J0 (p) =;. 2f ° o sin (p cosh 4,)d4 Yo(p) =f o cos (p cosh 4,)dq,.
25. Bessel's Functions with Imaginary Argument.—The functions with purely imaginary argument are of such importance in connexion with certain differential equations of physics that a special notation ;
.~I
sin (54_0 I1z8P 12 .32.5 3(8P2)3 1* — V . ..
when m is an integer,
Km(r)=(1),, / e_' 1+4m212+(4m212)(4}232) +... 2r I.8r I.2(8r2
I",(r)= I e (j I4m212+(4m212)(4m232) i~2ar (  1.8r 1.2(802
27. The Basset's functions of degree half an odd integer are of special
importance in connexion with the differential equations of physics. The two equations
at = kV2u, ata = k'v2u,
are reducible by means of the substitutions u=ek'v, u=e,"v to the form V2v+v=o. If we suppose v to be a function of r only, this last differential equation takes the form
dl (sr) dra {vro,
so that v has the values
sin r/r, cos tit.;
in order to obtain more general solutions of the equation pav+v=o, we may operate on
sin r/r, cos r/r
with the operator
y'' ( a a a 1
ax' ay' az '
where Y,, (x, y, z) is any spherical solid harmonic of degree n. The result of the operation may be at once obtained by taking, Y„ (x, y, z) for Mx, y, z) in the theorem (7'), we thus find as solutions, of viv+v=o, the expressions
d^ sin r ~, d" cos r
Y"(xy z)d(ra)" , "(x, z
y )d(ra)m r
By recurring to the definition of the function Jm(r), we see that
2p i — r2 + r4 2 sin r
Ji(r)=~ir 2.3 2.3.45—... =~~ thus
r iJi(r)='V sin 7 r
Using the relation between Bessel's functions whose orders differ by an integer, we have
Li++(r ) = 2) "r"+id(rad)"Ji(Jrr) ( 2)nN/xr 2"+id(dra)sn sinr g
(— =— .
It may be shown at once thatthat the function K"(z) has no real zeros unless n=2k+i where k is an integer, when it has one real negative zero; and that K"(z) has no purely imaginary zeros, and no zero whose real part is positive, other than those at infinity. When i> n> o, K"(z) has no zeros other than those at infinity, when 2>n> i, it has one zero whose real part is negative, and when m+I>n> m where m is .an integer, there are m zeros whose real parts are negative. When n is an integer, K"(z) has n zeros with negative real parts.
29. Spheroidal Harmonics.—For potential problems in which the boundary is an ellipsoid of revolution, the coordinates to be used are r, 0, o where in the case of a prolate spheroid
x=cJra—i sin9cos¢, y=cJr i sin0sin0, z=crcos0,
the surfaces r=ro, B=Bo, 4,=0o are confocal prolate spheroids, confocal hyperboloids of revolution, and planes passing through the axis of revolution. We may suppose r to range from i to so , 0 from o to r, and d, from o to 2,r, every point in space has then unique coordinates r, 0, O.
For oblate spheroids, the corresponding coordinates are r, 0, ¢ given by
x=cJr2} isin0cos0, y=cJr2+isin0sin0, z=crcos0, where
O — i, and that all the real zeros of J"(z)/z" when n is real and <i, and not an integer, are associated with the essential singularity at infinity. When n is. a negative integer — m, J"(z)/z" has, in addition„ 2m real zeros coincident at the origin. When n = —m —v, in being a positive integer, and i>v>o, J"(z)/z" has a finite number 2m 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, and two 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 J"(z)/z" are separated by those of J,,+i(P)/z", one zero of the latter, and one only, lies between two consecutive zeros of J"(z),/z". When n is real and > —i, all the zeros of J"(z)/z" are given by a formula due to Stokes; the m°' positive zero in order of magnitude is given by
a=4i i 4(4+22 — i) (28n0 3i) &c., 8a 3. (8a) 2
where a=}ir(2n+4m—i). It. has been. shown by Macdonald
The following expressions may be given for the toroidal functions:
P " (cosh p) = (— II (n z) "A cos in4) "} a II(n m— J)J5 (cosh p+sinh p cos ,p),do'
= n(n+m a) (' (cosh p + sinh p cos ¢)" mspdsp.
