**for**

**2.4 Branches of
Functions**

In Section 2.2 we defined the principal square root function and investigated some of its properties. We left unanswered some questions concerning the choices of square roots. We now look at these questions because they are similar to situations involving other elementary functions.

In our definition of a function in Section 2.1, we specified that each value of the independent variable in the domain is mapped onto one and only one value in the range. As a result, we often talk about a single-valued function, which emphasizes the "only one" part of the definition and allows us to distinguish such functions from multiple-valued functions, which we now introduce.

Let
denote a function whose domain is the set D
and whose range is the set R. If w is
a value in the range, then there is an associated inverse function
that assigns to each value w the
value (or values) of z in
D for which the equation
holds. But unless f takes
on the value w at most once in D,
then the inverse function g is
necessarily many valued, and se say that g
is a __multivalued
function__. For example, the inverse of the
function
is the square root function . For
each value z other than ,
then, the two points z and
-z are mapped onto the same point
;
hence g is in general a two-valued
function.

The study of limits, continuity, and derivatives loses all meaning if an arbitrary or ambiguous assignment of function values is made. For this reason, in Section 2.3 we did not allow multivalued functions to be considered when we defined these concepts. When working with inverse functions, you have to specify carefully one of the many possible inverse values when constructing an inverse function, as when you determine implicit functions in calculus. If the values of a function f are determined by an equation that they satisfy rather than by an explicit formula, then we say that the function is defined implicitly or that f is an implicit function. In the theory of complex variables we present a similar concept.

Let
be a multiple-valued function. A branch of f
is any single-valued function
that is continuous in some domain (except, perhaps, on the
boundary). At each point z
in the domain, assigns one of the values of . Associated
with the branch of a function is the __branch
cut__.

We now investigate the branches of the
__square
root__ function.

**Example 2.20.** We
consider some branches of the two-valued square root
function , (where
). Define
the principal square root function as

(2-28) ,

where and
so that . The
function
is a branch of . Using
the same notation, we can find other branches of the square root
function. For example, if we let

(2-29) ,

then

so
can be thought of as "plus" and "minus" square root
functions. The negative real axis is called a branch cut
for the functions . Each
point on the branch cut is a point of discontinuity for both
functions .

**Example 2.21.** Show
that the function is
discontinuous along the negative real axis.

Solution. Let denote
a negative real number. We compute the limit as
z approaches
through the upper half-plane
and the limit as z approaches
through the lower half-plane
. In polar coordinates these limits are given
by

, and

.

As the two limits are distinct, the function
is discontinuous at .

**Remark
2.4** Likewise,
is discontinuous at . The
mappings , , and
the branch cut are illustrated in Figure 2.18.

** (a)** The
branch (where ).

** (b)** The
branch (where ).

** Figure
2.18** The branches and of .

We can construct other branches of the
square root function by specifying that an argument of z given
by is
to lie in the interval . The
corresponding branch, denoted ,
is

(2-30) ,

where and .

The branch cut for is the ray , which includes the origin. The point , common to all branch cuts for the multivalued square root function, is called a branch point. The mapping and its branch cut are illustrated in Figure 2.19.

** Figure
2.19** The branch of .

**The Riemann Surface
for**

A __Riemann
surface__ is a construct useful for visualizing a
multivalued function. It was introduced by __Georg
Friedrich Bernhard Riemann__ (1826-1866) in
1851. The idea is ingenious - a geometric construction
that permits surfaces to be the domain or range of a multivalued
function. Riemann surfaces depend on the function being
investigated. We now give a nontechnical formulation of
the Riemann surface for the multivalued square root function.

** Figure
2.A** A graphical view of the Riemann surface
for .

Consider , which has two values for any . Each function in Figure 2.18 is single-valued on the domain formed by cutting the z plane along the negative x axis. Let and be the domains of , respectively. The range set for is the set consisting of the right half-plane, and the positive v axis; the range set for is the set consisting of the left half-plane and the negative v axis. The sets are "glued together" along the positive v axis and the negative v axis to form the w plane with the origin deleted.

We stack directly above . The edge of in the upper half-plane is joined to the edge of in the lower half-plane, and the edge of in the lower half-plane is joined to the edge of in the upper half-plane. When these domains are glued together in this manner, they form R, which is a Riemann surface domain for the mapping . The portions of that lie in are shown in Figure 2.20.

** (a)** A
portion of and
its image under .

** (b)** A
portion of and
its image under .

** (c)** A
portion of R and its image
under .

** Figure
2.20** Formation of the Riemann surface
for .

The beauty of this structure is that it makes this "full square root function" continuous for all . Normally, the principal square root function would be discontinuous along the negative real axis, as points near but above that axis would get mapped to points close to , and points near but below the axis would get mapped to points close to .

**Exercises
for Section 2.4. Branches of Functions**

__Graphics
for Complex Functions__

**The Next Module
is**

**The
Reciprocal Transformation w=1/z**

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell