Page:EB1911 - Volume 25.djvu/672

Of all particular integrals of Laplace's equation, these are of the greatest importance in respect of their applications, and were the only ones considered by the earlier investigators; the solutions of potential problems in which the bounding surfaces are exactly or approximately spherical are usually expressed as series in which the terms are these spherical harmonics. In the wider sense of the term, a spherical harmonic is any homogeneous function of the variables which satisfies Laplace's equation, the degree of the function being not necessarily integral or real, and the functions are not necessarily rational in x, y, z, or single-valued; the functions may, when necessary, be termed ordinary spherical harmonics. For the treatment of potential problems which relate to spaces bounded by special kinds of surfaces, solutions of Laplace's equation are required which are adapted to the particular boundaries, and various classes of such solutions have thus been introduced into analysis. Such functions are usually of a more complicated structure than ordinary spherical harmonics, although they possess analogous properties. As examples we may cite Bessel's functions in connexion with circular cylinders, Lamé's functions in connexion with ellipsoids, and toroidal functions for anchor rings. The theory of such functions may be regarded as embraced under the general term harmonic analysis. The present article contains an account of the principal properties of ordinary spherical harmonics, and some indications of the nature and properties of the more important of the other classes of functions which occur in harmonic analysis. Spherical and other harmonic functions are of additional importance in view of the fact that they are largely employed in the treatment of the partial differential equations of physics, other than Laplace's equation; as examples of this, we may refer to the equation ${\frac {\partial u}{\partial t}}=k\nabla ^{2}u$, which is fundamental in the theory of conduction of heat and electricity, also to the equation ${\frac {\partial ^{2}u}{\partial t^{2}}}=k\nabla ^{2}u$, which occurs in the theory of the propagation of aerial and electro-magnetic waves. The integration under given conditions of more complicated equations which occur in the theories of hydro-dynamics and elasticity, can in certain cases be effected by the use of the functions employed in harmonic analysis.

1. Relation between Spherical Harmonics of Positive and Negative Degrees.—A function which is homogeneous in x, y, z, of degree n in those variables, and which satisfies Laplace's equation

${\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0$, or $\nabla ^{2}V=0$,

is termed a solid spherical harmonic, or simply a spherical harmonic of degree n. The degree n may be fractional or imaginary, but we are at present mainly concerned with the case in which n is a positive or negative integer. If x, y, z be replaced by their values $r\sin \theta \cos \phi$, $r\sin \theta \sin \phi$, $r\cos \theta$ in polar co-ordinates, a solid spherical harmonic takes the form $r^{n}f_{n}(\theta ,\phi )$; the factor $f_{n}(\theta ,\phi )$ is called a surface harmonic of degree n. If V_n denote a spherical harmonic of degree n, it may be shown by differentiation that $\nabla ^{2}(r^{m}V_{n})=m(2n+m+1)r^{m-2}V_{n}$, and thus as a particular case that $\nabla ^{2}(r^{-2n-1}V_{n})=0$; we have thus the fundamental theorem that from any spherical harmonic V_n of degree n, another of degree $-n-1$ may be derived by dividing V_n by $r^{2n+1}$. All spherical harmonics of negative integral degree are obtainable in this way from those of positive integral degree. This theorem is a particular case of the more general inversion theorem that if $F(x,y,z)$ is any function which satisfies the equation (1), the function

$${\frac {1}{r}}F\left({\frac {x}{r^{2}}},{\frac {y}{r^{2}}},{\frac {z}{r^{2}}}\right)$$

also satisfies the equation.

The ordinary spherical harmonics of positive integral degree n are those which are rational integral functions of x, y, z. The most general rational integral function of degree n in three letters contains ${\frac {1}{2}}(n+1)(n+2)$ coefficients; if the expression be substituted in (1), we have on equating the coefficients separately to zero ${\frac {1}{2}}n(n-1)$ relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains ${\frac {1}{2}}(n+1)(n+2)-{\frac {1}{2}}n(n-1)$, or $2n+1$ independent constants. There exist, therefore, $2n+1$ independent ordinary harmonics of degree n; and corresponding to each of these there is a negative harmonic of degree $-n-1$ obtained by dividing by $r^{2n+1}$. The three independent harmonics of degree 1 are x, y, z; the five of degree 2 are $y^{2}-z^{2}$, $z^{2}-x^{2}$, $yz$, $zx$, $xy$. Every harmonic of degree n is a linear function of $2n+1$ independent harmonics of the degree; we proceed, therefore, to find the latter.

