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Online Encyclopedia
Originally appearing in Volume V25, Page 657 of the 1911 Encyclopedia Britannica.
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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 a2V 32V a2V dLV axe + art a,2 = o, 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 (x2+y2+z2)L 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; when the term spherical harmonic is used in the narrower sense, 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, Lames 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 al =k0'u, which is fundamental in the theory of conduction of heat and electricity, also to the equation ate = kV1u, 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. i. 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 ow ow OW axe +aye + aZa =o, or v2V =o, (I) is termed a solid spherical harmonic, or simply a spherical harmonic of degree n. The degree is 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 cos 4), r sin 0 sin 0, r cos B in polar co-ordinates, a solid spherical harmonic takes the form r"fn(B, ; the factor f(0, di) is called a surface harmonic of degree n. If Vn denote a spherical harmonic of degree n, it may be shown by differentiation that v2(r'"Vn) =m(2n + m + i)r"-2Vn, and thus as a particular case that v2(rr2"-,Vn) =o; we have thus the fundamental theorem that from any spherical harmonic V. of degree n, another of degree -n- i may be derived by dividing Vn by r2"+i 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 (I), the function rI (r' rr2' r~J 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 1(n +1 )(n +2) coefficients; if the expression be substituted in (i), we have on equating the coefficients separately to zero in(n - I) relations to be satisfied; the most general spherical harmonic of the prescribed type therefore contains 4(n+I)(n+2) -y-n(n-I), or 2n+I independent constants. There exist, there-fore, 2n+1 independent ordinary harmonics of degree n; and corresponding to each of these there is a negative harmonic of degree-n-i obtained by dividing by r2"+'. The three independent harmonics of degree i are x, y, z; the five of degree 2 are y2 z2, z2-x', yz, zx, xy. Every harmonic of degree n is a linear function of 2n+i 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 (I), if a, b, c are constants which satisfy the condition a2+b2-{-c2=o; in particular the equation is satisfied by (z+ix cos a+Ly sin a)". Taking n to be a positive integer, we proceed to expand this expression in a series of cosines and sines of multiples of a; each term will then satisfy (i) separately. , Denoting e'n by k, and y-fix by t, we have (z+Lx cos a+Ly sin a)" = (z+akt+zkt ak7.2 t)" which may be written as (2kt)-n1(z+kt)2-r2)} On expansion by Taylor's theorem this becomes 2n (2kt)-n --sy da^(z2-r2)", the differentiation applying to z only as it occurs explicitly; the terms involving cos ma, sin ma in this expansion are I S (y-l-ex)man+m (2_ 2)n..~(y+ix)-'" (2-se) m 2^ cos ma (n+m)! azn+m Z r (n±m)! az"-m z n+m n-m sir, sin ma (n-FL m (n—m _m a m}, azn+m+m (Z2—r2) ( "— n-)1 az^`TM (ZZ _r2) m where m= i , 2, ... n; and the term independent of a is I as 2_2)n 2"n! 0Z"(2 r On writing (y+Lx)m=L"rm(cos m¢-L sin m4')2 sin m0, (y+ix)-TM= ti 'r-"(cos m4)+L sin mip) sin-'"O and observing that in the expansion of (z+Lx cos a+Ly sin a)'" the expressions cos ma, sin ma can only occur in the combination cos rn(¢-a), we see that the relation timm sin m,anazn++m m(Z2_r2 sin --y-t'8 )! aaz""-`'" rn); ) _om (n (z2 r 2" must hold identically, and thus that the terms in the expansion reduce to I Lm an+m (n+m)!. 22n-irm cos ma cos m4i sin Bazn+,,,(z2-r2)n ry+m (n+ m)! 