# 1911 Encyclopædia Britannica/Spherical Harmonics

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​SPHERICAL HARMONICS, in mathematics, certain functions of fundamental importance in the mathematical theories of gravitation, electricity, hydrodynamics, and in other branches of physics. The term “spherical harmonic” is due to Lord Kelvin, and is primarily employed to denote either a rational integral homogeneous function of three variables x, y, z, which satisfies the differential equation
$\nabla ^{2}V={\frac {\partial ^{2}V}{\partial x^{2}}}+{\frac {\partial ^{2}V}{\partial y^{2}}}+{\frac {\partial ^{2}V}{\partial z^{2}}}=0,$
known as Laplace's equation, or a function which satisfies the differential equation, and becomes a rational integral homogeneous function when multiplied by a power of $(x^{2}+y^{2}+z^{2})^{\frac {1}{2}}$. ​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 ​$P_{n}(\mu )={\frac {(2n)!}{2^{n}n!n!}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+{\frac {n(n-1)(n-2)(n-3)}{2.4.2n-1.2n-3}}\mu ^{n-4}\dots \right\}$
${\begin{aligned}P_{n}^{m}(\mu )={\frac {(2n)!}{2^{n}n!(n-m)!}}(1-\mu ^{2})^{{\frac {1}{2}}m}&{\Bigl \\mu ^{n-m}\\&-{\frac {(n-m)(n-m-1)}{2.2n-1}}\mu ^{n-m-2}+\dots {\Bigr \}}\end{aligned}}$
$P_{n}^{n}(\mu )={\frac {(2n)!}{2^{n}n!}}(1-\mu ^{2})^{{\frac {1}{2}}n}.$
Every ordinary harmonic of degree n is expressible as a linear function of the system of $2n+1$ zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is $a_{0}P_{n}(\mu )+\sum _{m=1}^{n}(a_{m}\cos m\phi +b_{m}\sin m\phi )P_{n}^{m}(\mu )$
In the present notation we have
$(z+\iota x\cos \alpha +\iota y\sin \alpha )^{n}=r^{n}\left\{P_{n}(\mu )+2\sum _{1}^{n}\iota ^{m}{\frac {n!}{(n+m)!}}P_{n}^{m}(\mu )\cos m(\phi -\alpha )\right\}$
if we put $\alpha =0$, we thus have
$(\cos \theta +\iota \sin \theta \cos \phi )^{n}=P_{n}(\cos \theta )+2\sum _{1}^{n}i^{m}{\frac {n!}{(n+m!)}}P_{n}^{m}(\cos \theta )\cos m\phi ,$
from this we obtain expressions for $P_{n}(\cos \theta ),P_{n}^{m}(\cos \theta )$ as definite integrals
$P_{n}(\cos \theta )={\frac {1}{\pi }}\int _{0}^{\pi }(\cos \theta +\iota \sin \theta \cos \phi )^{n}d\phi$
$\iota ^{m}{\frac {n!}{(n+m)!}}P_{n}^{m}(\cos \theta )={\frac {1}{\pi }}\int _{0}^{\pi }(\cos \theta +\iota \sin \theta \cos \phi )^{n}\cos m\phi d\phi .$
4. Derivation of Spherical Harmonics by Differentiation.–The linear character of Laplace's equation shows that, from any solution, others may be derived by differentiation with respect to the variables x, y, z; or, more generally, if
$f\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$
denote any rational integral operator,
$f\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)V$
is a solution to the equation, if V satisfies it. This principle has been applied by Thomson and Tait to the derivation of the system of any integral degree, by operating upon $1/r$, which satisfies Laplace's equation. The operations may be conveniently carried out by means of the following differentiation theorem. (See papers by Hobson, in the Messenger of Mathematics, xxiii. 115, and Proc. Lond. Math. Soc., vol. xxiv.)
${\begin{aligned}f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right){\frac {1}{r}}&=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{2n+1}}}{\Bigl \1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}\\&+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\dots {\Bigr \}}f_{n}(x,y,z)\end{aligned}}$
which is a particular case of the more general theorem
${\begin{aligned}f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)F(r)&={\Bigl \2^{n}{\frac {d^{n}F}{d(r^{2})^{n}}}+{\frac {2^{n-2}}{1!}}{\frac {d^{n-1}F}{d(r^{2})^{n-1}}}\nabla ^{2}+\dots \\&+{\frac {2^{n-2s}}{s!}}{\frac {d^{n-s}F}{d(r^{2})^{n-s}}}\nabla ^{2s}+\dots {\Bigr \}}f_{n}(x,y,z)\end{aligned}}$
where $f_{n}(x,y,z)$ is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree $-n-1$ in the expression (7) is
$\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\dots \right\}f_{n}(x,y,z).$
It can be verified that even when n is unrestricted, this expression satisfies Laplace's equation, the sole restriction being that of the convergence of the series.
5. Maxwell's Theory of Poles.—Before proceeding to obtain by means of (7), the expressions for the zonal, tesseral, and sectorial harmonics, it is convenient to introduce the conception, due to Maxwell (see Electricity and Magnetism, vol. i. ch. ix.), of the poles of a spherical harmonic. Suppose a sphere of any radius drawn with its centre at the origin; any line whose direction-cosines are l, m, n drawn from theo rigin, is called an axis, and the point where this axis cuts the sphere is called the pole of the axis. Different axes will be denoted by suffixes attached to the direction-cosines: the cosine $(l_{\iota }x+m_{\iota }y+n_{\iota }z)/r$ of the angle between the radius vector r to a point $(x,y,z)$ and the axis $(l_{\iota },m_{\iota },n_{\iota })$ will be denoted by $\lambda _{\iota }$; the cosine of the angle between two axes is $l_{\iota }l_{\gamma }+m_{\iota }m_{\gamma }+n_{\iota }n_{\gamma }$, which will be noted by $\mu _{\iota \gamma }$. The operation
$l_{\iota }{\frac {\partial }{\partial x}}+m_{\iota }{\frac {\partial }{\partial y}}+n_{\iota }{\frac {\partial }{\partial z}}$
performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis $(l_{\iota },m_{\iota },n_{\iota })$, and is denoted by $\partial /\partial h_{\iota }$. The potential function $V_{0}=e_{0}/r$ is defined to be the potential due to a singular point of degree zero at the origin; $e_{0}$ is called the strength of the singular point. Let a singular point of degree zero, and strength $e_{0}$, be on an axis $h_{1}$, at a distance $a_{0}$from the origin, and also suppose that the origin is a singular point of strength $-e_{0}$; let $e_{0}$ be indefinitely increased, and $a_{0}$ indefinitely diminished, but so that the product $e_{0}a_{0}$ is finite and equal to $e_{0}$; the origin is then said to be a singular point of the first degree, of strength $e_{1}$, the axis being $h_{1}$. Such a singular point is frequently called a doublet. In a similar manner, by placing two singular points of degree, unity and strength, $e_{1}$, $-e_{1}$, at a distance $a_{1}$ along an axis $h_{2}$, and at the origin respectively, when $e_{1}$ is indefinitely increased, and $a_{1}$ diminished so that $e_{1}a_{1}$ is finite and $=e_{2}$, we obtain a singular point of degree 2, strength $e_{2}$ at the origin, the axes being $h_{1},h_{2}$. Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength $e_{n}$ at the origin, the singular point having any n given axes $h_{1},h_{2},\dots h_{n}$. If $e_{n-1}\phi _{n-1}(x,y,z)$ is the potential due to a singular point at the origin, of degree $n-1$, and strength $e_{n-1}$, with axes $h_{1},h_{2},\dots h_{n-1}$, the potential of a singular point of degree n, the new axis of which is $h_{n}$ is the limit of
$e_{n-1}\phi _{n-1}(x-l_{n}a,y-m_{n}a,z-n_{n}a)-e_{n-1}\phi _{n-1}(x,y,z)$
when
$La=0,Le_{n-1}=\infty ,Lae_{n-1}=e_{n};$
this limit is
$-e_{n}(l_{n}{\frac {\partial \phi _{n-1}}{\partial x}}+m_{n}{\frac {\partial \phi _{n-1}}{\partial y}}+n_{n}{\frac {\partial \phi _{n-1}}{\partial z}})$, or $-e_{n}{\frac {\partial }{\partial h_{n}}}\phi _{n-1}$.
Since $V_{0}=1/r$, we see that the potential V, due to a singular point at the origin of strength $e_{n}$ and axes $h_{1},h_{2},\dots h_{n}$ is given by
$V_{n}=(-1)^{n}e_{n}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}{\frac {1}{r}}.$
6. Expression for a Harmonic with given Poles.—The result of performing the operations in (8) is that $V_{n}$ is of the form
$n!e_{n}{\frac {Y_{n}}{r^{n+1}}},$
where $Y_{n}$ is a surface harmonic of degree n, and will appear as a function of the angles which r makes with the n axes, and of the angles these axes make with one another. The poles of the n axes are defined to be the poles of the surface harmonics, and are also frequently spoken of as the poles of the solid harmonics $Y_{n}r^{n},Y_{n}r^{-n-1}$. Any spherical harmonic is completely specified by means of its poles.
