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.

~ 77-79.

Multiplication of infinite series.

125

R. Sn~- Sn = (Q1 8Qn + Q2 Qn-l + Q n Q) + (Q2 n'+ Q3 Qn -1 - 1 - 9 2 Q ' S)
4 — (Q3 9n + 4Q Qf L-1 + * * * Qn Q3') + * * Qn Qn'
inasmuch as by the theorem just proved Lim (JRnRn- Sn) = 0, we
n= -
find, that the sum on the right also has zero as its limit.
This result enables us to prove the theorem regarding the product
in the following manner.
If Sn' be the sum of the first n +- 1 terms of series 5), we have:
PnQn -- Sn'= (p, qn + P2 n-1 + * * Pn c l)
+ (p2qn +p3n-1 + * * Pn 2)+ * * Pqn;
and it has to be shown, that the sum of the products on the right
has zero as its limit when n increases. This will be the case only
when the modulus of this complex expression converges to zero. Now
as the modulus of a sum of complex numbers is not greater than
the sum of the moduli of the summands, what we have stated is
proved, since
(Q1Qn'+ Q2Qfn-1+ Qn Ql )+(Q2Qn +3 n-1 + n* * n ) + *" QnQn
converges to zero.
Ve apply this theorem to the exponential function as defined ~ 73:
x x 2 x3 i y ' tJ2 I
e'ViiY~eXjcosY isinY)=l++ +
e27+l8=c~os,+t- l + 1 + |2 + -[3 +'')(' + Y + z W+**'
The product of the two infinite series is expressed by one infinite
series:
l X+iy x2 x y, ) i' i9 y X (iy)2 (iy)3,
1 _ y _ _ ____.x_ _ _ __ _ _ _ _ 1 [ -
1+ 1 2+ + 1 -i3 + 212 1 1 1 2 3
xn Xn-1 _ x- ( )2 -k 2 (iy1 k (iY)n
~\ j Inl- 1 -1 1i 2 ~ V- k Il / n
that can be contracted in the form:
+ x+iy [ (x+iy )2 (+iy)3 + i
1 + L --- + -+ * * * I, + *
This is the exponential series with a complex argument, it expresses the
simplest value of the exponential function ex+i. i
79. A quantity is called a complex variable when it is able to
assume different complex numterical values.
Whilst any set of real numbers can always be figured by the points
of a finite right line, a limited range of complex numbers is in
general presented to intuition by a "domain" of two dimensions of
the plane bounded by some curve. In each individual case it must be
specially assigned whether the points of the boundary curve themselves belong to the domain or not. Such a domain can in particular
cases reduce to a linear figure, a domain of one dimension: to the
points of a portion of a curve or of a finite right line.