2,-;:-_.,r"' sin ma sin m4i sin "Bazn+, (z2-r?)". We thus see that the spherical harmonics of degree n are of the form n+m r"sin mip sin med—n+m(t~2-I)" where i denotes cos a; by giving m the values o, t, 2 ...n we thus have the 2n+1 functions required. On carrying out the differentiations we see that the required functions are of the form A[(x+Ly)m-(x-Ly)m] ] z"--'" (n 2 )Zn-m— I)zm.i(x2+y2+Z2) +(n-In)(r2-m-I)(II—m—2)(n-m-3)z;,m-4(x2+y2+z2/ (2) . . . . 2.4; 2n— I :2n-3 where m=o, I , 2, 3, ... n. 3. Zonal, Tesseral and Sectorial Harmonics.—Of the system of 2n+ i harmonics of degree n, only one is symmetrical about the ;: axis; this is n r" n ! d 2 n. dn(/,L2 —I)n; r,, Pn(a)=- —Gi"—I)", we observe that P,,(u) has it zeros all lying between i, consequently the locus of points on a sphere r=a, for which Pn(u) vanishes is n circles all parallel to the meridian plane: these circles divide the sphere into zones, thus Pn(p.) is called the zonal surface harmonic of degree n, and r"P,(ti), r-"-'P,(µ) are the solid zonal harmonics of degrees it and -n---I. The locus of points on a +m sphere for which sn m4.sin m0 (te2-i)" 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 eiaaspa, 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 (I —µ2)i- dµ "(µ) by P, (ii), the tesseral surface harmonics are s n m4). Pry" (cos e), where m = i, 2, ... n - I, and the sectorial harmonics are 2; n4). P:(cos B). The functions Pe(i ),, P' Cu) denote the expressions writing _ ((2n) ! ' —n(n—I)µ„-!+n(n—I)(n—2)(n—3)µn_... (3) 2"n. R 2.27—I 2 .4 . 2n — I . 2n -3 P'' (A) n){(I —u2)ISm mn—'" 2''n! (n—m) (n—?n) (n—1n—I)tin_,,, t+... 2.2n—I Pn(ti) = (22,,',,),!(I —tit) In. Every ordinary harmonic of degree n is expressible as a linear function of the system of 2n-1-I zonal, tesseral and sectorial harmonics of degree n; thus the general form of the surface harmonic is n aoP,(A) +Z(am cos mcti +bm sin m0)P„0u). (5) ~n=l In the present notation we have (z+ix cos a+Iysin a)"=r" Pn(ti)+2EIm(n+m)! "(µ)cosm(,p —a) if we put a= o, we thus have n (cos 0+i sin 0 cos iy)"=P,(cos 0)+2Eim(n+m)IP" (cos 0) cos m¢, m from this we obtain expressions for Pn(cos 0), P" (cos 0) as definite integrals Pn(cos 0) =! f o (cos 0+i sin 0 cos ¢)"d¢ i.'"(n+m).P" (cos 0) =A f o (cos 0+o sin 0 cos 0)"cos m~d4,. 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 a a a f x ay az) denote any rational integral operator, a a a lv f ax' ay azJ is a solution of 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 I/r, which satisfies Laplace's equation. The operations may be conveniently carried out by means of the following diffe entiation theorem. (See papers by Hobson, in the Messenger of Mathematics, xxiii. 115, and Proc. Lond. Math. Soc. vol. xxiv.) r a a s I (2n)! I _ r2v2 f"\ax ay' az/ r°(—I)"2 . '4'4-1) 1 2.2n—i +2.¢.2n4I.2n—3—. fn(x, y, z) (7) which is a particular case of the more general theorem (a a a 2n-2 d"-IF 2 f ax' ay' az) F(r) d(r2)n+ I r d(r2)1V+ .. . +'7 d(r2)n_,d2a+... fn(x, y, z) (71), where fn(x, y, z) is a rational integral homogeneous function of degree n. The harmonic of positive degree n corresponding to that of degree —n — in the expression (7) is I _ r2 2.211—1+2.4.2n y9II.42n—3—.. , tfn(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, tessera( 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 1, m, n drawn from the origin, 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.x+m.y+n,z)/r of the angle between the radius vector r to a point (x, y, z) and the axis (l., ni), will be denoted by Iii; the cosine of the angle between two axes is,my+n,ny, which will be denoted by tier. The operation lea +m`ay+n` a performed upon any function of x, y, z, is spoken