In order to express $Y_{n}$ in terms of the positions of its poles, we apply the theorem (7) to the evaluation of $Y_{n}$ in (8). On putting
$f_{n}(x,y,z)=\prod _{r=1}^{r=n}(l_{r}x+m_{r}y+n_{r}z)$, we have
$Y_{n}={\frac {(2n!)}{2^{n}n!n!}}\cdot {\frac {1}{r^{n}}}\left(1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}\dots \right)\times \prod _{r}^{n}(l_{r}x+m_{r}y+n_{r}z).$
By $\sum (\mu ^{s}\lambda ^{n-2s})$ we shall denote the sum of the products of $s$ of the quantities $\mu$, and $n-2s$ of the quantities $\lambda$; in any term each suffix is to occur once, and once only, every possible order being taken. We find
$\prod (lx+my+nz)=\sum (\lambda ^{n})r^{n}\Delta ^{2}\prod (lx+my+nz)=2\sum (\mu ^{1}\lambda ^{n-2})r^{n-2},$
and generally
$\Delta ^{2m}\prod (lx+my+nz)=2^{m}m!\sum (\mu ^{m}\lambda ^{n-2m})r^{n-2m};$
thus we obtain the following expression for $Y_{n}$, the surface harmonic which has given poles $h_{1},h_{2},\dots h_{n}$;
${\begin{aligned}Y_{n}&=r^{n+1}{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}\cdot {\frac {1}{r}}\\&={\underset {m=0}{S}}\left\{(-1)^{m}{\frac {(2n-2m)!}{2^{n-m}n!(n-m)!}}\sum (\lambda ^{n-2m}\mu ^{m})\right\}\end{aligned}}$
where S denotes a summation with respect to m from $m=0$ to $m={\frac {1}{2}}n$, or ${\frac {1}{2}}(n-1)$, according as n is even or odd. This is Maxwell's general expression (loc. cit.) for a surface harmonic with given poles.
If the poles on a sphere of radius r are denoted by A, B, C. . ., we obtain from (9) the following expressions for the harmonics of the first four degrees:—
$Y_{1}=\cos PA,Y_{2}={\frac {1}{2}}(3\cos PA\cos PB-\cos AB),$
${\begin{aligned}Y_{3}={\frac {1}{2}}(15\cos PA\cos PB\cos PC&-\cos PA\cos BC-\cos PB\cos CA\\&-\cos PC\cos AB),\end{aligned}}$
${\begin{aligned}Y_{4}={\frac {1}{8}}(35\cos PA\cos PB\cos PC\cos PD&-5\sum \cos PA\cos PB\cos CD\\&+\sum \cos AB\cos CD).\end{aligned}}$
7. Poles of Zonal, Tesseral and Sectorial Harmonics.—Let the n axes of the harmonic coincide with the axis of z, we have then by (8) the harmonic
${\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}$
​applying the theorem (7) to evaluate this expression, we have
${\begin{aligned}{\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}&={\frac {(2n)!}{2^{n}n!n!}}{\frac {1}{r^{n}}}\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+{\frac {r^{4}\nabla ^{4}}{2.4.2n-1.2n-3}}-\right\}z^{n}\\&={\frac {(2n)!}{2^{n}n!n!}}\left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n-2}+\dots \right\},\end{aligned}}$
the expression on the right side is $P^{n}(\mu )$, the zonal surface harmonic; we have therefore
$P_{n}(\mu )={\frac {(-1)^{n}r^{n+1}}{n!}}{\frac {\partial ^{n}}{\partial z^{n}}}{\frac {1}{r}}.$
The zonal harmonic has therefore all its poles coincident with the z axis. Next, suppose n - m axes coincide with the z axis, and that the remaining m axes are distributed symmetrically in the plane of x, y at intervals $\pi /m$, the direction cosines of one of them being $\cos \alpha ,\sin \alpha ,0$. We have
${\begin{aligned}\prod _{0}^{m-1}\left\{\cos \left(\alpha +{\frac {r\pi }{m}}\right){\frac {\partial }{\partial x}}+\sin \left(\alpha +{\frac {r\pi }{m}}\right){\frac {\partial }{\partial y}}\right\}&={\frac {1}{2^{m}}}\prod {\Bigl \e^{i\left(\alpha +{\frac {r\pi }{m}}\right)}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\&+e^{-i\left(\alpha +{\frac {r\pi }{m}}\right)}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right){\Bigr \}}.\end{aligned}}$
Let $\xi =x+iy,\eta =x-iy$, the above product becomes
$\prod _{0}^{m-1}\left\{e^{i\left(\alpha +{\frac {r\pi }{m}}\right)}{\frac {\partial }{\partial \xi }}+e^{-i\left(\alpha +{\frac {r\pi }{m}}\right)}{\frac {\partial }{\partial \eta }}\right\}$
which is equal to
$e^{(m-1){\frac {i\pi }{2}}}\prod \left\{e^{mi\alpha }\left({\frac {\partial }{\partial \xi }}\right)^{m}-e^{-mi\alpha }\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}$; when $\alpha =0,{\frac {\pi }{2m}}$,
this becomes
$e^{(m-1){\frac {i\pi }{2}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}-(-1)^{m}\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}$ and
$e^{(m-1){\frac {i\pi }{2}}}\left[\left({\frac {\partial }{\partial \xi }}\right)^{m}+(-1)^{m}\left({\frac {\partial }{\partial \eta }}\right)^{m}\right]$.
From (7), we find
${\begin{aligned}{\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left({\frac {\partial }{\partial x}}\pm i{\frac {\partial }{\partial y}}\right)^{m}{\frac {1}{r}}&={\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{2n+1}}}\left[1-{\frac {r^{2}\Delta ^{2}}{2.2n-1}}+\dots \right]z^{n-m}(x\pm iy)^{m}\\&=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}{\frac {1}{r^{n+1}}}(\cos m\phi \pm i\sin m\phi )\sin ^{m}\theta {\Bigl \\cos ^{n-m}\theta \\&-{\frac {(n-m)(n-m-1)}{2.2n-1}}\cos ^{n-m-2}+\dots \theta {\Bigr \}}\end{aligned}}$
hence
${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left({\frac {\partial }{\partial x}}\pm i{\frac {\partial }{\partial y}}\right)^{m}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{r^{n+1}}}(\cos m\phi \pm i\sin m\phi )P_{n}^{m}(\cos \theta )$
as we see on referring to (4); we thus obtain the formulae
${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}+\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{2^{m-1}r^{n+1}}}\cos m\phi P_{n}^{m}(\cos \theta )$
${\frac {\partial ^{n-m}}{\partial z^{n-m}}}\left\{\left({\frac {\partial }{\partial \xi }}\right)^{m}-\left({\frac {\partial }{\partial \eta }}\right)^{m}\right\}{\frac {1}{r}}=(-1)^{n}{\frac {(n-m)!}{2^{m-1}r^{n+1}}}\sin m\phi P_{n}^{m}(\cos \theta )$
It is thus seen that the tesseral harmonics of degree n and order m are those which have $n-m$ axes coincident with the z axis, and the other m axis distributed in the equatorial plane, at angular intervals $\pi /m$. The sectorial harmonics have all their axes in the equatorial plane.
8. Determination of the Poles of a given Harmonic.—It has been shown that a spherical harmonic $Y_{n}(x,y,z)$ can be generated by means of an operator
$f_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)$ acting upon ${\frac {1}{r}}$,
the function $f_{n}$ being so chosen that
$Y_{n}(x,y,z)=(-1)^{n}{\frac {(2n)!}{2^{n}n!}}\left\{1-{\frac {r^{2}\nabla ^{2}}{2.2n-1}}+\dots \right\}f_{n}(x,y,z);$
this relation shows that if an expression of the form
$(x^{2}+y^{2}+^{2})f_{n-2}(x,y,z)$
is added to $f_{n}(x,y,z)$, the harmonic $Y_{n}(x,y,z)$ is unaltered; thus if $Y_{n}$ be regarded as given, $f_{n}(x,y,z)=0$, is not uniquely determined, but has an indefinite number of values differing by multiples of $x^{2}+y^{2}+z^{2}$. In order to determine the poles of a given harmonic, $f_{n}$ must be so chosen that it is resolvable into linear factors; it will be shown that this can be done in one, and only one, way, so that the poles are all real.
If x, y, z are such as to satisfy the two equations $Y_{n}(x,y,z)=0$, $x^{2}+y^{2}+z^{2}=0$, the equation $f_{n}(x,y,z)$ is also satisfied; the problem of determining the poles is therefore equivalent to the algebraical one of reducing $Y_{n}$ to the product of linear factors by means of the relation $x^{2}+y^{2}+z^{2}=0$, between the variables. Suppose
$Y_{n}(x,y,z)=A\prod _{s=1}^{m}(l_{s}x+m_{s}y+n_{s}z)+(x^{2}+y^{2}+z^{2})V_{n-2}(x,y,z),$
we see that the plane $l_{s}x+m_{s}y+n_{s}z=0$ passes through two of the 2n generating lines of the imaginary cone $x^{2}+y^{2}+z^{2}=0,$ in which that cone is intersected by the cone $Y_{n}(x,y,z)=0$. Thus a pole $(l_{s},m_{s},n_{s})$ is the pole with respect to the cone $x^{2}+y^{2}+z^{2}=0$, of a plane passing through two of the generating lines; the number of systems of poles is therefore $n(2n-1)$, the number of ways of taking the 2n generating lines in pairs. Of these systems of poles, however, only one is real, viz. that in which the lines in each pair correspond to conjugate complex roots of the equations $Y_{n}=0$, $x^{2}+y^{2}+z^{2}=0$. Suppose
${\frac {x}{\alpha _{1}+i\beta _{1}}}={\frac {y}{\alpha _{2}+i\beta _{2}}}={\frac {z}{\alpha _{2}+i\beta _{2}}}$
gives one generating line, then the conjugate one is given by
${\frac {x}{\alpha _{1}-i\beta _{1}}}={\frac {y}{\alpha _{2}-i\beta _{2}}}={\frac {z}{\alpha _{2}-i\beta _{2}}}$
and the corresponding factor $lx+my+nz$ is
${\begin{vmatrix}x&y&z\\\alpha _{1}+i\beta _{1}&\alpha _{2}+i\beta _{2}&\alpha _{3}+i\beta _{3}\\\alpha _{1}-i\beta _{1}&\alpha _{2}-i\beta _{2}&\alpha _{3}-i\beta _{3}\\\end{vmatrix}}$
which is real. It is obvious that if any non-conjugate pair of roots is taken, the corresponding factor, and therefore the pole, is imaginary. There is therefore only one system of real poles of a given harmonic, and its determination requires the solution of an equation of degree 2n. This, theorem is due to Sylvester (Phil. Mag. (1876), 5th series, vol. ii., "A Note on Spherical Harmonics").
