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.

336

The definite integral of a unique analytic function. Bk. IV. ch. I.

For a point of the second quadrant let us first proceed from
x= 1 parallel to ordinates to the point 1 - iyo, thence parallel to
abscisse to the point xo - iyo and lastly again parallel to ordinates
to x0 + iy0; thus:
z X0-iyo Xo Xo+io Xo- i o
adz ju( ds 2-d I (7o 2 2) + yl itan N - iz +, -
1 1 xo — Yo. Xo-z/o
But the last integral is equal to:
idy x - _ iy) dy ( - 2i tan- 1 y
x+iy J xo+ xo2
Xo- iYo -Yo
therefore for the points of the second quadrant:
z
J - = =2 + o2) + itan- _. -
Along the right side of the positive axis of ordinates the values of the
integral are - 1 (yo2) + -l i z, but along the left they are 1 (y2) -. i;
they differ by 2i7r.
3. The integral of an infinite series of powers.
Within the circle round the centre a, in which the complex series
of ascending positive integer powers:
ao + ac1 - a) + aj( - a)2 + ~ an ( - )n +..
converges, it expresses a unique and continuous function f(z) without
singular points; for every point z = Z within the circle of convergence
the series converges absolutely; and therefore it also converges
uniformly; i.e. a value of n can be assigned from which onwards
every remainder is of smaller amount than an arbitrarily small number
(, for every value of z such that abs [ - a] <abs[Z —a]. For,
abs [an (Z- a)n + a,+, (Z- a)n+l +..]< AAn En + An+ Rnl+1 +.*,
when A is written for abs a, and PR for abs[Z- a]. But since the
series:
Ao + AIR- + A2B2 +..
converges, a value can be assigned to n, for which the right side of the
above inequality will be always smaller than 6, a fortiori we shall have:
abs [an (z - a)n + anl ( -- oa)n+l.* ] < An r + An+i rn+' +. < 6,
when abs [z - a] = r < R.
From this results the following theorem:
When the complex series of powers is integrated from the point zo
up to the point Z, that both lie within its convergency, its integral is an