of as differentiation with respect to the axis (l., m,, ne), and is denoted by a/ah,. The potential function Vo=eo/r is defined to be the potential due to a singular point of degree zero at the origin ; eo is called the strength of the singular point. Let a singular point of degree zero, and strength eo, be on an axis hi, at a distance ao from the origin, and also suppose that the origin is a singular point of strength-eo; let eo be indefinitely increased, and ao indefinitely diminished, but so that the product eoao is finite and equal to eo; the origin is then said to be a singular point of the first degree, of strength ei, the axis being hi. Such a singular point is frequently called a doublet. In a similar manner, by placing two singular points of degree, unity and strength, el, —el, at a distance al along an axis h2, and at the origin respectively, when el is indefinitely increased, and al diminished so that is finite and =e2, we obtain a singular point of degree 2, strength e2 at the origin, the axes being hi, h2. Proceeding in this manner we arrive at the conception of a singular point of any degree n, of strength en at the origin, the singular point having any n given axes hi, h2,. .. hn. If en_I y.n_i (x, y, z) is the potential due to a singular point at the origin, of degree n—1, and strength en_i, with axes hi, the potential of a singular point of degree n, the new axis of which is hn, is the limit of en_1 QSn-i (x —lna, y—mna, z—nna) —en_i (x, y, z) ; when La=o, Len_I=o^, Laen_i=en; this limit is --en (1 ,, it+mna ~y I+nna a2 i) , or —enahn~n_I• Since ¢o=I/r, we see that the potential V, due to a singular point at the origin of strength en, and axes hi, h2, . . .h,, is given by Vn= (—t)"e"ahiahan..ahnr (8) 6. Expression for a Harmonic with given Poles.—The result of performing the operations in (8) is that V. is of the form n !e'' i, where Y. 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 is 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"r", Ynr -1. Any spherical harmonic is completely specified by means of its poles. In order to express Y. in terms of the positions of its poles, we apply the theorem (7) to the evaluation of V. in (8). On putting r=n fn(x, y, z) =II (l,x+mry+nrz), we have // r=I Y"= (2n!) I { I— r2,v2 + r4v4 ..~ X 2"n!n! f" \ 2.2n—I 2.4.2n—I .27—3' n II (l,x+mry+nrz). - By Z(u'X"-2') we shall denote the sum of the products of s of the quantities ti, and n—2s of the quantities X; in any term each suffix is to occur once, and once only, every possible order being taken. We find II (lx+my+nz) =E(X")r", 02II(lx+my+nz) =2Z(u1x"-2)rn-2, and generally o2i II (lx+my+nz) =2—ni ! (umxn-2m)rn~m . thus we obtain the following expression for Yn, the surface harmonic which has given poles h2,; yn=rn+1(—I)" a" . n! ahiah2. .ahn r = S (—I)m "(2n' 2m)! I~(~n-2mtim) 2"-'''n (n—m) m=0 where S denotes a summation with respect to m from m =o to m=in, or %(n—I), 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 de,ioted by A, B, C..., we obtain from (9) the following expressions for the harmonics of the first four degrees: (4) } (6) (9) YI = cos PA, Y2 =1(3 cos PA cos PB —cos AB), Ya ='-x (15 cos PA cos PB cos PC—cos PA cos BC—cos PB cos CA --cos PC cos AB), Y4 = 1(35 cos PA cos PB cos PC cos PD -5E cos PA cos PB cos CD + E cos AB cos CD), 7. Poles of Zonal, Tesseral and Sectorial Harmonics.—Let the is axes of the harmonic coincide with the axis of z, we have then by (8) the harmonic (— I) nrw+1 an t n! az" r' 652 applying the theorem (7) to evaluate this expression, we have (-1)"r"+1 a" 1 (2n)! I S r2v2 ,+, r4V4 )l I - - t Z n! aza r 2nn!n! ' 1 2 . 211- I 2 . 4.271- I . 