70 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX 



/•■'" J fix) 



INTEGRATION OF / 777- dx 



The discussion here is concerned only with integral values of ^ > 0. The 

 integral is not simply expressible in terms of known (i.e., tabulated) func- 

 tions, hence what amounts to a series expansion is used. The method 

 follows Ludinegg^ who gives the details for ^ = 1. 



The value of the integrand at :r = is first discussed. For ^ = 1 , /i(0) = 

 and /i(0) = 0.5, hence the integrand has the value zero. For I > \, 

 both numerator and denominator are zero, hence the value is indeterminate. 

 Evaluation by (f — 1) differentiations of numerator and denominator 

 separately leads to the result that the integrand (and the integral also) is 

 zero at X = for all C. 



We now introduce a constant p(. and a function 4>({x) which are such 

 that the following equation is satisfied, at least for a certain range of values 

 of x: 



Ji= -pcij'i-^^^^^^ + <i>tJl (1) 



Denote the desired integral by F(.{x), i.e.: 

 Then substitution of (1) into (2) yields: 



F( = -pC 



log 



For X = 0, J (/ x^ ^ is indeterminate, but evaluation by difTerentiating 

 numerator and denominator separately (/' — 1) times gives the value 

 iM^-l)! 



If we can now arrange matters so that 4>c remains finite in the range 

 (0, x), its integration can be carried out, a) by expansion into a power 

 series and integration term-by-term, or, b) by numerical integration. 



Solving (1) for (j)C one obtains 



«= ^, ^-^. (4): 



Jf 



Equation (4) becomes indeterminate at .v = 0, when (■ > \. Evaluation by 

 differentiating numerator and denominator separately € times shows </)^(0) = 0. 



> Uoclifrcqiicnztech. u. Elckhoak., V. 62, j)]). .VS-44, .Auk- 1943. 