7, II(n—) Jo
2 p cosh n4
P,.; (Cosh p) _ ~I o v2 cosh p — 2 cosh st,
sinh p cosh w) cosh mwdw
A net,
=(I)'"a2"'II(mz)n(z) sinh n'PJ cos d¢.
o (2 cosh p  2 cos
The relations between functions for three consecutive values of the degree or the order are
2n cosh pP,7 1 (cosh p) — (n —m+z)P,+ (cosh p)
 (n+m  2) P'" i (cosh p) = c
P (cosh p) + 2(m + i) coth pP+;[(cosh p)
(nm )(n+m+I)P:L(cosh p) =0,
with relations identical in form for the functions Q. , (cosh p). The function Q,,_ (cosh p) is expansible in the form
ti(n)
which is useful for calculation of the function when p is not small. P,. (cosh p) can also be expressed in terms of e.° by a somewhat complicated. formula.
31. Ellipsoidal Harmonics.—In order to treat potential problems in which the boundary surface is an ellipsoid, Lame took as coordinates the parameters p, p, v of systems of confocal ellipsoids, hyperboloids of one sheet, and of two sheets; these coordinates are three roots of the equation
x2 y2 „2 t"+t2—ly2'12_0=I (k>h);
we thence find that
pm, 1p2—h2Jp2h2vh2_v2 dp?k'LJk2p2A/lev2
x= 5= hvk2h2 ' z= Jk2_Iz2
where co > p2 >— h2 k2 p2 h", and k2 > y2 > o.
We find from these values of x, y, z
(dx)2 + (dy)2 + (dz) 2 (p2h2) (P2 k2)) (dp)2 + 22 15)) (dp) 2
(P2 v2) (p2 p2) + (2 — p2) (k2  y2 j (di') z,
and on applying the general transformation of Laplace's equation that equation becomes
(p2 —P2) N
a'YV ,32V
I .+. (y,”. _ p2) ay + (P2 ^ p2} 0,
where 4, n, { are defined by the formulae
It can now be shown that .Laplace's equation is satisfied by the product E(p)E(p)E(v), where E(o) satisfies the differential equation
d2E(P){n(n+I)p2(h2+k2)piE(p) o; de
and E(p), E(v) satisfy the equations d2E
+{11(11+ 1)p2p(h2+k2)1E(p) =0, d2E (v) In (n+ I) p2 ,.p (h2+k2)1 E (v) = o,
where n and p are arbitrary constants. 7 On substituting the valuesof the parameters I;, n, in terms of p,.p, v, we find that the equation satisfied by E(p) becomes
(p2
1t2)(PZ—k2) 1.(+P(2P2—h2k2) dE (P) dp
+{(h2+k2)pn(nf I)P2}E(p) o,
and E(p), E(v) satisfy equations in p, v respectively of identically the same form; this equation is known as Lame's equation.
If n be taken to be a positive integer, it can be shown that it is possible in 2ndI ways so to determine p that the equation in E(p) is satisfied by an algebraical function of degree n, rational in p, 1/(p2h22), v(p2k'2). The functions so determined are called Lame's functions, and the 2ndi functions of degree n are of one of the four forms.
K(p) =aop"+alp"~+. .
L(p) = J p2.h2(ao' 1"1 + + . . . ), M(p)=1Ipi._k2(ao"+a"1P" +. . ),
N(p) p2—k2 12—h2(a'P"2+ai pn4+...)
These are the four classes of Lime's functions of degree n; of the functions IK there are r+2n, or }(n+i), according as n is even or odd; of each of the functions L, M, there are sn, or 2(nI), and of the functions N, there are In, or i(n+i).
