Complex Analysis - Maths KU

Note
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5.6 Applications 289
If you have not had a course in differential equations, then the foregoing
manipulations may not impress you. But those readers with some knowledge
of that subject should recognize the result in (12) establishes that (11) is
equivalent to the exact differential d(ψ(x, y)) = 0. Integrating this last
equation shows that all solutions of (5) satisfy ψ(x, y) =c2. In other words,
the streamlines of the velocity field F(x, y) =P (x, y)i + Q(x, y)j obtained
from (5) are the same as the level curves of the harmonic conjugate ψ of φ
in (6).
Complex Potential Revisited Assume that F(x, y) =P (x, y)i
+ Q(x, y)j is the velocity field of the flow of an ideal fluid in some domain
D of the plane and that Ω(z) =φ(x, y) +iψ(x, y) is the complex velocity
potential of the flow. We know fromTheorem5.17(i) that fromthe complex
representation f(z) =P (x, y)+iQ(x, y) ofF we can construct an analytic
function g(z) =f(z) =P (x, y) − iQ(x, y). The two analytic functions g and
Ω are related. To see why this is so, we first write the gradient vector F in
(7) in equivalent complex notation as
f(z) = ∂φ
∂x
+ i∂φ.
(13)
∂y
Now by (9) of Section 3.2, the derivative of the analytic function Ω(z) =
φ(x, y)+iψ(x, y) is the analytic function
Ω ′ (z) = ∂φ
∂x
+ i∂ψ.
(14)
∂x
We now replace ∂ψ/∂x in (14) using the second of the Cauchy-Riemann equations
in (9):
Ω ′ (z) = ∂φ
∂x
− i∂φ.
(15)
∂y
By comparing (13) and (15) we see immediately that f(z) = Ω ′ (z) and,
consequently,
g(z) =Ω ′ (z). (16)
The conjugate of this analytic function, g(z) =f(z) =f(z), is the complex
representation of the vector field F whose complex potential is Ω(z). In
symbols,
f(z) =Ω ′ (z). (17)
Because f(z) is a complex representation of velocity vector field, the quantity
Ω ′ (z) in (17) is sometimes referred to as the complex velocity.
You may legitimately ask: Is (17) merely interesting or is it useful? Answer:
Useful. Here is one practical observation: Any function analytic in some
domain D can be regarded as a complex potential for the planar flow of an
ideal fluid.