1550251515-Classical_Complex_Analysis__Gonzalez_

298

y z

A

A

B

B

x

Fig. 5.42

v

--..._

' _ frrri

-5'1Ti

I

+Log z + 4'1Ti

I

~jLog z + 2'1Ti

I

Chapter 5

we need countably many replicas of the cut z-plane (denoted Gk, k = O,

±1, ±2, ... ) with each Gk on top of Gk-1' and the lower edge of the cut

in Gk glued (identified) to the upper edge of the cut in Gk-1·
The connection between the sheets of the Riemann surface for w = log z
is shown in Fig. 5.43 on a section made across the negative real axis (as
seen from the origin). The Riemann surface of the logarithmic function has
two boundary points, z = 0 and z = oo, which do not belong to the surface
since log z is not defined at those points. However, a circuit about either
point cause~ a branch Wk to turn into another: they are called logarithmic
branch points and are regarded as branch points of infinite order. By

G2

G, A

Go B A

G_, B

G_2--

L u

Fig. 5.43