9. Expression for the Zonal Harmonic with any Axis.—The zonal surface harmonic, whose axis is in the direction
${\frac {x'}{r'}},{\frac {y'}{r'}},{\frac {z'}{r'}}$, is $P_{n}\left({\frac {xx'+yy'+zz'}{rr'}}\right)$
or $P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos \phi -\phi ')$; this is expressible as a linear function of the system of zonal, tesseral, and sectorial harmonics already found. It will be observed that it is symmetrical with respect to $(x,y,z)$ and $(x',y',z')$, and must thus be capable of being expressed in the form
$a_{0}P_{n}(\cos \theta )P_{n}(\cos \theta ')+\sum _{1}^{n}a_{m}P_{n}^{m}(\cos \theta )P_{n}^{m}(\cos \theta ')\cos m(\phi -\phi ')$
and it only remains to determine the co-efficients $a_{0},a_{1},\dots ,a_{m},\dots a_{n}$ To find this expression, we transform $(x'x+y'y+z'z)^{n}$, where x, y, z satisfy the condition $x^{2}+y^{2}+z^{2}=0$; writing $\xi =x+iy,\eta =x-iy,\xi '=x'+iy',\eta '=x'-iy'$, we have
$(xx'+yy'+zz')^{m}=({\frac {1}{2}}\eta '\xi +{\frac {1}{2}}\xi '\eta +zz')^{n}$
which equals
$(xx'+yy'+zz')^{n}=(zz')^{n}+\sum \sum {\frac {n!}{a!b!(n-a-b)!}}\left\{{\frac {\eta '^{a}\xi '^{b}\eta ^{b}\xi ^{b}+\eta '^{b}\xi '^{a}\xi ^{b}\eta ^{a}}{2^{a+b}}}\right\}(zz')^{n-a-b}$
the summation being taken for all values of a and b, such that $a+b\leq n$, $a>b$; the values $a=0,b=0$ corresponding to the term $(zz')^{n}$. Using the relation $\xi \eta =-z^{2}$, this becomes
${\begin{aligned}(xx'+yy'+zz')^{n}=(zz')^{n}+\sum \sum &{\frac {(-1)^{b}}{2^{a+b}}}{\frac {n!}{a!b!(n-a-b)!}}(\eta '\xi ')^{b}z'^{n-a-b}\\&\{(\eta '\xi )^{a-b}+(\xi '\eta )^{a-b}\}z^{n-a+b},\end{aligned}}$
putting $a-b=m$, the coefficient of $\xi ^{m}z^{n-m}$ on the right side is
$\sum {\frac {(-1)^{b}}{2^{m+2b}}}{\frac {n!}{b!(m+b)!(n-m-2b)!}}(\xi '\eta ')^{b}\eta '^{m}z'^{n-m-2b},$
from $b=0$ to $b={\frac {1}{2}}(n-m)$, or ${\frac {1}{2}}(n-m-1)$, according as $n-m$ is even or odd. This coefficient is equal to
${\begin{aligned}&{\frac {n!}{2^{m}m!(n-m)!}}(x'-iy')^{m}{\Bigl \z'^{n-m}-{\frac {(n-m)(n-m-1)}{2.2m+2}}z'^{n-m-2}(x'^{2}+y'^{2})\\&+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2.4.2m+2.2m+3}}z'^{n-m-4}(x'^{2}+y'^{2})^{2}-{\Bigr \}}\end{aligned}}$
in order to evaluate this coefficient, put $z=1$, $x'=i\cos \alpha$, $y'=i\sin \alpha$, then this coefficient is that of $(i\cos \alpha +\sin \alpha )^{m}$, or of $i^{m}e^{-mi\alpha }$ in the expansion of $(z'+ix'\cos \alpha +iy'\sin \alpha )^{n}$ in powers of $e^{-i\alpha }$ and $e^{i\alpha }$, this has been already found, thus the coefficient is
${\frac {n!}{(n+m)!}}e^{-im\phi '}P_{n}^{m}(\cos \theta ')r'^{n}.$
Similarly the coefficient of $\eta ^{m}z^{n-m}$ is
${\frac {n!}{(n+m!)}}e^{+im\phi '}P_{n}^{m}(\cos \theta ')r'^{n};$
hence we have
${\begin{aligned}{\frac {1}{r'^{n}}}(xx'+yy'+zz')^{n}=z^{n}P_{n}^{m}(\cos \theta ')+n!\sum _{1}^{n}&P_{n}^{m}(\cos \theta ')\{\cos m\phi '(\xi ^{m}+\eta ^{m})\\&+i\sin m\phi '(\eta ^{m}-\xi ^{m})\}{\frac {z^{n-m}}{(n+m!)}}\end{aligned}}$
In this result, change x, y, z into
${\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}$
​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
$\int _{-1}^{1}\mu ^{k}P_{n}(\mu )d\mu ={\frac {1}{2^{n}n!}}\int _{-1}^{1}\mu ^{k}{\frac {d^{n}}{d\mu ^{n}}}(\mu ^{2}-1)^{n}d\mu ,$
​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.),
$\iint Y_{n}(x,y,z)f_{n}(x,y,z)dS={\frac {4\pi R^{2n+2}}{2n+1}}{\frac {1}{n!}}{\frac {\partial ^{n}}{\partial h_{1}\partial h_{2}\dots \partial h_{n}}}f_{n}(x,y,z)$
​where $h_{1}h_{2}\dots h_{n}$ are the axes of $Y_{n}$. Two harmonics of the same degree are said to be conjugate, when the surface integral of their product vanishes; if $Y_{n},Z_{n}$ are two such harmonics, the addition of conjugacy is
$Y_{n}\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)Z_{n}(x,y,z)=0.$
Lord Kelvin has shown how to express the conditions that $2n+1$ harmonics of degree n form a conjugate system (see B. A. Report, 1871).
16. Expansion of a Function in a Series of Spherical Harmonics.—It can be shown that under certain restrictions as to the nature of a function $F(\mu ,\phi )$ given arbitrarily over the surface of a sphere, the function can be represented by a series of spherical harmonics which converges in general uniformly. On this assumption we see that the terms of the series can be found by the use of the theorems (22), (23). Let $F(\mu ,\phi )$ be represented by
$V_{0}(\mu ,\phi )+V_{1}(\mu ,\phi )+\dots +V_{n}(\mu ,\phi )+\dots ;$
change $\mu ,\phi$ into $\mu ',\phi '$ and multiply by
$P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}}),$
we have then
${\begin{aligned}&\int _{0}^{2\pi }\int _{-1}^{1}F(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '\\&=\int _{0}^{2\pi }\int _{-1}^{1}V_{n}(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '\\&={\frac {4\pi }{2n+1}}V_{n}(\theta ,\phi )\end{aligned}}$
hence the series which represents $F(\mu ,\phi )$ is
${\frac {1}{4}}\pi \sum _{0}^{\infty }(2n+1)\int _{0}^{2\pi }\int _{-1}^{1}F(\mu ',\phi ')P_{n}(\cos \theta \cos \theta '+\sin \theta \sin \theta '\cos {\overline {\phi -\phi '}})d\mu 'd\phi '$
A rational integral function of $\sin \theta \cos \phi$, $\sin \theta \sin \phi$, $\cos \phi$ of degree n may be expressed as the sum of a series of spherical harmonics, by assuming
$f_{n}(x,y,z)=Y_{n}+r^{2}Y_{n-2}+r^{4}Y_{n-4}+\dots$
and determining the solid harmonics $Y_{n}$, $Y_{n-2}$, . . . and then letting $r=1$, in the result.
Since $\nabla ^{2}(r^{2s}Y_{n-2s})=2s(2n-2s+1)r^{2s-2}Y_{n-2s}$, we have
$\nabla ^{2}f_{n}=2(2n-1)Y_{n-2}+4(2n-3)r^{2}Y_{n-4}+6(2n-5)r^{4}Y_{n-6}+\dots$ $\nabla ^{4}f_{n}=2.4(2n-3)(2n-5)Y_{n-4}+4.6(2n-5)(2n-7)r^{2}Y_{n-6}+\dots$
the last equation being
$\nabla ^{n}f_{n}=n(n+1)(n-2)(n-1)\dots Y_{0}$, if n is even,
or
$\nabla ^{n-1}f_{n}=(n-1)(n+2)(n-3)n\dots Y_{1}$, if n is odd
from the last equation $Y_{0}$ or $Y_{1}$ is determined, then from the preceding one $Y_{2}$ or $Y_{3}$, and so on. This method is due to Gauss (see Collected Works, v. 630).
As an example of the use of spherical harmonics in the potential theory, suppose it required to calculate at an external point, the potential of a nearly spherical body bounded by $r=a(1+\epsilon u)$, the body being made of homogeneous material of density unity, and u being a given function of $\theta$, $\phi$, the quantity $\epsilon$ being so small that its square may be neglected. The potential is given by
$\int _{0}^{2\pi }\int _{-1}^{1}\int _{0}^{a(1+\epsilon u')}\{r^{2}+r'^{2}-2rr'\cos \gamma \}^{-{\frac {1}{2}}}dr'd\mu 'd\phi ',$
where $\gamma$ is the angle between r and r'; now let u' be expanded in a series
$Y_{0}(\mu ',\phi ')+Y_{1}(\mu ',\phi ')+\dots +Y_{n}(\mu ',\phi ')+\dots$
of surface harmonics; we may write the expression for the potential
${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}\int _{0}^{a(1+\epsilon u')}&{\Bigl \{\frac {1}{r}}+{\frac {r'}{r^{2}}}P_{1}(\cos \gamma )+\dots \\&+{\frac {r'^{n}}{r^{n+1}}}P_{n}(\cos \gamma )+\dots {\Bigr \}}r'^{2}dr'd\mu 'd\phi '\end{aligned}}$
which is,
${\begin{aligned}\int _{0}^{2\pi }\int _{-1}^{1}{\Bigl \{\frac {a^{3}}{3r}}(1+3\epsilon u')&+{\frac {1}{4}}{\frac {a^{4}}{r^{2}}}(1+4\epsilon u')P_{1}+\dots \\&+{\frac {1}{n+3}}{\frac {a^{n+3}}{r^{n+1}}}(1+{\overline {n+3}}\epsilon u')P_{n}(\cos \gamma ){\Bigr \}}d\mu 'd\phi '\end{aligned}}$
on substituting for u' the series of harmonics, and using (22), (23), this becomes
${\begin{aligned}4\pi a^{2}{\Bigl{\frac {1}{3}}{\frac {a}{r}}+\epsilon {\Bigl \{\frac {a^{3}}{3r^{2}}}Y_{1}(\mu ,\phi )&+{\frac {a^{3}}{5r^{3}}}Y_{2}(\mu ,\phi )+\dots \\&+{\frac {a^{n+1}}{(2n+1)r^{n-1}}}Y_{n}(\mu ,\phi )+\dots {\Bigr \}}{\Bigr ]}\end{aligned}}$
which is the required potential at the external point $(r,\theta ,\phi )$.
17. The Normal Solutions of Laplace's Equation in Polars.—If $h_{1}$, $h_{2}$, $h_{3}$ be the parameters of three orthogonal sets of surfaces, the length of an elementary arc ds may be expressed by an equation of
the form $ds^{2}={\frac {1}{H_{1}^{2}}}dh_{1}^{2}+{\frac {1}{H_{2}^{2}}}dh_{2}^{2}+{\frac {1}{H_{3}^{2}}}dh_{3}^{2}$, where $H_{1}$, $H_{2}$, $H_{3}$ are functions of $h_{1}$, $h_{2}$, $h_{3}$, which depend on the form of these parameters; it is known that Laplace's equation when expressed with $h_{1}$, $h_{2}$, $h_{3}$ as independent variables, takes the form
${\frac {\partial }{\partial h_{1}}}\left({\frac {H_{1}}{H_{2}H_{3}}}{\frac {\partial V}{\partial h_{1}}}\right)+{\frac {\partial }{\partial h_{2}}}\left({\frac {H_{2}}{H_{3}H_{1}}}{\frac {\partial V}{\partial h_{2}}}\right)+{\frac {\partial }{\partial h_{3}}}\left({\frac {H_{3}}{H_{1}H_{2}}}{\frac {\partial V}{\partial h_{3}}}\right)=0.$
In case the orthogonal surfaces are concentric spheres, co-axial, circular cones, and planes through the axes of the cones, the parameters are the usual polar co-ordinates r, $\theta$, $\phi$, and in this case $H_{1}=1,H_{2}={\frac {1}{r}},H_{3}={\frac {1}{r\sin \theta }}$, thus Laplace's equation becomes
${\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial V}{\partial r}}\right)+{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial V}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}=0.$
Assume that $V=R\Theta \Phi$ is a solution, R being a function of r only, $\Theta$ of $\theta$ only, $\Phi$ of $\phi$ only; we then have
${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)+{\frac {1}{\Theta \sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)+{\frac {1}{\sin ^{2}\theta .\Phi }}{\frac {d^{2}\Phi }{d\phi ^{2}}}=0.$
This can only be satisfied if ${\frac {1}{R}}{\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)$ is a constant, say $n(n+1)$, ${\frac {1}{\Phi }}{\frac {d^{2}\Phi }{\phi ^{2}}}$ is a constant, say $-m^{2}$, and $\Theta$ satisfies the equation
${\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\left(\sin \theta {\frac {d\Theta }{d\theta }}\right)+\left\{n(n+1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right\}\Theta =0,$
if we write u for $\Theta$, and $\mu$ for $\sin \theta$, this equation becomes
${\frac {d}{d\mu }}\left\{(1-\mu ^{2}){\frac {du}{d\mu }}\right\}+\left\{n(n+1)-{\frac {m^{2}}{1-\mu ^{2}}}\right\}u=0.$
From the equations which determine R, $\Theta$, $u$, it appears that Laplace's equation is satisfied by
${\begin{matrix}r^{n}\\r^{-n-1}\end{matrix}}{\begin{matrix}\cos \\\sin \end{matrix}}m\phi .u{\begin{matrix}m\\n\end{matrix}}$
where u is any solution of (26); this product we may speak of as the normal solution of Laplace's equation in polar co-ordinates; it will be observed that the constants n, m may have any real or complex values.
18. Legendre's Equation.—If in the above normal solution we consider the case $m=0$, we see that
${\begin{matrix}r^{n}\\r^{-n-1}\end{matrix}}u_{n}$
is the normal form, where $u_{n}$ atisfies the equation
${\frac {d}{d\mu }}\left\{(1-\mu ^{2}){\frac {du}{d\mu }}\right\}+n(n+1)u=0,$
known as Legendre's equation; we shall here consider the special case in which n is a positive integer. One solution of (27) will be the Legendre's coefficient $P_{n}(\mu )$, and to find the complete primitive we must find another particular integral; in considering the forms of solution, we shall consider $\mu$ to be not necessarily real and between $\pm 1$. If we assume
$u=\mu ^{m}+\alpha _{2}\mu ^{m-2}+\alpha _{4}\mu ^{m-4}+\dots$
as a solution, and substitute in the equation (27), we find that $m=n$, or $-n-1$, and thus we have as solutions, on determining the ratios of the coefficients in the two cases,
$\alpha \left\{\mu ^{n}-{\frac {n(n-1)}{2.2n-1}}\mu ^{n+2}+\dots \right\}$
and
$\beta \left\{{\frac {1}{\mu ^{n+1}}}+{\frac {(n+1)(n+2)}{2.2n+3}}{\frac {1}{\mu ^{n+3}}}+{\frac {(n+1)(n+2)(n+3)(n+4)}{2.4.2n+3.2n+5}}{\frac {1}{\mu ^{n+5}}}+\dots \right\}$
the first of these series is (n integral) finite, and represents $P_{n}(\mu )$, the second is an infinite series which is convergent when ${\text{mod}}~\mu >1$. If we choose the constant $\beta$ to be ${\frac {1.2.3\dots n}{3.5\dots 2n+1}}$, the second solution may be denoted by $Q_{n}(\mu )$, and is called the Legendre's function of the second kind, thus
${\begin{aligned}Q_{n}(\mu )={\frac {1.2.3\dots n}{3.5\dots 2n+1}}&\left\{{\frac {1}{\mu ^{n+1}}}+{\frac {(n+1)(n+2)}{2.2n+3}}{\frac {1}{\mu ^{n+3}}}+\dots \right\}\\&={\frac {1.2.3\dots n}{3.5\dots 2n+1}}{\frac {1}{\mu ^{n+1}}}F\left({\frac {n+1}{2}},{\frac {n+2}{2}},{\frac {2n+3}{2}},{\frac {1}{\mu ^{2}}}\right).\end{aligned}}$
This function $Q_{n}(\mu )$, thus defined for ${\text{mod}}~\mu >1$, is of considerable importance in the potential theory. When ${\text{mod}}~\mu <1$, we may in a similar manner obtain two series in ascending powers of $\mu$, one of which represents $P_{n}(\mu )$, and a certain linear function of the two series represents the analytical continuation of $Q_{n}(\mu )$, as defined above. The complete primitive of Legendre's equation is
$u=AP_{n}(\mu )+BQ_{n}(\mu ).$
By the usual rule for obtaining the complete primitive of an ordinary differential equation of the second order when a particular integral is known, it can be shown that (27) is satisfied by
$P_{n}(\mu )=\int ^{\mu }{\frac {d\mu }{(\mu ^{2}-1)\{P_{n}(\mu )\}^{2}}}$
the lower limit being arbitrary.
