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

482

THEORY OF BESSEL FUNCTIONS

[CHAP. XV

15*25. Lommel's theorem on the reality of the zeros of Jv (z).
An extension of a theorem due to Fourier*, that the function JO (z) has no
zeros which are not real, has been effected by Lommelt. The extended theorem
is that, if the order v exceeds - 1, then the function Jv (z) has no zeros which
are not real.
To prove Lommel's theorem, suppose, if possible, that a is a zero of J, (z)
which is not real. It follows from the series for J, (z) that a is not a pure
imaginary, because then
E (_) (I a)2
m r (v+m 1)
would be a series of positive terms.
Let a0 be the complex number conjugate to a, so that a0 is also a zero of
Jv (z), because J, (z) is a real function of z.
Since v > - 1, it follows from ~ 5'11 (8) that
f tfv ^tf /td x rJ() dJ, (ax) J dJ, (ax)]
ItJ (at ) ( dt= 2- 2 J(~a x)-clJ (x) _- J (Coo) x) (
and so, since a2 - a02,
f tJ (at) J, (a0t) dt = 0.
The integrand on the left is positive, and so we have obtained a contradiction.
Hence the number a cannot exist, and the theorem is proved.
Similar arguments + may be used to shew that, if A and B are real and v > - 1, the
function A J, (z)+ BzJ,' (z) has all its zeros real, except that it has two purely imaginary
zeros when (A B) + v <0.
These results follow from the series for d {zA/B Jv (z)} combined with the formula
I tJo (t) Jv (t) dt=o,
which is satisfied if / and /o are any zeros of AJ, (z) + BzJ,' (z) such that 2 2 002.
15'26. The analogue of Lommel's theorem for functions of the second kind.
It is not possible to prove by the methods of ~ 15-25 that~ Y (z) has no
complex zeros in the regionll in which I argz I <. But it has been proved
by Schafheitlin~ that Y, (z) has no zeros with a positive real part, other than
the real zeros.
* La Theorie Analytique de la Chaleur (Paris, 1822), ~ 308; see also Steam, Quarterly Journal,
xvII. (1880), p. 93.
f Studien iber die Bessel'schen Functionen (Leipzig, 1868), p. 69.
+ See A. C. Dixon, Messenger, xxxi. (1903), p. 7.
~ Or, more generally, Cv (z).
11 When arg z = - 7r, YI (z) = eT-ri Y ( - z) - 2i cos v7rJ (- z), and hence, by ~ 3 63 (1), Y, (z)
cannot vanish unless v is half of an odd integer. This type of reasoning is due to Macdonald,
Proc. London Math. Soc. xxx. (1899), pp. 165-179.
~ Archiv der Math. und Phys. (3) i. (1901), pp. 133-137. In this paper Schafheitlin also subjects the complex zeros of Y1 (z) to a similar treatment.