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>' u ">{p) satisfies Laplace's equation, it follows that

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