​ From this form it can be shown that
Q$_n$(μ)＝1/2 P$_n$(μ) log μ＋1/μ−1−W$_{n−1}$(μ),
where W$_{n−1}$(μ) is a rational integral function of degree n−1 in μ; it can be shown that this form is in agreement with the definition of Q$_n$(μ) by series, for the case mod μ>1. In case mod μ<1 it is convenient to use the symbol Q$_n$(μ) for
1/2P$_n$(μ) log 1＋μ/1−μ−W$_{n−1}$(μ),
which is real when μ is real and between Â±1, the function Qâ€ž(m) in this case is not the analytical continuation of the function Qt.(m) for mod m> x . but differs from it by an imaginary multiple of Pâ€ž(m). It will be observed that Q n (i), Q n ( â€” i) are infinite, and Qn(x) =o. The function Wn-i(ju) has been expressed by Christoffel in the form
2 "-?Pâ€ž_ lW + - 2 "-
â€¢w+|^ p â€” w+-
I . n ^' . ' 3 . n â€” I
and it can also be expressed in the form
~P (p.)P^ 1 (p.)+~P l (p)?^. 2 (p.) + . . . +Pâ€ž-,(m)Po(m).
It can easily be shown that the formula (28) is equivalent to
which is analogous to Rodrigue's expression for PnM- Another expression of a similar character is
It can be shown that under the condition mod \u â€” V (u- â€” 1)\ >mod \ti â€” V (v?â€” 1)|, the function il(n â€” u) can be expanded in the form 2(2Â»-|-i)PÂ«(tt)Qâ€ž(tt) ; this expansion is connected with the definite integral formula for Q n (v) which was used by F. Neumann as a definition of the function QÂ»(p.), this is
which holds for all values of \i which are not real and between Â± I. From Neumann's integral can be deduced the formula
d4>
â€¢â€¢00 Qn(M) = J ,
iM+vc^-o.cosh^r'
which holds for all values of p. which are not real and between ="= 1 , provided the sign of V (/* 2 â€” 1 ) is properly chosen ; when p is real and greater than 1, V (p 2 â€” 1) has its positive value. By means of the substitution.
Jm + V(m 2 â€” i).cosh \fr\\ii â€” V(m 2 â€” l).cosh x! = i,
the above integral becomes
Qâ€ž0) = J *V~V (m 2 - 1) cosh xNx. where xa = ^og c j~. '
This formula gives a simple means of calculating Qâ€ž(p) for small values of n; thus
If, in Legendre's equation, we differentiate m times, we find
. . n d m **u , . . d M+1 u , , ., , , .d m u
â€¢ c 11 1 d m u , m , , ,v m d m u
it follows that u = ^'hence Â«â€ž = 0* 2 -i)- - a -^-
2m'"
The complete solution of (26) is therefore
when m is real and lies between Â± 1 , the two functions
U-M - ) 4u'" ' U_M - ) C^" 1 are called Legendre's associated functions of degree <n, and ordur m, of the fhrt and second kinds respectively. When /j is not real and between =*=i, the same names are given to the two functions
Qi(p) =/Â»XÂ«-V (m 2 - I) -sinh Xo = M-^oSj^~ I.
Neumann's integral affords a means of establishing a relation between successive Q functions, thus
Â»QÂ»-(2rt-i)/Â«QÂ»-i + (n-l)(?Â»_2
"1 Â«P,(tt) + (n- i)Pâ€ž- 2 (h.) - (2w- p â€” u
pP n -i{u)du
-ITJ
= -|J _,(2Â»- l)PÂ»-l(") =0.
Again, it may similarly be proved that
19. Legendre Associated Functions. â€” Returning to the equation (26) satisfied by Â«â„¢ the factor in the normal forms â€ž_, â„¢ mtfr.uâ„¢,
we shall consider the case in which n, m are positive integers, and
n^m. Let Â« = (m 2 â€” i)* m v, then it will be found that v satisfies the equation
(m 2 -0
dp'" '
(V-i)
U d m Q*W
dp m
in either case the functions may be denoted by P n (m), Qn (m). It can be shown that, when p is real and between Â± 1
p:m =5^11 (S3 im ^io.-i)"^o.+i)-i
2Â»(n-m)l\l+p) dp" 1 ^ V " +l ' <â€¢ In the same case, we find
Pr +2 (cos 8)-2(m + i) cot $ Pâ„¢ +1 (cos 0)
+ (n â€” m)(n+m + l)Pâ„¢(cos 8)=o, (ra-Â«+2)Pr +2 (cos 0) - (2n+3) M Pr + i(cos 0)
+ (Â»+m + l)Pr(cos0)=o.
20. Bessel's Functions.â€” -If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are a, p, $, the cylindrical co-ordinates; in that case Hi = i, Hi = i, H 3 = i/p, and the equation (25) becomes
d"v yy 1 av 1 a-v d~ 2 + <v +p dp v a<^> 2 ~ '
To find the normal functions which satisfy this equation, we put V = ZRΦ, when Z is a function of z only, R of p only, and Φ of φ, the equation then becomes
I ^?j_Â± /^P:j_I ^E\ j.1 I 4!?_ Z dz- + K Up 2 + p dp/ V* d^~Â°-
1 2 Z That this may be satisfied we must have y~r^ constant, say =k',
tt yj conetant, say = â€” m-, and R, for which we write u, must satisfy the differential equation d 2 u , I dti
dp'
. 1 du /,â€ž pi-\
it follows that the normal forms arc e ^ z '-m<p.u(kp), where u(p) satisfies the equation
",7/ T n ii t n'"\
(29)
d 2 u , I du , / n7 ! \
d?+pd7 + { l -7) u=0 -
This is known as Bessel's equation of order m\ the particular case
d"-u , I du
d9 + pT P + u=0 '
(30)
corresponding to m = o. is known as Bessel's equation.
If we solve the equation (29) in series, we find by the usual process that it is satisfied by the series
+0
2.2OT+2 r 2.4.2/W-(-2.2W-|-4
the expression
P" \, P 2 ,
2"*II(m) ( 2.2m+2 2.4.2m+2.2m+4
...| s
or
(â€” i)V +2n
/ |) 2"' +2 "n(m+Â»)n(w' l
is denoted by J m (p). """i When m=o, the solution
i _pJ + _A_...
2 2 ' 2 2 . 4 2
of the equation (30) is denoted by J$_0$(ρ) or by J(ρ). ​ The function J m (p) is called Bessel-'s function of order m, and Jo(p) simply Bessel's function; the series are convergent for all finite values of p.
The equation (29) is unaltered by changing m intoâ€” m, it follows that J-m(p) is a second solution of (20). thus in general
Â« = AJ m (p)+BJ_ m (p)
is the complete primitive of (29). However, in the most important case, that in which m is an integer, the solutions J_ m (p), J m (p) are not distinct, for J~ m (p) may be written in the form
6 5?
mâ€” 1
M-V (-!)â– /aV V2/ ^^n(Â»-m)n(Â») W
n-0
i_VsV
now n(Â« â€” m) is infinite when m is an integer, and n < m ; thus the first part of the expression vanishes, and the second part is (-i)'"Jm(p), hence when m is an integer J_ ra (p) = ( â€” i) m ] m (p), and the second solution remains to be found.
Bessel's Functions of the Second Kind. â€” When m is not a real integer, we have seen that any linear function of ] m (p), F-m(p) satisfies the equation of order m. The Bessel's function of the second kind of order m is defined as the particular linear function
Tf m., J-â„¢(p) -cos rax . T m ( P ) t sin 2mir -
and may be denoted by Yâ€ž(p). This definition has the advantage of giving a meaning to Yâ€ž(p) in the case in which m is an integer, for it may be evaluated as a limiting form 0/0, and the limit will satisfy the equation (29). The only failing case is when m is half an odd integer; in that case we take cos mv Yâ€ž(p) as a second finite solution of the differential equation. When m is an integer, we have
on earning out the differentiations, and proceeding to the limit we find cc
n_0
+i(T2- nJ =sr i (r
â€”0
where t(Â») denotes ll'(n)/n(n).
When m = o wc have the second solution of (30) given by
21. Relations between Bessel's Functions of Different Orders.â€” -Since
sin m< t>-"m(p) satisfies the differential equation
d-u d-u
^, + ^â€”â€ž + u=o.
dx 1 dy-
(30 The linear character of this equation shows that if u is any solution
is also one, / denoting a rational integral function of the operators. Let i, tj denote x+iy, x-iy, then since pr"4â„¢Â«m(VÂ£';) satisfies the differential equation, so also does
thus we have
d(p--)^
Mm+p = Cp m+ "-
n Â»>t.(p)l,
d" [ Â«â€ž,(p)
d{?)Â»\ p,
where C is a constant. If u m (p)=] m (p), we have u m+p = J m +p(p), and by comparing the coefficients of p" l+ ", wc find C = (â€” 2) v , hence
Jm+p(p) = (-2)"p m+ " :
and changing m into â€”m, we find
d(p ! )" [ '
"Jm(p)(,
d" ',
J,_ m (p) = (-2) V -^[pÂ«J_ m (p)j.
In a similar manner it can be proved that
} m â€ž(p) = 2 Â»p^ 5 ^p|pÂ»'J m (p)j.
