_{0}$ we observe that when $\mu =1$,

$$P_{n}(\mu )=1,P_{n}^{m}(\mu )=0;$$

hence $a_{0}$ is equal to the value $V_{n}(0)$ of $V_{n}(\theta ,\phi )$ at the pole $\theta =0$ of $P_{n}(\mu )$. Multiply by $P_{n}(\mu )$ and integrate over the surface of the sphere of radius unity, we then have

$${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\theta ,\phi )P_{n}(\mu )d\mu d\phi &=a_{0}\int _{0}^{2\pi }\int _{-1}^{1}\{P_{n}(\mu )\}^{2}d\mu d\phi \\&={\frac {4\pi }{2n+1}}a_{0}={\frac {4\pi }{2n+1}}V_{n}(0),\end{aligned}}$$

if instead of taking $\mu =1$ as the pole of $P_{n}(\mu )$ we take any other point $(\mu ',\phi ')$ we obtain the theorem (23).

If $f(x,y,z)$ is a function which is finite and continuous throughout the interior of a sphere of radius R, it may be shown that

$${\begin{aligned}\iint Y_{n}(x,y,z)f(x,y,z)dS&=4\pi R^{2n+2}{\frac {2^{n}n!}{(2n+1)!}}{\Bigl \{}1+{\frac {R^{2}\nabla ^{2}}{2.2n+3}}\\&+{\frac {R^{4}\nabla ^{4}}{2.4.2n+3.2n+5}}+\dots \}Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)f(x,y,z)\end{aligned}}$$

where x, y, z are put equal to zero after the operations have been performed, the integral being taken over the surface of the sphere of radius R (see Hobson, "On the Evaluation of a certain Surface Integral," Proc. Land. Math. Soc. vol. xxv.).

The following case of this theorem should be remarked: If $f_{n}(x,y,z)$ is homogeneous and of degree n

$$\iint Y_{n}(x,y,z)f_{n}(x,y,z)dS=4\pi R^{2n+2}{\frac {2^{n}n!}{(2n+1)!}}Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)f(x,y,z)$$

if $f_{n}(x,y,z)$ is a spherical harmonic, we obtain from this a theorem, due to Maxwell (Electricity, vol. i. ch. ix.),

$$\iint Y_{n}(x,y,z)f_{n}(x,y,z)dS={\frac {4\pi R^{2n+2}}{2n+1}}{\frac {1}{n!}}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}f_{n}(x,y,z)$$