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.

346 Expansion of analytic functions in series of powers.

Bk. IV. ch. I-.

187. If we know that a function loses the property of a unique
analytic function at certain points c, c,... of the plane, and we
desire to expand this function in a series of powers in the neighbourhood
of a point a, the circle of convergence may only be so large as to pass
through one or more of the points c, but must include none of them.
For, the series of powers is a unique analytic function in the entire
circle, while by hypothesis the function loses this property in each point c.
As soon therefore as we know the properties of a function, we
are in a position to foretell the extent of the convergencies of its
elements. This we proceed to illustrate by the functions already studied.
I. It is known regarding the exponential function
e~+iY = ex (cos y + i sin y),
that it is analytic in the entire plane, for every finite value of x + iy.
There exists round each point a an expansion with an arbitrarily great
convergency, the series is:
e~ - a (z - a)2 +
e g =l(l+. — +....
II. The function i(z), which we have defined as the inverse of the
exponential function, is a many-valued function of z. Its values can be
uniquely coordinated to the points of a winding surface with infinitely
many leaves that are connected along branching sections from the
point 0 to oc. When we describe with a centre a in any leaf a
circle including the point zero, the logarithm is not a unique analytic
function within this -domain; moreover the domain itself is not
closed according to the idea we formed of the winding surface, for
its boundary circle crosses the branching section an odd number of
times and does not lead back to the original point, but into a
different leaf. But when we describe round a as centre a circle that
at the utmost passes through the branching point z = 0, the function
is everywhere analytic within this circle, and the circle itself is closed,
although parts of it may possibly be in different leaves. The function
l(z) must therefore admit of expansion by powers of ( - a), and
I -1 12 -L e
because its successive derived functions are -, - et,,.,
the series is, for abs [z - a] < abs a:
()-(a ) -a- ( k-)' (z-a)a (z-a)t
I ()2a' + a- 4+ a' - +... etc..
This series expresses that value of the logarithm into which the value
assumed for 1(a) changes continuously along any path within the
circle of convergence. When we extend the function 1(z) out beyond
this circle of convergence, by adopting some new point a' within it
as centre of the expansion:
I(Z)-] IC) \ + Z-a (i ) (-a'y (~-0)3a ' '<
1(z) --- + =-I) - l 7 -- a 3 ~ abs [z - a(] < abs a',
a' 2 a'2 + 3a' a — 3'< a