48 BELL SYSTEM TECHNICAL JOURNAL 



As stated above a large number of infinite integrals of the type 

 appearing in equation (9) have been worked out. Consequently the 

 solution of (9) can frequently be written down by inspection. When 

 this is not the case, however, the appropriate procedure is usually 

 to expand the function \/pH{p) in such a form that the individual 

 terms are recognizable as identical with infinite integrals of the re- 

 quired type. 



An interesting expansion of this kind and one which is applicable 

 to a large number of physical problems is as follows: 



Expand \/pH{p) asymptotically in the form of the divergent series 



l/pH{p)^^an/p^^\ 



This expansion is purely formal and the series is divergent. It is 

 summable, however, in the sense that it may be identified with its 

 generating function l/pH(p). It is also summable in accordance 

 with Borel's definition of the sum of a divergent series by the Borel 

 integral * 



poo 



J dt e-^^^y\ ant^/n\ 

 This suggests that these two series are equal and consequently that 



/oo 

 dt e-^'Va„/»/w! 



The solution is therefore 



h (/) = ^S\ ant^/n\ 



provided this series, which is called by Borel the associated function of 

 the divergent expansion, is itself convergent. This is the case in all 

 physical problems to which this form of expansion has been applied.^ 



The foregoing will be recognized as identical with Heaviside's 

 power series solution, obtained by the empirical rule of identifying 

 \/p^ with t"/n\ in the asymptotic expansion of l/H{p). 



Another form of solution of very considerable practical value 

 depends on a partial fraction expression which can be carried out in a 

 large number of physical problems. It is 



\/pH (p) - a + b/p + c/p'^ + 2) "^k/ip - pk) 



* See Bromwich, Theory of Infinite Series, pp. 267-269. 

 *See Appendix II. 



