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with a certain positive radius of convergence r called the
radius
of the analytic
element (f,c). The open disk of radius r about c will be called the
disk
of (f,c). Note
that the function f may be analytic over a larger region than its disk but the formula
(10.67) is valid only over that disk. How to extend the definition of f? Choose a
point d in the disk, |d - c| < r. The series in (10.67) can be rearranged into a new power
series
2
()
=+
(
)
+
(
)
fz b
bz d bz d
-
-
+ ...
(10.68)
2
0
1
2
about d. The radius of convergence of the series in (10.68) is at least r - |c - d|, but it
may be larger. If it is larger then since f and f
2
agree on an open set, they define an
analytic function with domain larger than that of f. In any case, the analytic element
(f
2
,d) is called a
direct analytic continuation
of the element (f,c). A point either in or
on the boundary of the disk for (f,c) which lies in the interior of the disk of a direct
analytic continuation of (f,c) is called a
point of continuability
of (f,c); otherwise, it is
called a
point of noncontinuability
or a
singular point
of (f,c). For example, one can
show that 0 is a singular point for the analytic elements (w
i
(z),1), i = 1, 2, where w
i
(z)
are the square root functions in (10.66).
If we repeat the process of continuation starting with f
2
, we shall get a
sequence of analytic functions f, f
2
, f
3
,.... See Figure 10.13. This general process of
extending functions by a sequence of direct analytic continuations is called
analytic continuation
. One can continue functions uniquely along curves, but if
two curves end up at the same point, the two functions we get at the end of the
curves by this process may not be the same. Weierstrass defined a
global analytic func-
tion
to be the totality of analytic elements that can be obtained by analytic continua-
tion from a given one. (Actually, Weierstrass did not use the adjective “global”. He also
dealt with holomorphic, that is, differentiable, functions. Although the terms “holo-
morphic” and “analytic” are often used interchangeably, “analytic” has a more general
connotation and can allow certain singularities.) One can prove that if two global ana-
lytic functions have an analytic element in common, then they are identical.
Given two analytic elements (f
1
,c
1
) and (f
2
,c
2
) belonging to the same global ana-
lytic function, one says that they determine the same
branch
at a point c that belongs
to the intersection of their disks if the functions f
1
and f
2
are identical in a neighbor-
hood of c. Determining the same branch at a point c is an equivalence relation on the
set of analytic elements that contain c in their disk. An equivalence class is called an
analytic branch at c
. The equivalence classes are in one-to-one correspondence with
power series in (z - c), which have a positive radius of convergence. For example, the
expansion of w
1
(z) about the point 1 is
Figure 10.13.
Analytic continuation.
