An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 164. 165.

Legendre's series for calculating F (a).

293

we have finally: o
ea 1 l-+-a
XIV) lrF(1 + a) -sna 2 1-a
X 2 sin ( ra 2 1-a
+ a( +C) — t - (S3l) a3- (815
(-1 < a < + 1).
From this formula the values of IF(a) can be calculated for the
arguments 0 to 2, when the quantities S3, S5... have been previously
determined. Legendre has given these values from S3 to S35 to 16
places of decimals. Also the value of C must be expressed, this is
done most rapidly by putting a- =, because:
1F () 3 (- (h) = (-4 i/=g)
is known. By the series:
31 ( 2 32. (S — 1) — (S7 - 1 l) +we obtain C =i - 0,577 215 664 901 532... ).
Accordingly the initial numerical terms of series XIV) are:
I r ( + a) 7a 1 11- a
tr(l + 2 sina 2 1 —a
+ 0,422 784 3a - 0,067 353 0a3 - 0, 007 385 5a5 - -
*) Euler obtained this number correctly to 15 places of decimals; annotating
Euler's Calc. integ., Mascheroni calculated it further, Legendre also gives 26
places but both differ in the 20th place from Gauss who had it recalculated by
Nicolai and gives the result to 40 places of decimals. Werke, Vol. III, p. 154.