292-3 (2n)! n(n-I),, , znn!n! (i 2.271 - I the expression on the right side is P"(u), the zonal surface harmonic; we have therefore (-I)nrn+i an I Pn(p) = n! g n 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 x/m, the direction cosines of one of them being cos a sin a, o. We have m-r r,r a r,r a - I `(a}m) a II cos (am.— ) + sin (a+ ) II 3 e (--1—) ax m ay) 2" ax ay +e ,(a m) (ax+lay/ Let i; =x+1y, n=x—ty, the above product becomes j o f m a m a't which is equal to (m_1)_ e 2Il { gmia (.P_) m—e m,a ( aan) ;when a=o, 2m' this becomes e(m-r)`2 (aim —=)' (a) m ( a El (.an eon-I)`2 L (a )m+(-x)m(±)m From (7), we find a"-" a a "'I (271 ! I r2L12 + . . 2n-m(x tty az"-'n (ax lay) r 2nn! r?n+l I _2 . 2n - I "(zn ! 1 =(-1) (cos mO = 1 sin m¢) sin'nB ~ cosn'n`B -(n-m)(n-971-I)cos" m 28+... 2.211-I hence I Oz"-z' (ax $ LI) mr = (- 1)" nrn+;n''(cos m4 f 1 sin m¢) P7 (cos 0) as we see on referring to (4) ; we thus obtain the formulae a n "' t (a)'+ ( 0 ) l ' r = (-I)n2n-iy +>cos m4,. PT (cos 0) n( ) (IO an-m rt \at) m— (a) m s (-1)?~misin m~.Pn (cos e) 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 ;r/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,,(x, y, z) can begenerated by means of an operator fn (ax' a' a) acting upon A, the function fn being so chosen that 71(271) r-p 2 Y,(x,y,z)=(^ 2n7l! 3I—2.2n—I+..: Sfn(x,y,z); this relation shows that if an expression of the form (x2+y2+2)fn-2(x, y, z) is added to fn(x, y, z), the harmonic Yn(x, y, z) is, unaltered; thus if Y,, be regarded as given, fn(x, y, z) =0, is not uniquely deter-mined, but) as an indefinite number of values differing by multiples of x2+y2+z2. In order to determine the poles of a given harmonic, fn must be so chosen that it is resolvable into linear facto's; 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,,(x, y, z) =0, x2+y2+z2 =0, the equation fn(x, y, z) is also satisfied ; the problem of determining the poles is therefore equivalent to the algebraical one of reducing Y. to the product of linear factors by means of the relation x2+y2+z2 =0, between the variables. Suppose m Y,,(x, y, z) =AII(1,x+m,y+n.z)+(x2-+2+z2)V,,-2(x, y, z),we see that the plane l,x+m,y+n.z =0 passes through two of the an generating lines of the imaginary cone x2+y2+z =0r in which that cone is intersected by the cone Yn(x, y, z) =0. Thus a pole (l„ m,, n,) is the pole with respect to the cone x2+y2+z2=0, of a plane passing through two of the generating lines; the number of systems of poles Is therefore n(2n-I), the number of ways of taking the 211 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„=0, x2+y2+z2=0. Suppose x qq z q a1+1131 a2+1132gq a3+1133 gives one generating line, then the conjugate one is given by x qq y z al - lhil -a2-02- a3 - lr ' and the corresponding factor lx+my+nz is xII qq 1 qq al Tll/3~i a2 +1Q2 a3+z 1133 al--1f31 a2-i/32 a3-1133 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 xx'+yy'+zz' or P,,(cos 0 cos' e'+ sin B sin 0' cos 4' -0') ; 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', 2'), and must thus be capable of being expressed in the form n aoP,, (cos 0)Pn(cos 0')+1amPn (cos 0) P',','' (cos 0') cos m(¢-4,'), I and it only remains to determine the co-efficients ¢o, al, To find this expression, we transform (x'x+y'y+z'z)n, where x, y, z satisfy the condition x2+y2+z2=0; writing l;=x+ty, n=x-1y, t;'=x'+1y', n'=x'-1y', we have which equals n I ,a ,b ab ,b ,a b'a (zz') +E~a!b!(n—a—b)! (n > 712-'-6 ' n (zz')n-a-6 the summation being taken for all values of a and b, such that a+b.n, a>b; the values a=0, b=0 corresponding to the term (zz')n. Using the relation to = -z2, this becomes (xx' +yy' +zz') n = (zz')"+FZ( a+66 I I nl t(t'~)6z'n-a-6 2 a.b.