A treatise on the theory of Bessel functions, by G. N. Watson.

3-51]

BESSEL FUNCTIONS

59

so that
V aJ. (z) (-)l-av(Z) = (V2 - n2) [J(z) - (-ajap ap L v av
+ 2v {J. ) -) (z-) J_- ()}.
Now make v -^ n. All the expressions in the last equation are continuous
functions of v, and so we have
( (Z) _ J(z) _o
where v is to be made equal to n immediately after the differentiations with
respect to v have been performed. We have therefore proved that
(3) Vn Yn () = 0,
so thlat Y, (z) is a solution of Bessel's equation for functions of order n.
It is to be noticed that
Y_ z(z)- = lim vJ (z) - (-)-n J-_ (z)
Y^^9(X = lim
= lim J- (Z)-(-)"( J (Z)
whence follows a result substantially due to Lonmmel*,
(4) Y_, )= (z -) = Y (Z).
Again,
Y,()w - [ ^ ~v=O ~ vJ=-O
while, because J. (z) is a monogenic function of v at v = 0, we have
aj_ (Z _ -JVZ1 = - J ()'
ap 'v =O a(-v)Jo ^a ^
and hence it follows that
(5) Y,(zo) = 2 UI (Z)
A result equivalent to this was given by Duhamelt as early as 1840.
3'51. The expansion of Yo (z) i n an ascending series.
Before considering the expansion of the general function Y,, (z), it is convenient to examine the function of order zero because the analysis is simpler
and the resulting expansion is more compact.
We use the formula just obtained,
a j (-)m (+ )v+2 n
Yo ()=2 T rom! (v +m+l)i o'
* Studien iiber die Bessel'schen Functionen (Leipzig, 1868), p. 87. Lommel actually proved
this result for what is sometimes called Neumann's function of the second kind. See ~ 3'58 (8).
t Cours d'Analyse, I. (Paris, 1840), pp. 122-124.