1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 515

y

arg z = c'

x

Fig. 7.32

are rays emanating from the origin (Fig. 7.32). The velocity at each point

z E D is given by

-- m 1 m

w = F'(z) = - :: = --z

271" z 27rlzl^2

so that

mx my

v= -----

27r(x2 + y2)

U= -----

27r(x2 + y2)'

Hence the flux integral for any simple closed curve about the origin has
the value

J

uy-vx=-do d m J x dy - y dx =m

271" x2 + y2

c+ c+

so the origin is a source if m > 0, a sink if m < 0. If m > 0, the velocity is

directed along the ray arg z = const. from the origin toward infinity. The
speed is very large near the origin and small far from the origin. Since

J u dy - v dx = -m

e-

we must regard oo as a sink. Thus we see that the branch points of
log z, namely, 0 and oo, correspond to a source and a sink, respectively.
Obviously, if m < 0, the roles of 0 and oo are interchanged.

When the source or sink is located at some point a f. 0, the potential is

m

F(z) =

2

71" log(z - a)