(n-a-b) { (n's)a-6+ (0'71) a-O) zn-a+6-, putting a-b=m, the coefficient of on the right side is 6 71-26 2m+2b b!(m+b)!(n—m—2b)! n) n ' from b=0 to b=I(n-m), or I(n-m-I), according as n-m is even or odd. This coefficient is equal to n! x, 1 , m Z,"-m- `n-m) `n-m - I)Z,n-m-2 '2 12 2"`m!(n—m)!( — y) 2.2m+2 +y ) L(n-m)(n-nt-I)(n-m-2)(n-m_3)+y'2 2 ; - -' 2.4.2m+2.2m+3 in order to evaluate this coefficient, put z=1, x' = 1 cos. a, y,=, in a, then this coefficient is that of (1 cos a+sin a)'", or of 11ne-"n"a in the expansion of (z'+1x' cos a+1y' sin a)" in powers of e 'a and eL', this has been already found, thus the coefficient is 1 , (n+m)!e ""¢ Pm (cos 0').r'n. Similarly the coefficient of is n! (n +m) Ie-{- m~b'P,, (cos 0')r'"; hence we have 4.(xx'+yy'+zz')"=z"P„(cos 9')+n!MP' (cos 0') {cos m4'(tm++,m) (xx'+yy'+2z')n= (2n't+It'n+zz')" In this result, change x, y, z into +1 sin mos'(n'"--E"')}(n+m)I. a a a 0x' : ay' 8z' and let each side operate on I/r, then in virtue of (to), we have (rr')"Pn (xx'+~Y'+zz') = P,, (cos 0 cos 0' +sin 0 sin 0' cos 4, -¢') =P,,(cos\0)Pn(cos0')+2Z(n+m)1P''(cos0)Pn(cos0')cosm(4-(11) which is known as the addition theorem for the function P,,. It has incidentally been proved that P cos 0) _ (n+m)! ,sin"0 cosh-'"0 (n-m)(n-m-' — cos"-'"20 sin20+ (I2) 2.2m+2 )) which is an expression for Pn (cos 0) alternative to (4). to. Legendre's Co ficients.—The reciprocal of the distance of a point (r, 0, 0) from a point on the z axis distant r' from the origin is (r2-2rr' µ+r'2)-I which satisfies Laplace's equation, µ denoting cos 0. Writing this expression in the forms r r2 (-I ) r' r'2 -I 1—2r,µ+Y'2 , r I—2yµ+y2 ' 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 h2(2µ-h)2<1 (I -2hµ+h') ' =1+h(2u-h)+~ :44h2(2µ-h)2+.. . +t 2.4.. 2ri 1h"(212—h)"+... and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of hn is then found to be I 1.2.3...n i µ„,_11(n _. 2.2n— 1 Iµn~+nt2t.¢2nn I?2nn-3 ),"—g... this is the expression we have already denoted by P,(µ) ; thus (1-2hµ+h2)-2=Po(n)+APl(µ)+...+hnPn(µ)+..., (13) the function P,(p) may thus be defined as the coefficient of hn 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. Ir•may be shown that the expansion is valid for all real and complex values of h and µ, such that mod. It is less than the smaller of the two numbers mod. (µ*dµ2-I). We now see that 2 -' is expressible in the form co E yn (~~ r'n+IPn V') 0 when r < r', or r'n rn+I Pn 0 when r' < r; it follows that the two expressions r"Pr()s), r-"-IPn(µ) are solutions of Laplace's equation. The values of the first few Legendre's coefficients are Po(µ)=1, PI(µ)=12, P2(µ)=1(31.42-1), Ps (A) =1(5A' - 3A) Pa(µ) =g(35µ4-3oµ2+3), Pa(µ) L(63µ5-70µa+15,) I 1 Pa(µ)=Iti(231µa-313124+1O5µ2-5), P7(A)=i6(429µ'-693µs +315,' - 3512) P„ (1) =1, Pn(—1) = (-On P,,(0)=0, or (—1)40.3.5...n—I 2.4...n according as n is odd or even; these values may be at once obtained from the expansion (13), by puttingµ =1, o, -1. ii. Additional Expressions for Legendre's Coefficients.—The expression (3) for Pn(s) may be written in the form (2n)! n n I—nn I 1 P. (µ) = 2rt 71, F ( 2 2 , 2 — n, 122 with the usual notation for hypergeometric series. On writing this series in the reverse order n! / 11 n+1, 1 Pn(p) =(—I}In2n (In) 1 (In) F 1 -2, 2 ' 2' 122) z z or n_I n! F n—I n 1 2 (—I) 2 2n-1n—I,nI (— 2 2+ , 3 ) 2 2 according as n is even or odd. From the identity (1-2h cos 6+10)-1 = (1 —heia)'f(1-he '°) i, it can be shown that P (cos B) = I'2 " 4 .6 2 zn I cos n0+1.2nn 1cos (n-2)0 + i.3.n(n-1) . 2 . (2n I) (2n - 3)cos (n-4)0+ (14) I By (13), or by the formula dn P,(,) = 2nrl i dµ" (1,2 1) n which is known as Rodrigue's formula, we may prove that Pn(cos 0) I—(n+I)nS1t120+(n+2)(n+I)n(n—1)s1n40 I2 2 I2,22 =F (7+I, —n, I, sin2l) Also that ))r P,,(cos 0)=cos'"~ ( S I—I2 tan22+n2In.2z )2. tan —. . .