Page:EB1911 - Volume 25.djvu/680

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SPHERICAL HARMONICS
658

infinite, and at the same time let n become infinite in such a way that n/a has a finite value λ. Then

${\displaystyle \operatorname {L} \sec ^{n}\theta =\operatorname {L} \left(\sec {\frac {\rho }{a}}\right)^{\lambda a}=1,\operatorname {L} \left(1+{\frac {z}{z}}\right)^{n}=e^{\lambda z}}$

and it remains to find the limiting value of Pn(cosθ). From the series (15), it may be at once proved that

${\displaystyle \operatorname {P} _{n}\left(\cos \theta \right)=1-{\frac {\left(n+1\right)n}{1^{2}}}\left(\sin {\frac {\theta }{2}}\right)^{2}+\cdots +{\left(-1\right)}^{m}\delta {\frac {\left(n+m\right)\cdots \left(n-m+1\right)}{1^{2}.2^{2}.m^{2}}}\left(\sin {\frac {\theta }{2}}\right)}$

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

${\displaystyle \operatorname {L} \operatorname {P} _{n}\left(\operatorname {cos} {\frac {\lambda \rho }{n}}\right)=1-{\frac {\lambda ^{2}\rho ^{2}}{2^{2}}}+{\frac {\lambda ^{4}\rho ^{4}}{2^{2}\cdot 4^{2}}}-\dots +\left(-1\right)^{m}\delta _{1}{\frac {\lambda ^{2m}\rho ^{2}m}{2^{2}\cdot 4^{2}\cdot \left(2m\right)^{2}}}}$

where 81 is less than unity Hence ${\displaystyle {\underset {n=\infty }{\operatorname {L} }}\operatorname {P} _{n}\left(\cos {\frac {\lambda \rho }{n}}\right)=\operatorname {J} _{0}\left(\lambda \rho \right)}$

Again, since ${\displaystyle P_{n}^{m}=\left(\cos \rho \right)=\sin ^{m}\theta {\frac {d^{m}\mathrm {P} _{n}\left(\cos \theta \right)}{d\left(\cos \theta \right)^{m}}},}$

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 ${\displaystyle \rho }$ is real and positive

${\displaystyle \mathrm {J} _{0}\left(\rho \right)={\frac {2}{pi}}\int _{0}^{\infty }\sin \left(\rho \cosh \phi \right)d\phi }$

${\displaystyle \mathrm {Y} _{0}\left(\rho \right)={\frac {2}{pi}}\int _{0}^{\infty }\cos \left(\rho \cosh \phi \right)d\phi }$

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