the circle (or contour) may not cross the branch cut. Although a bit complicated, the
Riemann surface can be shaped accordingly. Other choices of the branch cuts are
shown in Sect. 6.9.2 (vide infra).
When we consider logarithm functions, we have to deal with rather complicated
situation. For polar coordinate representation, we have z ¼ re
iθ . That is,
ln z ¼ ln r þ i arg z ¼ ln z
j j þ i arg z,
ð6:244Þ
where
arg z ¼ θ þ 2πn n ¼ 0, Æ1, Æ2, Á Á Á
ð
Þ :
ð6:245Þ
If n ¼ 0 is chosen, lnz is said to be the principal value. With the logarithm
functions, the Riemann surface comprises infinite planes each of which corresponds
to the individual branch whose argument is given by (6.245). The point z ¼ 0 is
called a logarithmic branch point. Meanwhile, the branch point for, e.g., w z
ð Þ ¼
ffiffi
z
p
is said to be an algebraic branch point.
6.9.2 Examples of Multivalued Functions
When we deal with a function having a branch point, we must remind that unless the
contour crosses the branch cut (either algebraic or logarithmic), a function we are
thinking of is held single valued and analytic (if that function is originally analytic)
and can be differentiated or integrated in a normal way. We give some examples
for this.
0
(a)
0
(b)
Fig. 6.28 Branch cut(s) for
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z À a
ð
Þ z À b
ð
Þ
p
. (a) A branch cut is shown by a line connecting z ¼ a
and z ¼ b. (b) Branch cuts are shown by a line connecting z ¼ a and z ¼ 1 and another
line connecting z ¼ b and z ¼ 1. In both (a, b) the branch cut(s) are depicted with doubled broken
line(s)
256
6 Theory of Analytic Functions
Riemann surface can be shaped accordingly. Other choices of the branch cuts are
shown in Sect. 6.9.2 (vide infra).
When we consider logarithm functions, we have to deal with rather complicated
situation. For polar coordinate representation, we have z ¼ re
iθ . That is,
ln z ¼ ln r þ i arg z ¼ ln z
j j þ i arg z,
ð6:244Þ
where
arg z ¼ θ þ 2πn n ¼ 0, Æ1, Æ2, Á Á Á
ð
Þ :
ð6:245Þ
If n ¼ 0 is chosen, lnz is said to be the principal value. With the logarithm
functions, the Riemann surface comprises infinite planes each of which corresponds
to the individual branch whose argument is given by (6.245). The point z ¼ 0 is
called a logarithmic branch point. Meanwhile, the branch point for, e.g., w z
ð Þ ¼
ffiffi
z
p
is said to be an algebraic branch point.
6.9.2 Examples of Multivalued Functions
When we deal with a function having a branch point, we must remind that unless the
contour crosses the branch cut (either algebraic or logarithmic), a function we are
thinking of is held single valued and analytic (if that function is originally analytic)
and can be differentiated or integrated in a normal way. We give some examples
for this.
0
(a)
0
(b)
Fig. 6.28 Branch cut(s) for
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z À a
ð
Þ z À b
ð
Þ
p
. (a) A branch cut is shown by a line connecting z ¼ a
and z ¼ b. (b) Branch cuts are shown by a line connecting z ¼ a and z ¼ 1 and another
line connecting z ¼ b and z ¼ 1. In both (a, b) the branch cut(s) are depicted with doubled broken
line(s)
256
6 Theory of Analytic Functions