`` (( =cos2tt-F (—n, —n, 1, —tan22 1 . (16) By means of the identity ( (I -2hµ+h2)-I = (I +h2( —µ2g ( (1-hµ) it may be shown that P„(cos0) =cos"o -n( 2- I)tan20+n(n-I)Zn.42)(n-3) tan 40-... =cos"pF(-2n, I-2n, 1, —tan2o). (17) Laplace's definite integral expression (6) may be transformed into the expression (µ+-122-1 cos $)(µ— yµ2— cos jf) =I. Two definite integral expressions for Pn(µ) given by Dirichlet have been put by Mehler into the forms Pn(ccs 0 __ 2 e_cos (n+2)0 d __ 2 n sin (n+3)~ d o112 COS 0—2 cos 0 7r,J 0 s 2 cos 0—2 cos ~ When n is large, and 0 is not nearly equal to o or to r, an approximate value of Pn(cos0) is {2/nir sin 01I sin {(n+*)0+17r}. 12. Relations between successive Legendre's Coefficients and their Derivatives.—If (I -2hµ+h2)-I be denoted by is, we find (I -2hµ+h2) ah+(h -µ)u =0; on substituting Zh"Pn for u, and equating to zero the coefficient of h", we obtain the relation nPn-(2n- 1),Pn-1+(n—I)Pn-2=0. From Laplace's definite integral, or otherwise, we find (,2 - 1) d--n = n (12Pn - P,_l) = - (n + 1) (,sP" — Pn+1) We may also show that dPs dPn_I µ dµ — dµ =nPn dPn dPn+1 (n+I)Ps=-µdµ+ d12 (2n+I)PP=d +1+I—d (2n+1), d (n+I)d i+nd (2n+I) ('2— 1) =n(n+1) (P.+1- P.-0 dP,, _ =(2n-1)Pn_1+(2n-5)Pn-a+(2n-9)Pn-a+. the last term being 3P1 or Po according as n is even or odd. 13. Integral ) rop,erties of Legendre's Coefficients.—It may be shown that if P,+a7s) be multiplied by any one of the numbers 1, µ, 122, ..-. and the product be integrated between the limits 1,- I with respect to As, the result is zero, thus f 1 µkPn(µ)dµ=0, a=0, I, 2, ... n-I. (18) To prove this theorem we have /.. 1 I do(~~ I,kPnV*)d12=2nn7 -Iµkdun(,2—I)nd12, We find also (15) 1 in d4, 0(12—dµ2—1 COS 4)n+1' by means of the relation on integrating the expression k times by parts, and remembering that. (p2 — On and its first n — I derivatives all vanish when p = =1, the theorem is established. This theorem derives additional importance from the fact that it may be shown that AP,(µ) is the only rational integral function of degree n which has this property; from this arises the importance of the functions P. in the theory of quadratures. The theorem which lies at the root of the applicability of the functions P„ to potential problems is that if n and n' are unequal integers Cr P.,(A)Pn(p)dp=0, (19) r 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' 1. If we choose the constant S to be 1 ' 2 . 3 . n the second 3.5...211+1' solution may be denoted by Qn(µ), and is called the Legendre's function of the second kind, thus Qn(tA)= 1.2.3...1 1 +(n+I)(11+2) 1 .+... 3.5...211+I µn+l+l 2.211+3 µn+3 1.2.3...15 -.1 F (n+1 n+2 211+3 (28) 3.5...211+I +, 2 2 , 2 , µ2 This function Qn(µ), thus defined for mod µ > I, is of considerable importance in the potential theory. When mod µ < i, we may in a similar manner obtain two series in ascending powers of µ, one of which represents P,,(µ), and a certain linear function of the two series represents the analytical continuation of Q„(µ) as defined above. The complete primitive of Legendre's equation is u=APn(s)+BQ,(µ). 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 PnGA)—f µ(µ2—I){Ps(tA)}2, the lower limit being arbitrary. where h,hi...h,, are the axes of Ys. Two harmonics of the same degree are said to be conjugate, when the surface integral of their product vanishes; if Yn, Z. are two such harmonics, the addition of conjugacy is an+t +(2n+I)r"-IYn(µ, 4')+.. . which is the required potential at the external point (r, 8, 4'). 17. The Normal Solutions of Laplace's Equation in Polars.