344 BELL SYSTEM TECHNICAL JOURNAL 



P\/a Ko Ki Kt Kz Ki Kb K^ 



.5 00 OOOOOOOOOOOO 



.6 1.43874 .61659 .31832 .22547 .15539 .12199 .09275 



.7 1.25423 .42303 .17355 .11508 .06971 .05353 .03704 



.8 1.17306 .33141 .11439 .07463 .04093 .03195 .02038 



.9 1.12748 .27575 .08261 .05431 .02737 .02214 .01325 



1.0 1.09861 .23761 .06305 .04244 .01976 .01675 .00938 



1.1 1.07891 .20953 .04994 .03471 .01509 .01341 .00705 



1.2 1.06476 .18785 .04075 .02935 .01192 .01126 .00548 



1.3 1.05421 .17050 .03391 .02539 .00977 .00956 .00443 



1.4 1.04610 .15626 .02871 .02243 .00807 .00838 .00365 



1.5 1.03972 .14434 .02467 .02013 .00682 .00745 .00315 

 2.0 1.02165 .10508 .01332 .01330 .00358 



2.5 1.01366 .08295 .00838 .01010 .00217 



APPENDIX II 



Functions of Almost Diagonal Matrices 



Let E be a matrix whose elements are small in comparison with unity. 

 It is then often possible to approximate a matrix defined as some function of 

 the matrix I -\- E, where / is the unit matrix, by the expansion 



/(/ + £) = 7/(1) + ~f'(l) + JV"(1) + • • • ■ (A2-1) 



Thus, for example, when we take /(a) to be z~^ we obtain 



(/ + £)-! = / - E + E2 - • . • . (A2-2) 



Here we shall give similar formal results for f(D + E) where now Z> is a 

 diagonal matrix 



D = 



di ' 

 di ' 



di, 



(A2-3) 



whose diagonal elements are unequal and the elements E^ and Eji are small 

 in comparison with the absolute value oi \ di — dj | . We shall restrict 

 ourselves to a first approximation of the non-diagonal terms of f(D + E) 

 and to a second approximation of the diagonal terms. The results are closely 

 related to the ones obtained from the perturbation theory used in wave 

 mechanics. 

 We assume that/(Z) + E) may be defined by the series 



fiD + E) = aoI+ ai{D -\- E) + a^^D + E)^ + • • • (A2-4) 



