98 Mr' Watson, Bessel functions of lar-ge oi'der 



The portion of this curve which is suitable for our purposes 

 consists of an arc* on the right of the imaginary axis in the ^-plane 

 with its vertex at the origin and with the lines I {t)= ±tt as 

 asymptotes. 



If we write t = u + iv, where tt, v are real, the equation of the 

 curve becomes 



cosh (a + u) = V cosec v cosh a. 



We shall put 



tanh a (cosh t - 1) + (sihh t — t) = — r, 



so that as t traverses the contour t diminishes from + x to 0, and 

 then increases to + oo ; and therefore 



Jn (^'^) = o-^ e«(t^'ii^*-«) I j + \ e""^ (dt/dr) dr ; 



in the first integral v ^0 and in the second integral v ^ 0. 

 Now define T by the equation j- 



1^2 tanh a + ^T' = -T. 



A contour in the T-plane on which t is real is a semi-hyperbola 

 touching the imaginary axis at the origin and going off to infinity 

 in directions inclined ± ^tt to the real axis. If we write 



T=U+iV, 



where U, V are real, the equation of the hyperbola becomes 



Utanha + ^U'' = iV-\ 



Taking the semi-hyperbola as the T-contour, we shall shew 

 that an approximation to 



foD+iri /• 00 exp ( Jtt?') 



e-'^-'dt is e-''^dT. 



J CO - Tii ■' 00 exp ( - JttO 



It is easy to see that the difference of these integrals is 

 D idt dT) , 



{] ^ Jo ] (dr dr] 



and so the problem before us is reduced to the determination of 

 an upper bound for \d{t— T)/dT |. 



* This curve is derived from the curve shewn in fig. 4 (p. 541) of Debye's first! 

 paper by turning it through a right angle and talcing the origin at the vertex. The! 

 degenerate case when a is zero is shewn in fig. 5. \ 



t Since T = ^t^ ta,nh a + lt^ + (t*) when | 1 1 is small, the curve in the T-plaue 

 closely resembles the curve in the i-plane near the origin ; and, the parts of the 

 curves near the origin being the most important when n is large, we are obviously 

 able to anticipate that the integrals under consideration are approximately equal. 



