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.

Second Chapter.
Complex series. Complex variable. Functions of a complex variable.
76. By the sum of an-infinite series, whose terms are complex:
Z(U + iv) ) +( ) (u1 + iv1) +.* * (un + ivn).
is to be understood the complex number U -- iV whose real part U
is equal to the sum of the infinite series u0 + ul - * * on - -*, and
its imaginary part iV to the sum of the series i (vo -- v + * vn ).
The complex series has therefore a determinate value and is said to
be convergent, only when both U and V have determinate finite
limiting values, that is to say, when both the series z-z, — + U. Un
and v - + vi + v * * converge.
Addition of two infinite series.
If 1) po +-t - p - p - + + p and 2) qo +- q+ -2.. +.-..
be two convergent infinite series having complex terms
(Pn = Un + V, n n qn= un + ivn')
and their sums respectively P and Q, the series
3) (po + qo) + (p, + q1) + (p2 + q2) + *... (Pn + n)
is convergent and its sum is P + Q. For, putting:
Pn = Po + Pi + * * -,7 Qn= Q o=q + q1 + 'qn
as n increases, Pn and Qn approximate to the limits P and Q, hence
(PO + qo) + (PI d- q,) + (- - -( + qn) -I P + Qn
the sum of the n + 1 first terms of series 3) has, as n increases,
the limiting value P + Q.
77. The complex series is said to be absolutely convergent, when
the sum of the positive terms of uo - u1 +- 2... and of ov + v1 + v2..-,
and likewise the sum of the negative terms of each, have finite
limiting values, or as this property is described: when the series u and
the series v are absolutely convergent.*) When each series consists
only of terms with one and the same sign, the only conception we
~) The conception of absolute convergence was introduced by Cauchy;
Dirichlet (1805-1869) noticed the contrast with infinite series in which the limit
of the sum depends on the order of the terms: Abhandlungen der Berliner Akademie, 1837.