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

52

THEORY OF BESSEL FUNCTIONS

[CHAP. III

Now the integrand is an integral periodic function of J, and so the limits of
integration with respect to r may be taken to be a and a + 2Sr, where a is an
arbitrary (complex) number. This follows from Cauchy's theorem.
We thus get
(1 V a'hr+27r
JV () (v)v j i sin Cos ^ cos2-1 0 sin, 6d^ cd
J ( 2( )iv4 27] 7 i sin cos (+a) cos2V-1 0 sin Od dO.
= -- ei~i~oo~ (~'i+S ) cos2-l 0 sin OdGdO.
7rl ()J) 0o
We now define a by the pair of equations
cos a =Z- z cos, sin a = z sin,
so that
J (O ) -= v( (2) exp [i (Z - z os ) sin 0 cos - iz sin if sin sin 0]
7rl () 0o Jo
cos^-1 0 sin Od# dO.
The only difference between this formula and the formula
J, ( (=)= r () J exp [iW sin 0 cos,] cos2'-1 0 sin Odfd0
7rl (P)Jo Jo
is in the form of the exponential factor; and we now retrace the steps of the
analysis with the modified form of the exponential factor. When the steps are
retraced the successive exponents are
i (Z - z cos d) I - iz sin 0. m,
i (Z - z cos b) n - iz sin q. 1,
i (Z- z cos b) cos 0 - iz sin Cb cos sin 0.
The last expression is
iZ cos 0 - iz (cos f cos 0 + sin G sin 0 cos A),
so that the result of retracing the steps is
vP (V) f|of exp [iZ cos 0- iz (cos q cos 0 + sin sin 0 cos )]
sin2v-l ~ sin2v Od fdO,
and consequently Gegenbauer's formula is established.
[NOTE. The device of using transformations of polar coordinates, after the manner of
this section, to evaluate definite integrals seems to be due to Legendre, MJem. de l'Acad. des
Sci., 1789, p. 372, and Poisson, Me'n. de I'Acad. des Sci. II. (1818), p. 126.]
3*4. The expression of J~(n+ ) (z) in finite terms.
We shall now deduce from Poisson's integral the important theorem that,
when v is half of an odd integer, the function J (z) is expressible in finite terms
by means of algebraic and trigonometrical functions of z.
It will appear later (~ 4'74) that, when v has not such a value, then J, (z)
is not so expressible; but of course this converse theorem is of a much more
recondite character than the theorem which is now about to be proved.