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

2-21]

THE BESSEL COEFFICIENTS

21

I {' r
Jo (z) == - cos nO cos (z sin 0) dO
(7) 2 (n even).
= -- cos nO cos (z sin 0) dO
If 0 be replaced by r - 71 in the latter parts of (6) and (7), it is found that
2 2^
(8) Jn (z) = - ()2 -) cos mn sin (z cos v) dq (n odd),
q7 ' o
o( odd),
2 HU
(9) J1g (z) = - (-)2 - cos nq1 cos (z cos V) d? (n even).
The last two results are due substantially to Jacobi*.
[NOTE. It was shewn by Parseval, femn. des savans etrangers, r. (1805), pp. 639-648,
that
a a a67r
12~2 + 2 A. 4 2 29 49 6 +... =- cos (a sinx) dx,
22 2+ ~ 42 - 22. 62 7r 02
and so, in the special case in which n=0, (2) will be described as Parseval's integral. It
will be seen in ~ 2'3 that two integral representations of Jn (z), namely Bessel's integral
and Poisson's integral become identical when n =0, so a special name for this case is
justified.]
The reader will find it interesting to obtain (after Bessel) the formulae ~ 2'12 (1) and
~ 212 (4) from Bessel's integral.
2'21. llodifications of Parseval's integral.
Two formulae involving definite integrals which are closely connected with Parseval's
integral formula are worth notice. The first, namely
(1) J { /(z2 -2)} - 1 ( eY COcos(z sin ) dO,
is due to Bessel t. The simplest method of proving it is to write the expression on the
right in the form
1r j ey cos 0+izsin 0 dO
2 7r - 7
expand in powers of y cos 0 + iz sin 0 and use the formulae
V(ycos 0 + sin )2n+I0=, | Iy cos 6 +iz sin 0)21 d0 =-2 (n + (n1) (y2
the formula then follows without difficulty.
The other definite integral, due to Catalan + namely
(2) Jo (2i %z) _ e(l+z) cos 0 cos {(1 - z) sin 0} dO,
is a special case of (1) obtained by substituting 1 - z and 1 +z for z and y respectively.
* Journalfur Math. xv. (1836), pp. 12-13. [Ges. Math. Werke, vi. (1891), pp. 100-102]; the
integrals actually given by Jacobi had limits 0 and 7r with factors 1/T7 replacing the factors 2/7r.
See also Anger, Neueste Schriften der Naturf. Ges. in Danzig, v. (1855), p. 1, and Cauchy,
Conmptes Rendus, xxxvIII. (1854), pp. 910-913.
t Berliner Abh., 1824 [published 1826], p. 37. See also Anger, Neleste Schriften der Naturf.
Ges. in Danzig, v. (1855), p. 10, and Lommel, Zeitschrift filr Math. und Phys. xv. (1870), p. 151.
+ Bulletin de l'Acad. R. de Belgique, (2) XLI. (1876), p. 938.