From the definition of Yâ€ž,(p), and applying the above analysis, wi, prove that
and
^ +1 ,(p) = (-2)Â»pâ€” ^{p-
'Y,(p)
Y m _ J) (p)=2'>p>'
d"
, d* d{p>)
,Yo(p)
^pfp m Y m (p)!.
As particular cases of the above formulae, we find
L,(p) = (-2p)*^|Lj (p), Y p (p) = (-2p)
J I ( p) = _%W,Y I (p) = -^ Y Â«(Â£). cp dp
22. Bessel's Functions as Coefficients in an Expansion. â€” It is clear that eipÂ°<Â«* = ei* or ew> si Â» * = g>Â» satisfy the differential equation (31), hence if these exponentials be expanded in series of cosines and sines of multiples of 0, the coefficients must be Bessel's functions, which it is easy to see are of the first kind. To expand e'psin *_ put <â– <* = /, we have then to expand eM'-' -1 ) in powers of t. Multiplying together the two absolutely convergent series
we obtain for the coefficient of t m in the product P'" S . P 2 p 4
(5)
hence
I-
elt>(.'-
2 .2m+i~*' 1 .4 ,2m + 2 .2m+4.~
â– ')_
Jo(p)-HTi(p) + ...+^j T m (p) + â€ž -^Ji(p) + â– -.+(- 1)â„¢*-â„¢ Â»â„¢J m (p)
^or J m (p),
JÂ»(p) 5
(32)
the Bessel's functions were defined by Schlomilch as the coefficients of the powers of I in the expansion of eip('~'~ \ and many of the properties of the functions can be deduced from this expansion. By differentiating both sides of (32) with respect to t, and equating the coefficients of /'" ' on both sides, we find the relation
Jm-i(p) +Jm+i(p) =-j-J m (p),
which connects three consecutive functions. Again, by differ- entiating both sides of (32) with respect to p, and equating the coefficients of corresponding terms, we find
â€ž djm(p) dp
= ]m 1 (/>),â€” Jm+lM.
In (32), let l = e"t>, and equate the real and imaginary parts, we

## have then

cos (p sin 0)=Jo(p)+2.J 2 (p) cos 20+2j 3 (p) cos 30+. . .
sin (p sm 0) =2ji(p) sin 0+2j 3 (p) sin 30+ ... we obtain expansions of cos (p cos 0), sin (p cos 0), by changing into 2-0. On comparing these expansions with Fourier's series, we find expressions for J,â€ž(p) as definite integrals, thus
Jn(p) = ~J cos (psin0)<20, J,â€ž(p) = I j^cos(p sin 0) cos m<jd<t>{m even)
]m(p) = â€ž. J sin (p sin 0) sin m0i0 (m odd).
It can easily be deduced that when m is any positive integer
jm(p) = _ I cos ('"0 â€” P s >" <p)d<j>.
23. Bessel's Functions as Limits of Legmdrc's Functions. â€” The system of orthogonal surfaces whose parameters are cylindrical co- ordinates may be obtained as a limiting case of those whose para- meters are polar co-ordinates, when the centre of the spheres moves off to an indefinite distance from the portion of space which is contemplated. It would therefore be expected that the normal forms
e _,t2 Jm(Xp)si' n s m0 would be derivable as limits of ^-^"(cos 6)^nt<t>..
and we shall show that this is actually the case. If O be the centre of the spheres, take as new origin a point C on the axis of 2, such that OC=a; let P be a point whose polar co-ordinates are r, 0, <f> referred to O as origin, and cylindrical co-ordinates p, z, referred to C as origin; we have
P = r sin 0, z = r cos 8 - a, hence (Â£) n Pâ€ž(cose) = sec"0 (1 +-) "Pâ€ž(cos 6) .
Now let move off to an infinite distance from C, so that a becomes ​infinite, and at the same time let n become infinite in such a way that n/a has a finite value λ. Then
$\operatorname {L} \sec ^{n}\theta =\operatorname {L} \left(\sec {\frac {\rho }{a}}\right)^{\lambda a}=1,\operatorname {L} \left(1+{\frac {z}{z}}\right)^{n}=e^{\lambda z}$
and it remains to find the limiting value of P$_n$(cosθ). From the series (15), it may be at once proved that
$\operatorname {P} _{n}\left(\cos \theta \right)=1-{\frac {\left(n+1\right)n}{1^{2}}}\left(\sin {\frac {\theta }{2}}\right)^{2}+\cdots +{\left(-1\right)}^{m}\delta {\frac {\left(n+m\right)\cdots \left(n-m+1\right)}{1^{2}.2^{2}.m^{2}}}\left(\sin {\frac {\theta }{2}}\right)$
where 5 is some number numerically less than unity and m is a fixed finite quantity sufficiently large; on proceeding to the limit, we have
$\operatorname {L} \operatorname {P} _{n}\left(\operatorname {cos} {\frac {\lambda \rho }{n}}\right)=1-{\frac {\lambda ^{2}\rho ^{2}}{2^{2}}}+{\frac {\lambda ^{4}\rho ^{4}}{2^{2}\cdot 4^{2}}}-\dots +\left(-1\right)^{m}\delta _{1}{\frac {\lambda ^{2m}\rho ^{2}m}{2^{2}\cdot 4^{2}\cdot \left(2m\right)^{2}}}$
where 81 is less than unity Hence ${\underset {n=\infty }{\operatorname {L} }}\operatorname {P} _{n}\left(\cos {\frac {\lambda \rho }{n}}\right)=\operatorname {J} _{0}\left(\lambda \rho \right)$
Again, since $P_{n}^{m}=\left(\cos \rho \right)=\sin ^{m}\theta {\frac {d^{m}\mathrm {P} _{n}\left(\cos \theta \right)}{d\left(\cos \theta \right)^{m}}},$
we have
P"(cosp) = sin"fl'
L *- mp Â»K)= L !
= (-2)"p'
hence
.( fPnCcosa)
d(cos B) m '
<â€¢â– *â€¢(Â»Â«â– Â£ )
'(-Â£)"
AJo(p)
Â«*(pÂ»)-
has been introduced for them. We denote the two solutions of the
equation
(Pit , 1 du
-f-jH â€” j â€” w = o
by I (r), K (r) when
1 /"Â»â–
= - I cosh (r cos </>)<2<K
K (r) = Yo(ir) +-nrJo(tr) = I Q e~r cos A<M<Â£ = I cos (r sinh tfdf.
The particular integral KoM is so chosen that it vanishes when r is real and infinite; it is also represented by
and
and by
rdv.
/"<Â° e~ ru ,
J 1 7I^T) & '
The solutions of the equation
du 2 , 1 du I , m'\ d^+-rTr-{ l +^) u =
are denoted by l m (r), K m (r), where
T m (
2.2W + 2 ' 2.4.2m + 2.2W+4
...J
It may be shown that Yo (p) is obtainable as the limit of Qâ€ž (cos - ) the zonal harmonic of the second kind ; and that
Y m (p)=Ln-Â»Q:(cosQ.
24. Definite Integral Solutions of Bessel's Equation.^-BeaseYs equation of order m, where m is unrestricted, is satisfied by the
r m~i
expression pâ„¢ I ev' (Pâ€”l) dt, where the path of integration is either
a curve which is closed on the Riemann's surface on which the integrand is represented, or is taken between limits, at each of which ew>'(< 2 â€” i) m+ i is zero. The equation is also satisfied by the expres- sion I e 2P ^ â€” 'r m ~ 1 dt where the integral is taken along a closed
path as before, or between limits at eacii of which e* p ^ )/->Â»-i
vanishes.
The following definite integral expressions for Bessel's functions are derivable from these fundamental forms.
J-to Â° n(-i)n(Â«-a (2) 7V ros * sin im ^
where the real part of m-\-\ if positive. Y m (p) + ^7n.e"" r 'sec mir.]â€ž(p)
where the real parts of m-\-\, are positive; if p is purely imaginary and positive the upper limit may be replaced by 00 .
Y m (p) â€” Jiri.e""" sec tmr.] m (p)
under the same restrictions as in the last case; if p is a negative
imaginary number, we may put Â» for the upper limit.
If $\rho$ is real and positive
$\mathrm {J} _{0}\left(\rho \right)={\frac {2}{pi}}\int _{0}^{\infty }\sin \left(\rho \cosh \phi \right)d\phi$
$\mathrm {Y} _{0}\left(\rho \right)={\frac {2}{pi}}\int _{0}^{\infty }\cos \left(\rho \cosh \phi \right)d\phi$
25. Bessel's Functions with Imaginary Argument. â€” The functions with purely imaginary argument are of such importance in connexion with certain differential equations of physics that a special notation
when m is an integer, and
KÂ»M = (aO-j^ICM =e~ im " 1 j Y m W) +|Â«rj m (,r) { . We find also
I-(0-
K m (r) =
I.3.5. ..(2W
(-l)"V
"^TTJo
e-r cosh cj> s j[ n h 2m <pd$
1.3.5. ..(2m-
-(-0-3.5... ssmr-/;^^^.. .