—If h1, h2, ha 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 ds2 = II2dh; + 1dh= + H dh21, where H1, H2, H3 are HARMONICS If, in Legendre's equation, we differentiate in times, we find 1 (I—t~E)dµ11+z-2(m+1)pd/2—R+(n—m)(n±m±I) ,~,=o; 656 From this form it can be shown that Q, (/A) =2 P,,(/A) log - -W,,_i(/A), where W,,_.l(/A) is a rational integral function of degree n-i in /A; it can be shown that this form is in agreement with the definition of Q,(p) by series, for the case mod µ>I. In case mod AI, but differs from it by an imaginary multiple of Pn(A). It will be observed that Q,(I), Q,,(—i) are infinite, and Qs(x) =o. The function \V,, i(/A) has been expressed by Christoffel in the form 211—I 2n—5 211—9 I .n Pn_1(!A) +3 1A — 1 Pn-a (µ) +5.n— 2 Pn_a (µ) + ... , and it can also be expressed in the form ' nPo(A)Pn-1(/)+n? IPI(/A)Pn—2(/A)+. +Pn_1(/A)Po(IA). It can easily be shown that the formula (28) is equivalent to T oo / (d n+1 Qn(FA)=2'n!J µJ ... f V*2-µ) I)n+1 which is analogous to Rodrigue's expression for P,,(,4). Another expression of a similar character is 2nn! do r ° dp. Q11(µ)_(—I)" zn)! die (µ2—I)nJ µ(/A'2—1)nt1 It can be shown that under the condition mod {u—11(u22—i)J >mod {µ—/(µ2—I)1, the function 1J(µ—u) can be expanded in the form E(2n+I)P,,(u)Q,,(u); this expansion is connected with the definite integral formula for Qs(A) which was used by F. Neumann as a definition of the function (MA), this is 1 P,,(u) Q=(/-) f I udu, which holds for all values of /A which are not real and between = From Neumann's integral can be deduced the formula . Qn(/A) = f i {1A+1l (lA2— I).cosh ,,tJn+1, which holds for all values ofµ which are not real and between 1, provided the sign of (A2—1) is properly chosen; when µ is real and greater than 1, J (µ2—I) has its positive value. By means of the substitution. IA +Al (IA2—I).cosh iGJ{µ—~ (µ2—.cosh xi=1, the above integral becomes Q, ,(A) =f xa°{p —,J (/A2 — I) . cosh xlndx, where xo = 2logeµ: This formula gives a simple means of calculating Q,(µ) for small values of n; thus Q.(/A) = fox°dx = 2log,.— . Ql(A)=Axe—NI (A2-I).sinh xo=p.. 210gµ+i—I. Neumann's integral affords a means of establishing a between successive Q functions, thus n+— (22n — I )4Qn-1 +(n- 1)Q-2 l fPn(u)+(n—I)Pn-2(u)—(211-I) 2 -1 p—u µP _1(u)du =—2 If 1(2n—I)P,, i(11)=0. Again, it may similarly be proved that dQn+l dQn _(211+1)Qn 19. Legendre Associated Functions.—Returning to the equation n (26) satisfied by uT the factor in the normal forms r yn_1 ni . un , we shall consider the case in which n, m are positive integers, and n'a m. Let u=(/A2—I)'A'nv, then it will be found that v satisfies the equation (I —µ2)--2(m+1)µdu+(n—m)(n+m+I) follows that v=d =d--e— u-^hence u' =GP—01147f. The complete solution of (26) is therefore n=(/A2—I)im Aa d n(,A)+Bd u ? when µ is real and lies between i, the two functions (I —µ2)+md 11(/4) (I_ µ2)IlmdmQ (µ) are called Legendre's associated functions of degree n, and order m, of the first and second kinds respectively. When /4 is not real and between. i, the same names are given to the two functions (lA2—kmdPn(FA), (k2—I) mdmQ,(p)d in either case the functions may be denoted by P7 (A), Q','`(p.). It can be shown that, whenµ is real and between I f— I)m I+µ Ic'n do Pn(u)=211(11—m)!~IdµnJ(A—I)n+m(!A+I)n—'1 _ I I —µ\) im dn' n—m/ n+m 211(11—ni)! \I+pJ dµnl(K—I) ./2+I) ). In the same case, we find Pn+2(cos 6)—2(112+I) cot 0 Pm11(cos 0) +(n—m)(n+m+I)P,, (cos 0) =o, (n—m+2)1'„+2(cos 0) — (2n+3)µI'01 l(cos 0) +(n+m+1)Pn (cos 0) =o. 20. Bessel's Functions.