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.

~ 133 —134. Trigonometric series for calculating elliptic integrals.

231

A _-+ c 1.3 2 1 1.3. 5 13 *.3.5.7 16.3.5 1.3.5.7.9 +
2 2 2.4 ' 2 2.4.6 24' 2.4.6.8 2.4.6 ' 2.4.6.8.10
A 14 —c{l1.3.5c3 1 3.5.7 5+ 1.3 1.3.5.7.9
3 2.4.6C 2 2.-'4.6.8 2.4 2.4.6.8.10 o
These numbers A, Al 22, 3 A3, 4A4, etc., decrease continually
and have zero as limit, for:
A1 < cA, 2A2 < cA,, 3A3 < 2cA2, 4A4 < 3cA,...
consequently:
A1 < cA, 2A2 < c2A, 3A3 < c3A, 4A4 < cA,... etc..
Hence, as c < 1, series 3) converges even when we give each of its
terms the absolutely greatest value it admits of; therefore 3) converges
uniformly (~ 127) and expresses a continuous function in the interval
from zero to zt. Accordingly we obtain by integration:
4) 1() =J j — - Acp - A - sin 29 +- A2sin 49 - A3sin 6 6P- +
In particular for p =- ~z:
I
5) A 2 f d (P = 2 _F.
0
But the other coefficients of this series can also be expressed as
definite integrals. For, if we multiply series 3) in turn by:
cos 2g, cos49p, cos 6,...
and integrate these products between zero and — v, since for n ~ n:
7t 4t
f cos 2 m cos 2n Tc ==- {cos2m + np + cos2m - n cp dcq = 0,
o o
and form = n:. (cos 2mn9)2d9 =2. 2 2, we thus obtain:
6 /os 2 p d _A 7 cos 4 dp.2A2 rcos 6 qd 34A c
A)J-c d - ~-=-o- - 2*
j - 2 A 1I 2 2 A 22J A 32 2
0 0
The calculation of A, introduces the values F(-) and E (), for:
7) n Ik-( - 2 sin2 )d p 2
Recurring formulas can be found for the other coefficients by
differentiating series 3). Since the series thus obtained:
k2 ssin cos
inCo3 - 4 A1 sin 2 9 - 1G A2 sin 4 p +- 36 A sin 6 -..