The normal forms of . solution of Laplace's equation, applicable to the space inside the ellipsoid, are the 211+1 products E (p) E(p) E(v). It can be shown that the 2n+1 values of p are 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=po can be expressed as the sum of products of Lame's functions of p and v, in the form
00 211+t I 2=I
the potential function for the space inside the ellipsoid, which has the arbitrarily given value over the surface of the ellipsoid, ie consequently`
E En(P)EnI(pp) En(v)
c" En Wl) It can be shown that a second solution of Lame's equation is
F"(P) where
cosh p
a d,y o J 1  k12 sin2t,ti
= fP dp Cm. d is
PZ —h2 P2 — ks — J p2 ha k2 _ 2'
_ lc)" di'
J o v 112 v2
which are equivalent to memoir, Theorie des attractions des spheroides et de la figure des
pkdn(k4, ki), p=kdn(K—kn, ki), v=ksn(kE, ki'), planktes, published by the Paris Academy in Y785. Laplace was where kit, kl'2 denote the quantities t h2/0, h2/k2 and K denotes the first to consider the functions of two angles, which functions
the complete elliptic integral , have consequently been known as Laplace's functions; his investi
gations on these functions are given in the Mecanique celeste, tome ii. livre iii., tome v. livre xi., and in the supplement to vol. v. The notation'P'n) was introduced by Dirichlet (see Crelle's Journal, vol. xvii., " sur les series dont In terme general depend de deux angles" &c. ; see also his memoir, " Ueber einenneuen Ausdruck zur Eestimmung der Dichtigkeit einer unendlich dfinnen Kugelschale," in the Abhandlungen of the Berlin Academy, 185o). The name " Kugelfunctionen " was introduced by Gauss (see Collected' Works, vi. 648). A direct investigation of the expression for the reciprocal of the distance between two points in spherical surface harmonics was given by Jacobi (Crelle's Journal, vol. xxvi., see also vol. xxxii.). The functions of the second kind were first introduced by Reine (see his " Theorie der Anziehung eines Ellipsoides," Crelle's Journal, vol. xlii., 1851). The abovementioned investigators employed almost entirely polar coordinates; the use of Cartesian coordinates for the expression of spherical harmonics was introduced by Kelvin in his theory of the equilibrium of an elastic spherical shell (see
F"(p) (2n+I)En(P)j~ dp
p{ (P)}Zdp2h2pki
this f unction F"(p) vanishesat infinity as p^', and is therefore adapted to the space outside the bounding ellipsoid. The external potential which has at the surface p = pi, the value
EcnEi(p)E,I(v) isE EcnF(P)E"(p)En(v)•
32. History and Literature.—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 of the attractions of solids of revolution: Legendre's investigations are contained in a memoir of the Paris Academy, Sur'l'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 potetial function was introduced by Laplace, who also first obtained the equation which
i bears his name; he applied spherical surface harmonics to the
determination of the potential of a nearly spherical solid, in his
Phil. Trans. Roy. Soc., 1862), and also independently by Clebsch (see his paper, " Ueber die Reflexion an einer Kugelflache," Crelle's Journal, vol. lxi., 063). 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 Theorie 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 l'ecole polyt., 1823, cah. 19). The toroidal functions were introduced by C. Neumann (Theorie der Elektricitats and Warmevertheilung in einem Ringe, Halle, 1864), and independently by Hicks (Phil. Trans. Roy. Soc., 1881). The ellipsoidal harmonics were first investigated by Lame 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 coordinates 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, Lame and Bessel (Cambridge, 1875); Thomson and Tait, Natural Philosophy (1879), App. B: Haentzschel, Reduction der Potentialgleichung auf gewohnliche Differentialgleichungen (Berlin, 1893) ; F. Neumann, Beitrage. zur Theorie der Kugelfunctionen (Leipzig, 1878) ; C. Neumann, Theorie der Bessel'schen Functionen (Leipzig, 1867) ; Ueber die nach Kreis,Kugel and Cylinderf unctionen fortschreitenden Entwickelungen (Leipzig, 1881); Lommel, Studien fiber die Bessel'schen Functionen (Leipzig, 1868) ; Mathieu, Cours de physique mathematique (Paris, 1873); Pockels, Ueber die partielle Differentialgleichung Au+k2u=o (Berlin, 1891); Bucher, Ueber die Reihenentwickelungen der Potentialtheorie (Leipzig, 1894); Gray and Mathews, Treatise on Bessel's Functions; Dini, Serie di Fourier e altre rappresentazione . (Pisa, 188o) ; Graf and Gabler, 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 Di erentialgleichungen der Physik (Bremen, 1900) ; W. E. Byerly, Fourier's Series and Spherical, Cylin
drical and Ellipsoidal Harmonics (Boston, 1893). (E. W. H.)
End of Article: J0 (p)
