53 



APPENDIX II 

 EVALUATION OP THE AUXILIARIES INTEGRALS* 



The integral to be computed is given by: 



r ~ _ii2L (y/Vj 2 

 W?,, = I = Cl e Y o ° M.(y)M.(y)dy 



U °\ j/(y/y o ) 2 -l x ^ 



The functions e r o and M.(y)M.(y) are well-behaved in the entire interval of 

 integration. However, the algebraic function (y/y) I V(Y/y) z — 1 causes some 

 difficulty at the lower limit of integration, i.e., at y = y . In the neigh- 

 borhood of y = y , the contribution of I is far from negligible and therefore 

 an investigation was carried out to determine the asymptotic behavior of the 

 integral as a function of the upper limit. Specifically, the following func- 

 tion was examined: 



Ke) = J M(y)f(y)dy 6 > 



where 

 and 



It was found that 



ki 



My) = e "o M i (y)M j (y) 



f(y) 



nr/y ) 2 -i 



l(e)~e" 4 'o M 1 (y)M j (y) y V / 27|l + ^e + 0(€ 2 )| 



This asymptotic expression was used to determine the interval of 

 integration, Ay far a numerical integration. This interval was too small to 

 be practicable, even allowing for subsequent changes in Ay. 



A new approach to the problem was sought in a suitable transforma- 

 tion. The following transformation very quickly presented itself: 



Y = Z 2 + y Q 

 dy = 2Z dZ 



The original integral was transformed as given by: 



*By J. Blum, National Bureau of Standards 



