1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

ll8 CHAPTER 3 • ANALYTIC ANO HARMONIC FUNCTIONS

Theorem 3.8 implies that </> (x, y) is a harmonic function. Using the vector
interpretation of a complex number, the gradient of</> can be written as

grad</> (x, y) = </>:r: (x, y) + i</>y (x, y).

The Cauchy-Riemann equations applied to F (z) give </>y (x, y) = -1/J., (x, y);

ma.king this substitution in the preceding equation yields

grad </>(x,y) = </>,,(x,y)-i't/J., (x ,y) = </>,,(x,y) +i't/J., (x,y).

Equation (3-14) says that </>:r (x, y) + i't/J:r; (x, y) = F' (z), which by the pre-
ceding equation and Equation (3-32) implies that

grad</> (x,y) = F' (z) = f (z).

Finally, from Equation (3-29), </> is the scalar potential function for the fluid

flow, so

V(x,y) = grad <f>(x,y).

The curves given by { ( x, y) : </> ( x, y) = constant} are called equipotentials.
The curves {(x,y): 1/J(x,y) =constant} are called streamlines and describe the
path of fluid flow. In Chapter 10 we show that the family of equipotentials is
orthogonal to the family of streamlines, as depicted in Figure 3.5.

• EXAMPLE 3. 14 Show that the harmonic function (x,y) = x^2 - y^2 is the
  scalar potential function for the fluid flow expression V (x, y) = 2x - i2y.

Solution We can write the fluid flow expression as

V (x, y) = f (z) = 2x + i2y = 2z.

An antiderivative off (z) = 2z is F (z) = z^2 , and the real part of F (z) is the

desired harmonic function:

(x,y) = Re[F(z)) = R.e [x^2 -y^2 + i2xy] = x^2 -y^2 •

Note that the hyperbolas <P (x,y) = x^2 -y^2 =Care the equipotential curves

and that the hyperbolas 1/J (x, y) = 2xy = C are the streamline curves; these

curves are orthogonal, as shown in Figure 3.6.