Complex Analysis - Maths KU

5.6 Applications 287
Indeed, by rewriting f as
f(z) = K
�
x − x0
2πi (x − x0) 2 2 + i
+(y − y0)
y − y0
(x − x0) 2 +(y − y0) 2
and using z0 = x0 + iy0 and z = x + iy, you should be able to recognize f(z)
is the same as
f(z) = K z − z0
2πi |z − z0| 2 or f(z) = K 1
.
2πi ¯z − ¯z0
Hence fromTheorem5.17(i), the complex function g(z) defined as the conjugate
of f(z) is a rational function,
g(z) =f(z) =− K 1
,K>0,
2πi z − z0
and is analytic in a domain D of the z-plane not containing z0.
Any analytic function g(z) can be interpreted as a complex representation
of the velocity field F of a planar fluid flow. But in view of Theorem5.17(ii), it
is the function f defined as the conjugate of g, f(z) =g(z), that is a complex
representation of a velocity field F(x, y) =P (x, y)i + Q(x, y)j of the planar
flow of an ideal fluid in some domain D of the plane.
EXAMPLE 2 Analytic Function Gives a Vector Field
The polynomial function g(z) =kz = k(x + iy),k > 0, is analytic in any
domain D of the complex plane. From Theorem 5.17(ii), f(z) =g(z) =k¯z =
kx − iky is the complex representation of a velocity field F of an ideal fluid
in D. With the identifications P (x, y) =kx and Q(x, y) =−ky, we have
F(x, y) =k(xi − yj). A quick inspection of (2) verifies that div F = 0 and
curl F = 0.
Streamlines Revisited We can now tie up a few loose ends between
Section 2.7 and Section 3.4. In Section 2.7, we saw that if F(x, y) =
P (x, y)i + Q(x, y)j or f(z) =P (x, y) +iQ(x, y) represented the velocity
field of any planar fluid flow, then the actual path z(t) =x(t) +iy(t) ofa
particle (such as a small cork) placed in the flow must satisfy the system of
first-order differential equations:
dx
= P (x, y)
dt
dy
= Q(x, y).
dt
The family of all solutions of (5) were called streamlines of the flow.
�
(5)