2. Determination of Harmonics of given Degree.—It is clear that a function $f(ax+by+cz)$ satisfies the equation (1), if a, b, c are constants which satisfy the condition $a^{2}+b^{2}+c^{2}=0$; in particular the equation is satisfied by $(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}$. Taking n to be a positive integer, we proceed to expand this expression in a series of cosines and sines of multiples of $\alpha$; each term will then satisfy (1) separately. Denoting $e^{\iota \alpha }$ by k, and $y+\iota x$ by t, we have

$$(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}=\left(z+{\frac {1}{2}}kt+{\frac {z^{2}}{2kt}}-{\frac {r^{2}}{2kt}}\right)^{n}$$

which may be written as $(2kt)^{-n}\{(z+kt)^{2}-r^{2}\}^{n}$. On expansion by Taylor's theorem this becomes

$$(2kt)^{-n}\sum _{0}^{2n}{\frac {k^{3}t^{3}}{s!}}{\frac {\partial ^{3}}{\partial z^{3}}}(z^{2}-r^{2})^{n},$$

the differentiation applying to z only as it occurs explicitly; the terms involving $\cos m\alpha$, $\sin m\alpha$ in this expansion are

$${\frac {1}{2^{n}}}\cos m\alpha \left\{{\frac {(y+ix)^{m}}{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}+{\frac {(y+ix)^{-m}}{(n+m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{m}\right\}$$

$$i{\frac {1}{2^{n}}}\sin m\alpha \left\{{\frac {(y+ix)^{m}}{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}+{\frac {(y+ix)^{-m}}{(n-m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{m}\right\}$$

where $m=1,2,\dots ,n$; and the term independent of $\alpha$ is

$${\frac {1}{2^{n}n!}}{\frac {\partial ^{n}}{\partial z^{n}}}(z^{2}-r^{2})^{n}.$$

On writing $$(y+ix)^{m}=i^{m}r^{m}(\cos m\phi -i\sin m\phi )2\sin ^{m}\theta ,(y+ix)^{-m}=i^{-m}r^{-m}(\cos m\phi +i\sin m\phi )2\sin ^{-m}\theta$$

and observing that in the expansion of $(z+\iota x\cos \alpha +\iota y\sin \alpha )^{m}$ the expressions $\cos m\alpha ,\sin m\alpha$ can only occur in the combination $\cos m(\phi -\alpha )$, we see that the relation

$$i^{m}r^{m}{\frac {\sin ^{m}\theta }{(n+m)!}}{\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}=i^{-m}r^{-m}{\frac {\sin ^{-m}\theta }{(n-m)!}}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}(z^{2}-r^{2})^{n}$$

must hold identically, and thus that the terms in the expansion reduce to

$${\frac {1}{(n+m)!}}{\frac {i^{m}}{2^{n-1}}}r^{m}\cos m\alpha \cos m\phi \sin ^{m}\theta {\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}$$

$${\frac {1}{(n+m)!}}{\frac {i^{m}}{2^{n-1}}}r^{m}\sin m\alpha \sin m\phi \sin ^{m}\theta {\frac {\partial ^{n+m}}{\partial z^{n+m}}}(z^{2}-r^{2})^{n}$$

We thus see that the spherical harmonics of degree n are of the form

$$r^{n}~{\stackrel {\cos }{\sin }}~m\phi \sin ^{m}\theta {\frac {d^{n+m}}{d\mu ^{n+m}}}(\mu ^{2}-1)^{n}$$

where $\mu$ denotes $\cos \theta$; by giving m the values $0,1,\dots ,n$ we thus have the $2n+1$ functions required. On carrying out the differentiations we see that the required functions are of the form

$${\begin{aligned}&A[(x+iy)^{m}=(x-iy)^{m}]{\Bigl \{}z^{n-m}-{\frac {(n-m)(n-m-1)}{2.2n-1}}z^{n-m-2}(x^{2}+y^{2}+z^{2})\\&+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2.4.2n-1.2n-3}}z^{n-m-4}(x^{2}+y^{2}+z^{2})^{2}+\cdots {\Bigr \}}\end{aligned}}$$

where $m=0,1,2,3,\dots ,n$.

3. Zonal, Tesseral and Sectorial Harmonics.–Of the system of $2n+1$ harmonics of degree n, only one is symmetrical about the z axis; this is

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

writing

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

we observe that $P_{n}(\mu )$ has n zeros all lying between =1, consequently the locus of points on a sphere $r=a$, for which $P_{n}(\mu )$ vanishes is n circles all parallel to the meridian plane: these circles divide the sphere into zones, thus $P_{n}(\mu )$ is called the zonal surface harmonic of degree n, and $r^{n}P_{n}(\mu ),r^{-n-1}P_{n}(\mu )$ are the solid zonal harmonics of degrees n and $-n-1$. The locus of points on a sphere for which ${\stackrel {\cos }{\sin }}~m\phi .\sin ^{m}\theta {\frac {d^{n+m}}{d\mu ^{n+m}}}(\mu ^{2}-1)^{n}$ vanishes consists of $n-m$ circles parallel to the meridian plane, and m great circles through the poles; these circles divide the spherical surface into quadrilaterals or τέσσερα, except when $n=m$, in which case the surface is divided into sectors, and the harmonics are therefore called tesseral, except those for which $m=n$, which are called sectorial. Denoting $(1-\mu ^{2}){\frac {{\frac {1}{2}}^{m}d^{m}P_{n}(\mu )}{d\mu ^{m}}}$ by $P_{n}^{m}(\mu )$, the tesseral surface harmonics are ${\stackrel {\cos }{\sin }}~m\phi .P_{n}^{m}(\cos \theta )$, where $m=1,2,\dots ,n-1$, and the sectorial harmonics are ${\stackrel {\cos }{\sin }}~n\phi .P_{n}^{n}(\cos \theta )$. The functions $P_{n}(\mu ),P_{n}^{m}(\mu )$ denote the expressions