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.

The integral of a series of powers.

337

analytic function and presents itself as the difference between the
values for the arguments Z and z0 of the series of powers that arises
on integrating the given series term by term:
z
Jf()d = (Z + (Z -- )) + 2 (
- ).t>(o - ") + (a (( )2 a ) + }2..
As a particular case we have:
z
J(C) dsa ((Z- a)+ _a (Z-)2 -(Z-... (Z- a)"+, + etc..
(C
The circle of convergence of this new series can be neither smaller
nor larger than that of the original. It is however possible that
this second series may still converge (conditionally or unconditionally)
upon the circle itself, while the original series diverges in the points of
the circumference. In this case the series expresses the integral of
the function f(s) for the point on the circle of convergence also.
For, when Z denotes such a point, we have:
z Z- -
f (t ) ( - = Limj}(z)c d
a a
Lim ao (Z —a) + a1(Z -- )2+ 2 (Z-8- a)3-
JF=o 22
Since a series of powers remains continuous provided it converges in
the points of the limiting circle (~ 83), this right hand limiting value
passes over continuously into the series:
ao(Z - a) + a (Z - a)2 + a2 (Z - a)3 +.. + etc.
When the infinite plane is the convergency of the series, the
value of the integral is expressed by the series of powers for values
of Z however great in amount. For the point Z-= o, the integral,
meaning thereby the limiting value along a determinate path, may
possibly remain finite, only its value can no longer be expressed by
a series of powers.
This is exemplified in the case of the integral Jezdz, whose
value for every finite value of Z is:
z z
0 0
Z2 ZC + Z n +.
=z+ 12 1+-2+... c13 —;

HARNACK, Calculus.

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