26. JTje Asymptotic Series for Bessel's Functions. â€” It may be shown, by means of definite integral expressions for the Bessel's functions, that
J-Cp)-\^!p Â«-(==+ J-p)+Q-(^4J-p) |
Yâ€ž(p) = ^Â« sec Â«, j P sin (^+J-p) -Q cos (^+J-p) \ where P and Q denote the series
n_. (4m*-i')Um*-3') 1. 2.(8,0*
â– ( 4 m 2 - 1 2 ) (4m 2 - 3 2 ) ( 4 m 8 - 5 2 ) ( 4 m* - 7') ^ i.2.3-4(8p) 4 ! " '
4 W 2 -I 2 ( 4OT 2-lÂ»)( 4w i- 3 Â«)( 4OT 2_.;Â»)
v i.8p i.2.3.(8p)Â» i "-"
These series for P, Q are divergent unless m is half an odd integer, but it can be shown that they may be used for calculating the values of the functions, as they have the property that if in the calculation we stop at any term, the error in the value of the function is less than the next term ; thus in using the series for calculation, we must stop at a term which is small. In such series the remainder after n terms has a minimum for some value of Â», and for greater values of n increases beyond all limits ; such series are called semi-convergent or asymptotic.
We have as particular cases of such series : â€”
"Vâ„¢"" (4-') irg-,'It(Jri-+--i
when m is an integer,
27. The Bessel's functions of degree half an odd integer are of special ​
importance in connexion with the differential equations of physics. The two equations
are reducible by means of the substitutions u = e~ k 'v, u = e lk 'v to the form v*Â»+Â»=o. If we suppose v to be a function of r only, this last differential equation takes the form
0,
so that v has the values
sin r/r, cos rjr; in order to obtain more general solutions of the equation v*Â»+d=o, we may operate on
sin rjr, cos rjr with the operator
Y fÂ±. Â± -Â±) In Va*' dy' dzj '
where Yâ€ž (x, y, z) is any spherical solid harmonic of degree n. The result of the operation may be at once obtained by taking Yâ€ž {x, y, z) for /â€ž(*, y, z) in the theorem (7'), we thus find as solutions, of V*v+f = o, the expressions
Y.(x, y. z)j^ By recurring to the definition of the function J m (r), we see that
sin r v , s d" cosy
-7â€”, 'Â»ft y< z -><Z( r 2)Â» r '
j*wVÂ£i'-
r 2 r
2.3 2-3-
4-5
-â– M
2 sin r
thus
r-ijl^
, /2 sin r
Using the relation between Bessel's functions whose orders differ by an integer, we have
Mr).(-2)^,^-(-.)VV' d " Sinr
<Z(r 2 )"
It may be shown at once that
that the function Kâ€ž(z) has no real zeros unless Â«=2fe + l where k is an integer, when it has one real negative zero; and that K n (z) has no purely imaginary zeros, and no zero whose real part is positive, other than those at infinity. When l>re>0, Kâ€ž(z) has no zeros other than those at infinity, when 2>n>l,it has one zero whose real part is negative, and when m+i>-n>m where m is ah integer, there are to zeros whose real parts are negative. When n is an integer, Kâ€ž(z) has n zeros with negative real parts.
29. Spheroidal Harmonics. â€” For potential problems in which the boundary is an ellipsoid of revolution, the co-ordinates to be used are r, 0, <t> where in the case of a prolate spheroid
the surfaces r = r , 0=0o, 4>=<t><s are confocal prolate spheroids, confocal hyperboloids of revolution, and planes passing through the axis of revolution. We may suppose r to range from 1 to 00 , 8 from o to ir, and <j> from o to 2ir, every point in space has then unique co-ordinates r, 8, <j>.
For oblate spheroids, the corresponding co-ordinates are r, given by
8, 4,
Â»+i
d n cos r
d{r*)Â» r
a second solution of 4>) denotes a surface
x = c^r z + i sin 8cos<t>, y = c\V 2 + l sin 8 sin <t>, z = cr cos 8,
where
0< r < w , o < < ff, o < tfr < 2jt ;
these may be obtained from those for the prolate spheroid by chang- ing c into â€” ic, and r into ir.
Taking the case of the prolate spheroid, Laplace's equation becomes
3V\ , r 2 -cos'0 a 2 v
d { ,â€ž ,aV) . 1 a / . â€žav\ .
m<t>.
dr S ' sin 8 08 V*" " 68/ " r (^-i)sinÂ»9 dtf and it will be found that the normal solutions are
pÂ»)p;T(cos0);
Q:(r))Q:(cos8)
For the space inside a bounding spheroid the appropriate normal
m are positive integers, and for the external space
forms are P"(r)Vâ„¢ (cos 8)^?*m<t>, where n,
is a second solution of Bessel's equation of order n-f-j; thus the differential equation y*v+v = o is satisfied by the expression
y.(*. y, 3) J -^tV
and by the corresponding expression with Bessel's equation instead of J^+i (r) ; if Sâ€ž(/j, harmonic of degree n, the expression
S.O., *)^ w l(r)
is a solution of the equation v 2 ti-{-v = o.
The Bessel's functions of degree half an odd integer are the only ones which are expressible in a closed form involving no trans- cendental functions other than circular functions. It will be observed that in this case the semi-convergent series for J m becomes a finite one as the expressions P, Q then break off after a finite number of terms.
28. The Zeros of Bessel's Functions. â€” The determination of the position of the zeros of the Bessel's functions, and the values of the argument at which they occur, have been investigated by Hurwitz {Math. Ann. vol. xxxiii.j, and more completely by H. M. Macdonald (Proc. Lond. Math. Soc. vols. xxix.,xxx.). It has been shown that the zeros of JÂ»(z)/z* are all real and associated with the singular point at infinity when n is real and > â€” 1 , and that all the real zeros of JÂ»(z)/z" when n is real and <â€” I, and not an integer, are associated with the essential singularity at infinity. When n is. a negative integer â€” to, Jr.(z)/z" has, in addition,, 2w real zeros co- incident at the origin. When n â€” â€” mâ€” v, m being a positive integer, and i>f>o, LCzWz" has a finite number 2tn of zeros which are not associated with the essential singularity. If n is real, and starts with any positive value, the zeros nearest the origin approach it as n diminishes, two of them reaching it when n= -i-,i, and t\vo more reach it whenever n passes through a negative integral value; these zeros then become complex for values of n not integral. The zeros of Jâ€ž(z)/z" are separated by those of JÂ»+i(z)/z n , one zero of the latter, and one only, lies between two consecutive zeros of Jn(z),/z". When n is real and >â€” I, all the zeros of Jn(z)/z" are givenby a formula due to Stokes; the m th positive zero in order of magnitude is given by
q:wpâ€ž-(cos0)^>*.
5Â£r
For the case of an oblate spheroid, P"('f), Q7("')i take the place
of K(r), QT(r).
30. Toroidal Functions. â€” For potential problems connected with the anchor-ring, the following co-ordinates are appropriate: If A, B are points at the extremities of a diameter of a fixed circle, and P is any point in the plane PAB which is perpendicular to the plane of the fixed circle, let P = log(AP/BP), 0=/_APB, and let <t> be the angle the plane APB makes with a fixed plane through the axis of the circle. Let be restricted to lie between â€” ir and ir, a discontinuity in its value arising as we pass through the circle, so that within the circumference is ir on the upper side of the circle, and â€” ir on the lower side ; 8 is zero in the plane of the circle outside the circumference; p may have any value between â€”00 and co . and <j> any value between o and 2ir. The position of a point is then uniquely represented by the co-ordinates p, 8, <t>, which ate the parameters of a system of tores with the fixed circle as limiting circle, a system of bowls with the fixed circle as common rim, and a system of planes through the axis of the tores. If x, y r z are the co-ordinates of a point referred to axes, two of which x, y are in the plane of the circle and the third along its axis, we find that
a sinh p . , _ ,0. sin
,sin<p, Z =
cosh pâ€” cos 8
4nÂ»â€” 1 4(4^â€” i)(28w 8 â€” 31)
â€” &C.,
I
sinh p dtf " can be shown
that this
a sinh p x ~ cosh p - cos COS *' y = cosh p - cos C
where a is the radius of the fixed circle. ' Laplace's equation reduces to
_a ( sinhp aV ) _a \ sinhp 3V dp\ P 2 dp ) + d9\ P 2 d8 )
when P denotes V(cosh pâ€” cos 0). It equation is satisfied by
1 1 u . PÂ»_i(cosh p) cos â€ž cos ,
V (cosh p-cos ^Q-^cosh p) sin "" sin m *'
the functions P"_ } (cosh p), Qr_|(cosh p) required for the potential problems, are associated Legendre's functions of degree n â€” |, half an odd integer, of integral order Â«, and of argument real and greater than unity; these are known as toroidal functions. For the space
external to a boundary tore the function Qâ„¢_ i (cosh p) must be
where (i = iir(2Â»+4Â»Â»-i). It has been, shown by Macdonald | used > al Â» d for the internal space P'^cosh p). ​
The following expressions may be given for the toroidal functions: â€”
P^j(coshp)
ir H(Â»â€” mâ€” j)J a (cosh p + sinh p cos .
cos m(j>
JTP^
I n(re+m â€” J) f , i , -i ,nâ€ž_i ,j,
= â€” jtH r~- (coshp -f sinhp cos 0)" 5 cos m$d<t>.
w ii\n~i) Jo
P^.(coshp)=2J^_
cosh n<t>
=d<t>.
0V2 cosh p â€” 2 cosh <Â£
Q^j(coshp) = (-1)" n (â€ž_|) j .. â– ( cosh P
â– â€” sinh p cosh w)"~3 cosh mwdw
^(-^TUm-mi-i) -nh " P f; (2 cosh ;Â°_ S f cos0) ^^.
