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Multiple-valued functions, branch points, branch lines, Riemann surfaces

Multiple-valued functions, branch points, branch lines

Example 1. Suppose
we are given the function
w = z^{1/2} and ask
ourselves how this
function will map points
taken along curve C
which encircles the
origin as shown in Fig. 1. We pick points along the curve and let z make a complete circuit
(counterclockwise) around the origin. For insight into what is happening we express z as z = re^{iθ}.
The image of z in the w plane is then given by w =
e^{iθ/2}. Thus each point P(r, θ) is mapped
into its image P'(
, θ/2). Point P_{1 }(r_{1}, θ_{1}) is imaged into P_{1}'(
, θ_{1}/2). P_{2}, P_{3} and P_{4} are imaged
into points P_{2}', P_{3}' and P_{4}' as shown in the figure. Each point P is mapped into its image P' in the
w plane where P' has half the amplitude of P. We make a complete circuit around curve C and
arrive back at point P_{1} again. The amplitude of P_{1} is now θ = θ_{1} + 2π. Point P_{1}(r_{1}, θ_{1} + 2π) is
imaged into point
(r_{1}, θ_{1}/2 + π) shown in the figure in the third quadrant of the w plane, where
we are denoting the images of points from the second circuit around C with an over-bar. As we
continue points P_{2}, P_{3}
and P_{4} map into points
and
as
shown in the figure.
Finally we arrive at
point P_{1} for the third
time and the amplitude
is now θ = θ_{1} + 4π.
This time point P_{1} maps
into the same point that
it did on the first trip
through, point P_{1}', and
the process starts over
again.

We can describe the above process by saying that if 0
θ < 2π we are on one branch of
multiple-valued function z^{1/2}, while if 2π
θ < 4π we are on the other branch of the function.

The above phenomenon in which a point P on C maps into two points on C' occurs because curve
C encircles the origin. If C did not encircle the origin it wouldn’t happen. A point on C would
then map into a single point on C'. Let us consider this case. See Fig. 2. In Fig. 2 we see that the
amplitude of z varies between θ_{1} and θ_{2}. The amplitude of w will then vary between θ_{1}/2 and θ_{2}
/2. Point P_{1} will be mapped only into point P_{1}', P_{2} will be mapped only into point P_{2}', etc. The
mapping is different and one-to-one.

Example 2. Let us now
consider how the function
w = (z - a)^{1/2} maps a
curve C that encircles
point a. See Fig. 3.
is
the position vector to
point z on the curve.
is
the position vector to
point a. As point z moves
counterclockwise around
curve C the vector
,
extending from point a to
z, winds around point a.
On the first trip around
the curve, point P_{1} is
mapped into point P_{1}'. Assume the amplitude of the vector
is θ_{1} at P_{1}. Note that arg w =
arg (z - a) so the amplitude of w will be θ_{1} /2. When z has made a complete circuit and arrives
at P_{1} the second time the amplitude of
will be θ_{1} + 2π and the amplitude of w will be θ_{1}/2
+ π. P_{1} will map into
. When z proceeds on around the circuit again and arrives at P_{1} for the
third time, the amplitude of
will be 4π, the amplitude of w will be θ_{1} /2 + 2π, and P_{1} will
map into P_{1}' again.

Branch point. If different values of a function f(z) are obtained by successively encircling
some point z_{0} in the complex plane, as occurred in examples 1 and 2 above, then the point z_{0} is
called a branch point. In Example 1 the origin O is a branch point and in Example 2, the point a
is a branch point.

A branch point represents a singularity of a multi-valued function, however, it has a different character from the points ordinarily called singular points.

Example 3. f(z) = (z - 2)^{1/3 } has a branch point at z = 2.

Example 4. f(z) = ln (z^{2} + z - 6) has branch points where z^{2} + z - 6 = 0, i.e. at z = 2 and z =
-3.

Branch line. A branch line is a line extending out from a branch point defining a boundary between branches. When a variable z crosses a branch line the function f(z) switches from one branch to another. The heavy line OX in Fig. 1 extending from the branch point O to infinity is a branch line.

Riemann surface. Let z_{0} be any complex number. How many complex numbers w will
satisfy the equation w^{5} = z_{0}? Answer: There are five different complex numbers that will satisfy
this equation. If we pick any number z_{0} in the complex plane there are five different fifth roots of
z_{0}. Thus we say that the function w = z^{1/5} is a 5-value function. Five different numbers satisfy it.
The function w = z^{1/5} associates with each point z in the complex plane five different numbers.
Let us designate these five numbers as s_{1}, s_{2}, ... , s_{5}. We can now conceive of five “sheets of
values” overlaying the z plane, one stacked on top of another, going from Sheet 1 up to Sheet 5.
The first sheet consists of the s_{1} numbers for all (x, y) number pairs in the plane i.e. all points of
the plane. The second sheet consists of the s_{2} numbers for all (x, y) number pairs in the plane.
Etc. for all five sheets. A function w = z^{1/2} has two sheets associated with it. A function w = z^{1/8}
has eight sheets associated with it. The five sheets associated with the 5-value function w = z^{1/5}
represent the five branches of the function. They represent the five sheets of a 5-sheet Riemann
surface. The Riemann surface consists of a collection of five sheets where each sheet represents
a single-valued function. The sheets in a Riemann surface are conceived of as connected to one
another. To understand how they are connected consider the case of a 2-sheet Riemann surface
where the branch point is the origin O as in Example 1 above. Imagine the two sheets, Sheet 1
and Sheet 2, overlaying the z plane. Now cut both sheets along OX and imagine that the lower
edge of the bottom sheet is joined to the upper edge of the top sheet. Then starting in the bottom
sheet and making one complete circuit around the branch point we arrive in the top sheet when
we cross the branch line (or branch cut) OX. We must now imagine the other cut edges joined
together so that by continuing the circuit we go from the top sheet back to the bottom sheet.

References

Spiegel. Complex Variables (Schaum)

Hauser. Complex Variables with Physical Applications

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