1550251515-Classical_Complex_Analysis__Gonzalez_

514 Chapter^7

and sources or sinks correspond to isolated singular points of the complex
potential F( z ).

If there are isolated vortices, sources, or sinks in the flow region D, let Do

be the multiply connected subregion obtained by deleting those points from
D. Assuming that the flow is irrotational and solenoidal in every simply

connected subdomain of Do, if follows that w = u - iv is a single-valued

analytic function i:n Do. However, the complex potential

F(z)= r(u-iv)dz

jzo

is in general a multiple-valued function in Do with single-valued analytic
branches on every simply connected subdomain of D 0 • All these branches
have the same derivative, namely, u -iv. Again, the vortices and sources or
sinks correspond to isolated singular points of F (some of which are possibly
branch points). Conversely._ any mu~tiple-valued function F with isolated
singular points in a domain D can be regarded as the complex potential
of a flow in D 0 = D - S (S being the set of its singular points), which
is irrotati.onal and solenoidal in every simply connected subdomain of D 0 ,
provided that F has single-valued analytic branches on such subdomains.

Example Consider the multiple-valued function

m

F( z) =

2

7r log z

defined in the domain D = { z: 0 < /z / < oo}, m -::/:- 0 being a real constant.
This function has the single-valued derivative

F'(z) = m ~

27r z

so it may be taken as the complex potential of a steady plane-parallel flow
of an ideal fluid. The real velocity potential is given by

m

U(x, y) =

2

7r ln /z/

and the stream function by

m

V(x, y) = - argz

27r

The equipotential lines

m

• ln /z/ = const.
  27r

or /z/ = const.

are circles with center at the origin, and the streamlines

arg z = const.