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

4-43]

DIFFERENTIAL EQUATIONS

105

the other terms on the right cancelling by a use of equation (1). This is, of
course, the expression for J-N_+ (icz) in finite terms with a different notation.
For Barnes' proof of Kummer's formulae, by the methods of contour integration, see ~ 6'5.
4'43. Sharpe's differential equation.
The equation
dZy dy
(1).Zg+, d ~ +(Z+A)y=0,
which is a generalisation of Bessel's equation for functions of order zero,
occurs in the theory of the reflexion of sound by a paraboloid. It has been
investigated by Sharpe*, who has shewn that the integral which reduces to
unity at the origin is
(2) y C cos (z cos 0 + A log cot 0) dO,. o
where
7rr
(3) 1 = C cos (A log cot I0) dO.
J0
This is the appropriate modification of Parseval's integral (~ 2'3). To investigate its convergence write cos 0 = tanh b, and it becomes
(4) cosh d
(a~) C ~j cos (A q f+ z tanh 4) dob.
It is easy to see from this form of the integral that it converges for (complex)
values of A for which I(A) l < 1, and-t
C= cosh -7rA.
The integral has been investigated in great detail by Sharpe and he has
given elaborate rules for calculating successive coefficients in the expansion of
y in powers of z.
A simple form of the solution (which was not given by Sharpe) is
y = eiZ 1F ( 7 — iA; 1; + 2iz).
The reader should have no difficulty in verifying this result.
* Messenger, x. (1881), pp. 174-185; xII. (1884), pp. 66-79; Proc. Camb. Phil. Soc. x. (1900),
pp. 101-136.
t See, e.g. Watson, Complex Integration and Cauchy's Theorem (1914), pp. 64-65.