—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 z, p, ¢, the cylindrical co-ordinates; in that case H1= I, Ho= 1, Ho= and the equation (25) becomes ,32V a2V i aV i a2V a2+ap2+p ap+'°2 a02=o. To find the normal functions which satisfy this equation, we put V=ZR'h, when Z is a function of z only, R of p only, and shof 0, the equation then becomes i d2Z I (d2R i dR\ i i d24, Zd+K dp2+ dp)+p2 doe=o. d2Z That this may be satisfied we must have 2 0 constant, say =k2, 2- constant, say = —m2, and R, for which we write u, must satisfy the differential equation d2u ids (, 1112 ap2+p ap+ k P' U=0, it follows that the normal forms arc e 3ks' inm~. u(kp), where u(p) satisfies the equation 1216 I Lit 117 dp2 +pdp+ ~L— p2~71=0. (29) This is known as Bessel's equation of order m; the particular case den. I du dpz+v dp+u=0, 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 p2 p4 pm I—2.2ni+2-1-2.4.2m+2.2m+4—... the expression pm p2 + p4 2n`II(m) I—2.201+2 2.4.201+2.201+4—or y (—i)npm+211 211,+zniI(m+n)II(n) is denoted by Jm(p)• When m=o, the solution p2 p4 222=.42 of the equation (30) is denoted by Jo(p) or by J(p). relation (30) The function J,(P) is called Bessel's function of order in, 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 into-m, it follows that J_,,,(p) is a second solution of (2n). thus in general n = AJm(P) + BJ-m (P) is the complete primitive of (29). However, in the most important case, that in which nz is an integer, the solutions J_,,,(p), J,,,(p) are not distinct, for J_,,,(p) may be written in the form sz, +(-I)m(P ( - I)P p 2p z) II(m-11p)II(p) ~2J P=0 now II(n -nz) is infinite when m is an integer, and n < m; thus the first part of the expression vanishes, and the second part is (-I)'^J,,,(p), hence when m is an integer )"Im(p), and the second solution remains to be found. Bessel's Functions of the Second Kind.—When in is not a real integer, we have seen that any linear function of Jm(p), J_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 vrem+n'J-m(P) - cos Mir. Jm(P) sin 2ma - and may be denoted by Ym(p). This definition has the advantage of giving a meaning to Ym(p) in the case in which m is an integer, for it may be evaluated as a limiting form o/d, 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 mzr Y,s(P) as a second finite solution of the differential equation. When m is an integer, we have Ym(P)_(-I)m dJdE-F_(_I)-ddF+! e=o on carrying out the differentiations, and proceeding to the limit we find Ym(p) =J,,(P)logP+2 (P) m }t(n)+t(m.+n)!II ((- 1)17 (P) 2, z 2 )II(n) \2 =o +-(2~-m(1 11(-n)- 1) (2)217 °=o where t(n) denotes II'(n)/II(n). When m=o we have the second solution of (30) given by sz, 1 o(p) =Jo(P)logp+ IIn)IIf (720) P 2" 0 2I. Relations between Bessel's Functions of Different Orders .—Since e' sin satisfies Laplace's equation, it follows that sin m~.u", (P) satisfies the differential equation ar,+may- I u=o. The linear character of this equation shows that if u is any solution f /0 a r' ay) u is also one, f denoting.a rational integral function of the operators. Let t, i denote x+zy, x-ty, then since mum(J0n) satisfies the differential equation, so also does t''Z( p 'um(\I07)}, ~m+pd(pz) p{P m0i,i,(P) }' u,,,,+P=CPm+nd(P2)p c up(P) where C is a constant. If um(p)=J,„(p), we have um+p=Jm+P(P). and by comparing the coefficients of pet,' . we find C=(-2)P, hence Jm+p(P) = (-2)PP,"+Pd(P )P{P mJm(P)}, and changing in into -in, we find Jp-m (p) _ (- 2) npn-"'d (PPl P}PmJ-m (P) }HARMONICS 57 In a similar manner it can be proved that Jm p(P) =2PPT "`d(pz)P}Pm.Jm(P}• From the definition of Y,,,(p), and applying the above analysis, wt, prove that Ym+r(P)(-2)r'P''' PZ)p{P mYm(P)} and Ym-,,(p) = 2ppp-md(d;) p{PmYm(P) }. As particular cases of the above formulae, we find Jr(e) _ (-2P)pd(N o'P )5Jo(P), Yp(P) = (-2P)Pd(pz)PYo(P)
SPHEROID (Gr. c4aipa-etbis, like a sphere)

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