1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.3 • HARMONIC FUNCTIONS 117

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  • Figure 3.4 The vector field V (x, y) = p (x, y) + iq (x , y), which can be considered as a
    fluid flow.
  
  
  
  

the same. Situations such as this occur when .fluid is flowing in a deep channel.

The velocity vector at the point (x, y) is

V {x,y) =p(x,y) +iq(x,y), (3-29)

which we illustrate in Figure 3.4.
The assumption that the flow is irrotational and has no sources or sinks
implies that both the curl and divergence vanish; that is, Qx - p 11 = 0 and

Px + Qy = 0. Hence p and q obey the equations

p., (x, y) = -qy (x, y) and p 11 (x, y) = q., (x, y). {3-30)

Equations {3-30) are similar to the Cauchy- Riemann equations and permit

us to define a special complex function:

f (z) = u (x,y) +iv (x,y) = p(x,y) -iq (x, y). (3-31)

Here we have u., = Px, Uy =Pin Vx = -q.,, and v 11 = -q 11 • We can use

Equations (3-30) to verify that the Cauchy- Riemann equations hold for/:

u., (x, y) = Px (x,y) = -q 11 (x, y) =Vy (x,y) and

Uy (x, y) = p 11 (x, y) = q., (x, y) = -v., (x, y).

Assuming that the functions p and q have continuous partials, Theorem 3.4
guarantees that function f defined in Equation {3-31) is analytic a nd that t he
fluid flow of Equation (3-29) is the conjugate of an a nalytic function; that is,

V (x,y) = f (z).

In Chapter 6 we prove that every analytic function f has an analytic an-

tiderivative F; assuming this to be the case, we can write

F(z) = ¢(x, y) + i'l/J(x,y), {3-32)

where F' (z) = f (z).