236 Bessel Functions of Equal Order and Argument. 



seems necessary to make use of Schlomilch's contour- 

 integral 



~. I e iz{t~Ht) t -n-l dt 



or the derived Bessel-Schlafli integral 



W. 



rco+) , 



Z7T? ) 



Bessel-Schlafli integi 

 J„(^)= M %os (nO-z sin e)dd- S ^^ 7T f V^+^nh*), 

 from which can be derived the formula 



J n (z) = -L\ e «ie-izsinO d 



the integral being taken round a contour consisting of three 

 sides of a rectangle whose corners are 



— 7T+ 00 2, 



7T, 77", 7r + °0&. 



As describes this contour, .x = n(0 — sin 0) describes a 

 similar contour whose corners are 



— 7Z7T+GO i 



wit. 



nir. 



hit + go i ; 



to avoid a branch point at the origin, it is convenient to 

 suppose that the 6 contour passes above it, and then the 

 x contour encircles it one and a half times. 



But a contour of this nature is easily deformable into- 

 the contour used by Debye in his investigations of Bessel 

 functions of high order ; and the problem of obtaining the 

 range of validity of the expansion seems to be much more 

 difficult with the contour consisting of three straight lines- 

 than with Debye's contour. 



The fact is that BessePs integral is extremely ill-adapted 

 for obtaining the asymptotic expression of Z n {z) when n is 

 large ; while Debye's beautiful contour integrals, which 

 (like BesseFs integral) are special cases of Kchlomilch's- 

 contour integral, yield theoretically complete expansions 

 without great difficulty. The real reason for this lies in the 

 fact that the integrands in Debye's imegrals are positive 

 and monotonic while the integrand in BessePs integral is 

 oscillatory. 



I might remark that I have obtained the first two terms 

 of the expansions of J n (n) and J»'(?i) by using BessePs. 

 integral, though it was necessary to use Tannery's theorem * 

 for integrals in the course of the analysis. To obtain the- 



* Broinwich, ' Infinite Series/ p. 443. 



