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

15-25-15-27]

ZEROS OF BESSEL FUNCTIONS

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For let 3 be a complex zero of Y0 (z), and let,8/ be the conjugate complex,
so that /o is also a zero of Yo (z). Then, by ~~ 5'11 (8) and 3'51 (1),
f tY,(i3t) Yo (3ot) dt
"-~T~- log" - 0Po Y, (k= dY( Y(/3 d o ) - -7r2(og
/92 302 dx o d/x dro2 (/2 _ Y(02)BO0
and so, if / = peiw, we have
tYo (St) Yo (30t)t= -rp2 Si 2 '
and the expression on the left is positive while the expression on the right is
negative when o is an acute angle.
15'27. The theorems of Hurwitz on the zeros of J, (z).
The proof which was given by Fourier that the zeros of J, (z) are all real
was made more rigorous and extensive by Hurwitz*, who proved (i) that when
v > - 1, the zeros of J, (z) are all real, (ii) that, if s is a positive integer or
zero and P lies between -(2s + 1) and - (2s+ 2), J (z) has 4s + 2 complex
zeros, of which 2 are purely imaginary, (iii) that, if s is a positive integer and
v lies between - 2s and - (2s + 1), Jv (z) has 4s complex zeros, of which none
are purely imaginary. To establish these results, we use the notation of 9'7.
We take the function g,,,,v(') which has, in the respective cases (i) in
positive zeros, (ii) n - 2s - 1 positive zeros, 1 negative zero and 2s complex
zeros, (iii) n - 2s positive zeros and 2s complex zeros.
so (/_)n Cn
We now prove that, iff, (): ( 1' then the function f ()
n=on! i (v + n + )'n
has at least as many complex zeros as g21,. ('). After Hurwitz, we write
M _ ( + ') gD, v (",+, (~"), ( ), v (, v )
where I, 7 are real and [ = d + iy, [' = -ii. The terms of highest degree in
Om (I:,?) are easily shewn to be
Inm (Qn + - 1) (/; + mi) ( + - ) {(n + m) (2., + 1) + m- 1} (2_ +,2)m —1;
and since g2,,^ is a real function, it follows that if C is a complex zero of
gm,v (), so also is r'; and therefore the complex zeros satisfy the equation
dn (~, v) = o.
Again, it is not difficult to deduce from the recurrence formulae (~ 9 7) that
Om+il ( ( ) + 2in + 2) 21+h, v (g^) g2m+i, v (':) + 2 + ( q2) Om (, q).
* Math. Ann. xxxIII. (1889), pp. 246-266;' cf. also Segar, Messenger, xxii. (1893), pp. 171-181,
for a discussion of the Bessel coefficients. The analysis of this section differs in some respects from
that of Hurwitz; see Watson, Proc. London Miath. Soc. (2) xix. (1921), pp. 266-272.
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