Page 136 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 12: Fundamental Theorem of Algebra

can be shown to have a complex analogue (in which values of the multi-functions involved have Notes
to be appropriately selected) but the formula
x x = (x x ) , x , x , a  R
a
a
2
2
1
2
1
2
1
has no universally complex generalization.
Branches, Branch Points and Branch Cuts
We recall that a multifunction w defined on a set S   is an assignment to each z  S of a set
[w(z)] of complex numbers. Our main aim is that given a multifunction w defined on S, can we
select, for each z  S, a point f(z) in [w(z)] so that f(z) is analytic in an open subset G of S, where G
is to be chosen as large as possible? If we are to do this, then f(z) must vary continuously with
z in G, since an analytic function is necessarily continuous.
Suppose w is defined in some punctured disc D having centre a and radius R i.e. 0 < |z - a| < R
and that f(z)  [w(z)] is chosen so that f(z) is at least continuous on the circle g with centre a and
radius r (0 < r < R). As z traces out the circle g starting from, say z , f(z) varies continuously, but
0
must be restored to its original value f(z ) when z completes its circuit, since f(z) is, by hypothesis,
0
single valued. Notice also that if z – a = r e i(z) , where (z) is chosen to vary continuously with z,
then (z) increases by 2p as z performs its circuit, so that (z) is not restored to its original value.
The same phenomenon does not occur if z moves round a circle in the punctured disc D not
containing a, in this case (z) does return to its original value. More generally, our discussion
suggests that if we are to extract an analytic function from a multi-function w, we shall meet to
restrict to a set in which it is impossible to encircle, one at a time, points a such that the definition
of [w(z)] involves the argument of (z-a). In some cases, encircling several of these ‘bad’ points
simultaneously may be allowable.
A branch of a multiple-valued function f(z) defined on S   is any single-valued function F(z)
which is analytic in some domain D  S at each point of which the value F(z) is one of the values
of f(z). The requirement of analyticity, of course, prevents F(z) from taking on a random selection
of the values of f(z).
A branch cut is a portion of a line or curve that is introduced in order to define a branch F(z) of
a multiple-valued function f(z).
A multivalued function f(z) defined on S   is said to have a branch point at z when z describes
0
an arbitrary small circle about z , then for every branch F(z) of f(z), F(z) does not return to its
0
original value. Points on the branch cut for F(z) are singular points of F(z) and any point that is
common to all branch cuts of f(z) is called a branch point. For example, let us consider the
logarithmic function
log z = log r + iq = log |z| + i arg z (1)

If we let  denote any real number and restrict the values of q in (1) to the interval  <  <  + 2,
then the function
log z = log r + i (r > 0,  <  <  + 2) (2)

with component functions
u(r, ) = log r and v(r, ) =  (3)
is single-valued, continuous and analytic function. Thus for each fixed a, the function (2) is a
branch of the function (1). We note that if the function (2) were to be defined on the ray  = a, it
would not be continuous there. For, if z is any point on that ray, there are points arbitrarily close
to z at which the values of v are near to a and also points such that the values of v are near to

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