1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

462 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

¢'=- 120 ?=-60 ?=0 ?=60 ?=120

Figure 11.37 The electric field produced by two charged half-planes that are perpen-

dicular to the complex plane.

is mapped onto the negative iv.axis and C 2 is mapped onto the positive u-axis.

The potential <l> ( u, v) in the upper half-plane that satisfies the new boundary

values

(u, O) = 80, for u < 0 and (u , 0) = 0, for u > 0,

is given by

80 80 v

\I> (u, v) = - Argw = -Arctan-.

7r 7r u

(11-29)

A straightforward calculation shows that

. (x - 1)
2

• (y - 1)
  2
  
  
  • 1 + i (1 -x^2 - y^2 )
    u+iv=S(z)= 2
    (x - 1) + y2
  
  
  
  

We substitute the real and imaginary parts, u and v from this equation, into
E.quation (11-29) to obtain the desired solution:

80 1-x^2 -y2

<f>(x, y) = - Arctan 2 2 ·

7r (x - 1) + (y -1) - 1

T he level curve \I> ( u, v) = a in the upper half-plane is a ray emanating from the

origin, and the preimage </> (x, y) = a in the unit disk is an arc of a circle that

passes through the points 1 and i. Several level curves are illustrated in Figure

11.38.