w 1 , respectively, so that each branch can be single valued. In other words, w(z) ¼ z
1/2
is a single-valued function that is defined on the whole Riemann surface.
There are a variety of Riemann surfaces according to the nature of the complex
functions. Another example of the multivalued function is expressed as
w z
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z À a
ð
Þ z À b
ð
Þ
p
,
where two branch points are located at z ¼ a and z ¼ b. As before, we assume that
z À a ¼ r a e
iθ a and z À b ¼ r b e
iθ b :
Then, we have
w θ a , θ b
ð
Þ¼z
1=2
¼
ffiffiffiffiffiffiffiffi
r a r b
p
e
i θ a þθ b
ð
Þ =2
:
ð6:239Þ
In the above, e.g., the argument θ a can be decided graphically as shown in
Fig. 6.26, where θ a is an angle between a line connecting z and a and another line
drawn in parallel with the real axis. We can choose w(θ a , θ b ) of (6.239) for the
principal branch and define this as
w 0 θ a , θ b
ð
Þ
ffiffiffiffiffiffiffiffi
r a r b
p
e
i θ a þθ b
ð
Þ =2
:
ð6:240Þ
Also, we define w 1 (θ a , θ b ) as
w 1 θ a , θ b
ð
Þ
ffiffiffiffiffiffiffiffi
r a r b
p
e
i θ a þθ b À2π
ð
Þ =2
:
ð6:241Þ
Then, as in (6.235) we have
w 1 θ a , θ b
ð
Þ¼w 0 θ a À 2π, θ b
ð
Þ¼w 0 θ a , θ b À 2π
ð
Þ¼À w 0 θ a , θ b
ð
Þ:
ð6:242Þ
−
−
0
Fig. 6.26 Graphically
decided argument θ a , which
is an angle between a line
connecting z and a and
another line drawn in
parallel with the real axis
254
6 Theory of Analytic Functions