The relations between functions for three consecutive values of the degree or the order are
2Â»t cosh pP ? ^,(cosh p) â€” (Â« â€” m + 1) Pâ„¢ , (cosh p) â€” (n-\-m â€” l)P^_i (cosh p) = o. P^?(cosh p) + 2(m + 1) coth pP^Hcosh p) â€” (n.â€” Â»Â»;â€” j)(Â» + Â»Â»4-5)P^_, (cosh p) = 0,
with relations identical in form for the functions Q^, (cosh p). The function Qâ€ž_;(cosh p) is expansible in the form
3^g2^W+npF(i, M + h -n+.i, Â«-Â«p),
which is useful for calculation of the function when p is not small. PÂ»_i(cosh p) can also be expressed in terms of tr* by a somewhat complicated, formula.
31. Ellipsoidal Harmonics. â€” In order to treat potential problems in which the boundary surface is an ellipsoid, Lame took as co- ordinates the parameters p, p, v of systems of confocal ellipsoids, hyperboloids of one sheet, and of two sheets; these co-ordinates are three roots of the equation
pTp-h^P-U
+ Â«
j = l, (k>h);
we thence find that
_ppv Vp'-^VM^-gVtf-.- 2 _Vp 2 -Â£ 2 Vfe 2 -p. 2 V'& 2 -y 2
where 00 >p 2 >h 2 , W- < p. 2 < h- , and Â£ 2 >f 2 >o.
We find from these values of *, y, z
(p 2 -^)(m 2 -, 2 )
and on applying the general transformation of Laplace's equation that equation becomes
(M 2 -^)-g|2 +(p'-^)y^ -Hp 2 -/* 2 )-^ =0, where Â£, if, f are defined by the formulae
t ff dÂ» r)=C* d Â»
<z*
of the parameters Â£, jj, f in terms of p, p, 1/, ^e find that the equation satisfied by E(p) becomes
(p 2 -^)(p 2 -* 2 )^+p(V-ft 2 -* 2 )^
+\(h 2 +k 2 )p-n(n + l)p 2 }E(p) =0,
and E(p.), E(i>) satisfy equations in p, v respectively of identically the same form ; this equation is known as Lamp's equation.
If n be taken to be a positive integer, it can be shown that it is possible in 2Â« + i ways so to determine p that the equation in E(p) is satisfied by an algebraical function of degree n, rational in p, V (p 2 â€” h 2 ), V(p 2 â€” k 2 ). The functions so determined are called Lamp's functions, and the 2Â» + I functions of degree n are of one of the four forms. -
K(p) = flo p" + a ip"^ + â– â€¢ â– > L(p) = Vp 2 -A 2 (oo' p"" 1 + o'lp"" 3 + ...), M(p) = V 7^F (o o"p"- 1 + a"i pÂ»- 3 +...), N(p) = yl7^k\^7^Â¥{<"p n ^+a["p"- i + . . .).
These are the four classes of Lame's functions qf degree, n ; of the functions K there are l+|Â», or i(n-\-l), according as n is even or odd ; of each of the functions L, M, there are in, or i(n â€” 1), and of the functions N, there are -|Â», or i(n + i).
The normal forms of. solution of Laplace's equation, applicable to the space inside the ellipsoid, are the 2m + i products E (p) Kip.) E(v). It can be shown that the 2ra + i values of p a,re real and unequal.
It can be shown that, subject to certain restrictions, a function of p and v, arbitrarily given over the surface of the' ellipsoid p = pi, can be expressed as the sum of products of Lame's functions of p and c, in the form
00 2M+I
2 2 C jEjl(p.)E;(x); i s-i
the potential function for the space inside the ellipsoid, which has the arbitrarily given value over the surface of the ellipsoid, is consequently
22
, e^(p)e;( m )eâ€žw EJKpO
Wtf-vVF-?'
which are equivalent to
p^kdn{kÂ£, ki),p~=kdn(K~kr), ki), v-ksn(k(, hi), where ki 2 , k\ n denote the quantities iâ€”h 2 jk', VjV and K denotes the complete elliptic integral
Vi-fe ! sinV
It can now be shown that Laplace's equation is satisfied by the product E(p)E(p,)E(>'), where E(o) satisfies the differential equation
^^-MÂ«+l)p 2 -(ft 2 + fe 2 )Â£!E(p)=o ;
and E(jj), E(v) satisfy the equations
^|^+[Â»(n4 i) M t-.0(AÂ»+tf)]EGi)Â«O,__
,^|^~ [n{n+ iy -p{W+V)]E{S) =0, where n and p are arbitrary constants. 'On substituting the values
It can be shown that a second solution of Lame's equation it, Fâ€ž(p) Where
Fâ€ž( P ) = (2B+I
)En{ P )j' i
dp
plEâ€ž(p)l 2 Vp 2 -^-Vp 2 -fe 2 '
this function Fâ€ž(p) vanishes'at infinity as p~" -1 , and is therefore adapted tq the space outside the bounding ellipsoid. The external potential which has at the surface p = pi, the value
22^ E *) E Â»Â« is S2 c "
'FÂ« P i)
El(p)El(v).
32. History and Literature. â€” The first investigator in the subject was Legendre, who introduced the functions known by his name, and at present also called zonal surface harmonics; he applied them to the determination bf the attractions Of solids of revolution. Legendre's investigations are contained in a memoir Of the Paris Academy, Sur V attraction des spheroides, published in 1785, and in a memoir published by the Academy in 1787, Recherches sur la figure des planktes; his investigations are collected in his Exercices, and in his Traite des functions elliptiques. The potential function was introduced by Laplace, who also first obtained the equation' which tears his name; he applied spherical surface harmonics to the
j determination of the potential of a nearly spherical solid, in his memoir, ThSorie des attractions des spheroides el de la figure des planUes, published by the Paris Academy in ^785. Laplace was the first to consider the functions of two angles, which Junctions have consequently been known as Laplace's functions; his investi-
' gationson these functions are given in the Mecanique celeste, tome ii.
Phil. Trans. Roy. Soc., 1862), and also independently by Clebsch (see his paper, “Ueber die Reflexion an einer Kugelfläche,” Crelle's Journal, vol. lxi., 1863). The general theory of spherical harmonics of unrestricted degree, order and argument has been treated by Hobson (Phil. Trans., 1896) ; see also a paper by Barnes in the Quar. Journ. Math. 39, p. 97. The functions which bear the name of Bessel were first introduced by Fourier in his investigations on the conduction of heat (see his Théorie analytique de la chaleur, 1822) ; they were employed by Bessel in the theory of planetary motion (see the Abhandlungen of the Berlin Academy, 1824). The functions which are now known as Bessel's functions of degree half an odd integer were employed by Poisson in the theory of the conduction of heat in a solid spherical body (see the Journ. de école polyt., 1823, cah. 19). The toroidal functions were introduced by C. Neumann (Theorie der Elektricitäts- und Wärmevertheilung in einem Ringe, Halle, 1864), and independently by Hicks (Phil. Trans. Roy. Soc., 1881). The ellipsoidal harmonics were first investigated by Lamé in connection with the stationary motion of heat in an ellipsoidal body (see Liouville's Journal, 1839, pt. iv. The external ellipsoidal harmonics were introduced by Liouville and Heine (see Liouville's Journal, vol. x., and Crelle's Journal, vol. xxix.). The ellipsoidal harmonics have been considered as expressed in Cartesian co-ordinates by Green (see Collected Works), by Ferrers (see his treatise), and by W. D. Niven (Phil. Trans. Roy. Soc., 1892). A method of representing ellipsoidal harmonics in a form adapted for actual use in certain physical problems has been developed by G. H. Darwin (Phil. Trans., vol. 197).
The following treatises may be consulted: Heine, Theorie der Kugelfunctionen (2nd ed., 1878, vol. i.; 1881, vol. ii.); this treatise gives much information as to the history and literature of the subject; Ferrers, Spherical Harmonics (Cambridge, 1881); Todhunter, The Functions of Laplace, Lamé and Bessel (Cambridge, 1875); Thomson and Tait, Natural Philosophy (1879), App. B; Haentzschel, Reduction der Potentialgleichung auf gewöhnliche Differentialgleichungen (Berlin, 1893); F. Neumann, Beiträge zur Theorie der Kugelfunctionen (Leipzig, 1878); C. Neumann, Theorie der Bessel'schen Functionen (Leipzig, 1867); Ueber die nach Kreis-, Kugel- und Cylinder-functionen fortschreitenden Entwickelungen (Leipzig, 1881); Lommel, Studien über die Bessel'schen Functionen (Leipzig, 1868); Mathieu, Cours de physique mathématique (Paris, 1873); Pockels, Ueber die partielle Differentialgleichung Δu + k$^2$u＝0 (Berlin, 1891); Bôcher, Ueber die Reihenentwickelungen der Potentialtheorie (Leipzig, 1894); Gray and Mathews, Treatise on Bessel's Functions; Dini, Serie di Fourier e altre rappresentazione . . . (Pisa, 1880) ; Graf and Gubler, Einleitung in die Theorie der Bessel’schen Functionen (Berne, 1898) ; Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904) ; Whittaker, A Course of Modern Analysis (Cambridge, 1902); H. Weber, Die partiellen Differentialgleichungen der Physik (Bremen, 1900) ; W. E. Byerly, Fourier's Series and Spherical, Cylindrical and Ellipsoidal Harmonics (Boston, 1893). (E. W. H.)