228 SOME CALCULATIONS IN ILLUSTRATION OF FOURIER'S THEOREM
The integrals in (2) are afc once expressible by what is called the i integral, defined by
and if the sine-integral were thoroughly known there would be scai anything more to do. For moderate .values of 6 the integral may be cs lated from an ascending series which is always convergent. For la values this series becomes useless; we may then fall back upon a descem series of the semi-convergent class, viz.,
Si(6} = ?-cos0\l~1'2
+
1 1.2.3
smtf •(-£------2ii~"
0"
DT Glaisher* has given very complete tables extending from 6 = I 0 = 1, and also from 1 to 5 at intervals of O'l. Beyond this point he g the function for integer values of 6 from 5 to 15 inclusive, and afterw only at intervals of 5 for 20, 25, 30, 35, &c. For my purpose these do suffice, and I have calculated from (5) the values for the missing inte up to 6 = 60. The results are recorded in the Table below. In each < except those quoted from Glaisher, the last figure is subject to a. s: error.
For the further calculation, involving merely subtractions, I have sele the special cases 7^ = 1, 2, 10. For fc: = 1, we have
<£(») = Si (0 + 1)-Si (0-1)......................0
6 Si(fl) 6 Si (6) 0 8i(0) e Si(6>)
16 1-63130 28 1-60474 39 '•. 1 -56334 50 1-55162
17 1-59013 29 1-59731 40 1-58699 51 1 -55600
18 1-53662 30 1-56676 41 1 '59494 52 1 -57357
19 1-51863 31 1-54177 42 j 1-58083 53 1-58798
20 1-54824 " 32 1-54424 43 1 -55836 54 1-58634
21 1-59490 33 1-57028 44 1-54808 55 1-57072
22 1-61609 34 1-59525 45 1-55871 56 1-55574
23 1-59546 35 1-59692 46 1-57976 57 1-55490
24 1-55474 36 1-57512 47 j 1-59184 58 1-56845
25 1-53148 37 1-54861 48 i 1 -58445 59 1 -58368
26 ' 1 -54487 38 1 -54549 49 1-56507 60 1-58675
27 1-58029
In every case $(#) is an even function, so that it suffices to consid positive.
* Phil. Trans. Vol. CLX. p. 367 (1870). 207.hich m is written for 27dS cos fa. It is to be observed that in this solution there is no limitation upon the value of R if (K — I)2 is neglected absolutely. In practice it will suffice that (K— 1) R/\ be small, X (equal to 2?r/&) being the wave-length.