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.

150

The vanishing values of a series of powers. Bk. II.'ch. II1l

therefore the bracketed factor vanishes. Hence follows as before:
`(,) = 0.
In like manner from the vanishing of the product:
2 ( i () (D)r ()++ )
it results that f""(z) vanishes, and similarly it is found that:
f"' (z) =0,... t/'( ) (-0,... etc..
Thus f(z + h) and all its derivates vanish for all values of z + h
that lie within the circle of convergence of series 1). In this circle a
point a' for which mod ' < mod z, being chosen as centre, gives
rise to an expansion the radius of whose convergency
H' = - - mod [z'] is > H.
The coefficients in this new expansion all vanish, i. e. the function
is zero everywhere in this greater circle also. Taking a point 2"
within this circle as centre, we obtain a new circle, and this process
can be continued till we reach a circle that includes the origin
z: O. Since for this point the function and all its derivates vanish,
the same is true for all points of the original circle of convergence
of the series:
f(z) = C0 + Ca1 + Ct 2 + * *. a + * *.
that is, we have
f'(O) = ao0 - 0 f'(0) =- a,= 0, f"(0)= a21 =0,.. f n (0) =an L =0, etc..
By means of this proposition we can generalise the Theorem
proved at the close of last Chapter concerning the unique expression
of a function by a series of powers. For, from it follows that:
When the values of two series of powers are the same even only
in infinitely many points of a domain, the series are identical throughout
the entire common part of their convergencies. For let these series be:
a +- a (a - a) + (a -- )2 +.. and bo +- b (z -/) + b2 ( — 13) +...,
then their difference can be expanded for a point y, within the domain
common to their circles of convergence, in a series of powers
CO + c1( - Y) + c2 ( - )2 +... etc..
This series vanishes in infinitely many points, accordingly it is zero
within its entire circle of convergence. But from this series we can
attain to any other point lying at a finite distance however small
inside the boundaries, by adopting a new point within its circle of
convergence as centre of an expansion, that in like manner must
vanish, and continuing this process.
88. Suppose a domain of convergence with the radius X is to
be investigated; since there is only a finite number of vanishing
points in each finite part of the plane, we can assume, that none of