ON CALCULATION OF MATHEMATICAL TABLES. 243 
Lommel-Weber Functions (2,(<) and Q_;(z). 
The Lommel-Weber function Qy (z) is defined by the integral > )sin (% sin @ 
J 0 
—vo)do. 
The function Ey (%)= —7Q4(x) was employed by H. F. Weber * in his paper on 
‘The Theory of Fresnel’s Interference-phenomena.’ 
For small values of the argument (21(x) and ©_3(x) were calculated from the 
ascending series, viz. : 
Q4y(x) = X4(x) — ITy (x) 
and Q._4(%) = X3(x) + II;(2). 
. _2(2u  (22)3 (20) ) 
as x2) = 215 My Fa 
‘ _2( (Qn)? (Qx)h (2x8 ) 
mde) ileg 5 Use To Sere eee | 
For large values, the two functions were computed from the relations 
7 
| X4 (20) = C(x)I4(x) — S(x) .T_4(z) 
. and IJ}(x) = C(a)J_4(x) + S(x).J4(x). 
O4(2) = — T_a(z) +2 { Bate) + Aula) 
O24(x)= Iy(x) + 2 {Ba(e)— Ave) } 
BN (py else te asa heo eet ) 
where By(z) = 7/1 (xy (2x)8 ~ a6 --E A OE 
and Ay(z) — 1 {1 1.3.5, 1.3.5.7.9__ | 
x|2Qe  (2x)% - (2x)5 (Nagel 
: An independent calculation was made from the asymptotic expansions, 
| 
_ Asymptotic series begin by converging, but eventually become divergent. If the 
"remaining terms after the smallest be omitted, the sum of the terms already found 
4 presents the value of the function with an error less than the last included term, 
a it is generally supposed that the degree of approximation cannot be carried beyond 
this point. In the case of asymptotic series where the signs of the coefficients are- 
alternately positive and negative, a much closer degree of accuracy can be secured by 
breaking up the divergent part of the asymptotic expansion into more tractable series, 
whose summation can be effected by Euler’s method. By this method, a ‘ converging 
factor ’ can be found which usually takes the form } + a + at thine 
The product of the least term and the ‘ converging factor’ is equivalent to the 
divergent part of the series. Even for small values of x, three or four places of decimals 
can be added to the value of the function obtained by confining the calculation to the 
convergent terms. 
For the asymptotic series, when x=2n-+«, 
WSOPE HUT) 1. TM 
0) al aa), eR (a 
* Vierteljahrsschrift der Naturforschenden Gesellschaft in Ziirich. Band 24, 1879. 
Rk 2 
