ON CALCULATION OF MATHEMATICAL TABLES. 277 



For small values of the argument, the series in ascending powers of x were used 

 in calculating M (1 . J . x) and M ( — ^ . J . a;) to nine places of decimals. 



^ ^ ' 3 3-5 3.5.7^ 



M(-i.^.x) = 1 



3.2! 5.3! 7.4! 



Since M (0 . ^ . x) = 1 and M (^ . ^ . z) = e'^, the recurrence relations may be 

 applied at once to construct the rest of the table for these values of x and other values 

 of the parameters, a and y- 



For larger values of x, the asymptotic series were employed, viz. : 



\T /I JL \ r — r 1 1.3,1.3.5 1.3.5.7, 1 , , 



^ - ^ ' l2x^(2a;)2 (2a;)» (2a:)* ^- ' ' j ^"' 



Both these series begin by converging, but eventually become divergent, and it was 

 maintained that the approximation to the value of the functions could not be carried 

 beyond the point where the terms begin to diverge, i.e. the calculation must not be 

 taken beyond the least term. 



' When the argument is at all large, the series at first are rapidly convergent, but 

 they are ultimately in all cases hypergeometrically divergent. Notwithstanding this 

 divergence, we may employ the series in numerical calculation, provided we do not 

 take in the divergent terms.' § 



' Series of this kind are, strictly speaking, not convergent at all, for when carried 

 sufiiciently far, the sum of the series may be made to exceed any assignable quantity. 

 But, though ultimately divergent, they begin by converging ; and when a certain 

 point is reached, the terms become very small. Calculations founded on these 

 series are, therefore, only approximate ; and the degree of approximation cannot be 

 carried beyond a certain point. If more terms are included, the result is made worse 

 instead of better. In numerical calculations, therefore, we are to include only the 

 convergent part.' || 



In the case of M (1 .\ .x), where the signs of the terms are alternately positive and 

 negative, if a; = v-j-^ and v = n-\-\, it can be shown that the asymptotic series 

 becomes 



1 _. 1.3| 1.3.5 _ , 1.3.5 (2w-l) -. . 



2a; (2a;)2 (2a;)'* - (2a;)» ^ '5'^'' 



where cpi is the ' converging factor.' The first six terms of ^j are 



When X is about 10, it is possible to add eight or nine decimal places to the result 

 obtained when the divergent terms are omitted in the calculation from the asymptotic 

 series (a). 



For the function M ( — i .\ .x), where the signs of the terms in the asymptotic 

 series are all positive, a 'converging factor' may also be found. If a;=v+A and 

 v=n+^, the descending power series in (6) is 



1 , 1.3 ,1 .3.5 , 1.3.5 (2/t— 1) -, . 



2a; "^ (2a;)2 "^ (2a;)-'' "^ (2a;)" ^ "^ 'P''' 



§ Stokes. Mathematical and Physical Papers, vol. 2, p. 237. 



II Rayleigh. Scientific Papers, vol. 1, p. 190. Also Glaisher, Phil. Tninn. 

 London, vol. 160, pp. 367-387. Borel, Series JDivergentes, p. 3. 



