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.

232

Integration of an infinite series.

Bk. III. ch. IV.

is uniformly convergent, the differentiation is admissible (~ 129). If
we multiply the left side of this equation by k2, and the right by
the equal value:
2 -- k2
-72 -- cos 2(p == 4- +cos 2 9p
and arrange by sines of multiples of qp, since:
2sin 2 m cos2 p = sin 2m -+- 19 - + sill 2 - 1 9,
we shall have:
sin2 -- (4A1i - 8A2)sin2( - (16A2A - 2A1 -- 18A3)sin49
AT
+(36A3 - 8A, - 32A)sin69p - (64A4 - 18A3 - 50A5)sin8
(- 1) ((2 - 2)2 A - (2n - 4)2 A (2 2 A) sin 2m -
2 - m-2 2
On the other hand we obtain by multiplying equation 3) by sin 2p:
sin 2q = (A - 2A2)sin2p- (Al- - 3A3)s 4 (A —si4 q (n2 -4 6
ATS
~ - *- ( - 1)2) 2 ( - 2),) sin2 -^) 1* *
This series must be identical with the previous one; and since both
series converge uniformly, the only way in which this identity can subsist,
is, that the coefficients of corresponding sines coincide in both. This
becomes evident when, as in the deduction of equations 6), we express
each coefficient by a definite integral. Hence:
A-2A2 — 4A1A - 8A2, A1 -3A3 = - 16A2; - 2A - 18A3,
(m - 2)A,-2- mA,= (2m-2)2A Aml — (2m -2 Am-2 2 (2 )2 A,
2 2
or:
8) 2 m (2 rn - 1 ) A,, ==2 (2 n - 2)2 A-1 - (2 2i -3 ) (2m - 4) A,,-2.
Accordingly the coefficients of series 4) are determined by the equations:
A2 2 4A11 A 2 A 7 2 7 J 4AiZ-A
A = F(2) A {AF()- {2 -E( -2)} A2== — A -
A3 - 16A- 3A etc..
3 15 '
Similarly an explicit expression is got for the integral E(<q).
The third normal integral -TT() requires special investigations,
upon which we do not here enter since these series can in general be
replaced by more rapidly convergent developments, investigations that
demand a detailed theory of elliptic integrals.