﻿the Partition of Energy. 463 



Arguments of tins type — asymptotic expansions by steepest 

 descents — are, of course, well known in pure mathematics. 

 Consider a contour integral of the form 



isj/WW-)]"* .... (6-1) 



subject to the following conditions : — 



(i.) <\>{z) is an analytic function of z, which can be ex- 

 panded in a series of ascending powers of z. 



(ii.) This series starts with some negative power. 



(iii.) Every coefficient is real and positive. 



(iv.) Its circle of convergence is of radius unity. (This 

 condition is quite unessential to the mathematics, 

 but makes the statement simpler, and is physically 

 true.) 



(v.) The powers that occur in the series cannot all be put 

 in the form a + /3r- where a and ft are any given 

 integers and r takes all integer values. 



(vi.) F(r) is an analytic function with no poles in the unit 

 circle, except perhaps at the origin, 

 (vii.) 7 is a closed contour going once counterclockwise 

 round the origin. 



The problem is to obtain the asymptotic value as E tends 

 through integer values to infinity. 



We shall first study the properties of <j>(z). Considering 

 real values, it must have one and only one minimum between 

 and 1. For it is continuous and tends to + co at both 

 and 1, and so must have at least one minimum between. 

 Further, to find minima we differentiate, and then all the 

 negative powers will have negative coefficients and all the 

 positive positive. It follows that minima are given where 

 two curves cut, one of which decreases steadily between 

 and 1, while the other increases steadily. These curves can 

 only cut in one point, and so there is only one minimum. 

 Xext, for the complex values, consider a circle of any radius 

 r less than 1. As the modulus of a sum is never greater 

 than the sum of the moduli, it follows that at no point on this 

 circle can \<j>(z)\ be greater than <f>(r). Moreover, it can only 

 equal 4>(r) provided that condition (v.) is broken, and in that 

 case there will be /3 points at each of which \<j>(z)\ = (j>(r). 

 We can thus see that on account of the large exponent it is 

 only the part of <y near the real axis that contributes effec- 

 tively to the asymptotic value of the integral. This suggests 



