1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

442 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

x

Figure 11.14 The graph of u = 4> (x, y) with the boundary values 4> (x, 0) = x, for

lxl < 1, and 4> (x, 0) = 0, for !xi > 1.

Using techniques from calculus and Equations ( 11 - 15 ), we write the solution as

y (x-1)^2 +y^2 x y x y

<f>(x, y) = - In + -Arctan--- -Arctan--.

2n (x+1)^2 +y2 n x - 1 n x+l

The function <f>(x, y) is continuous in the upper half-plane, and on the boundary

</> (x, 0), it has discontinuities at x = ±1 on the real a.xis. The graph in Figure

1 1. 14 shows this phenomenon.

• EXAMPLE 11.13 Find </>(x,y) that is harmonic in the upper half-plane
  Im (z) > 0 and that has the boundary values </> (x, 0) = x, for lxl < 1, </> (x, 0) =
  -1, for x < -1, and </>(x,O) = 1, for x > 1.
  Solution Using techniques from Section 11.2, we find that the function
  1 y 1 y

v(x, y) = 1--Arctan--- -Arctan--

7r x + l n x - 1

is harmonic in the upper half-plane and has the boundary values v (x, 0) = 0, for

lxl < 1, v (x, 0) = -1, for x < -1, and v (x, 0) = 1, for x > 1. This function can

be added to the one in Example 11.1 2 to obtain the desired result :

y (x 1)^2 + y^2 x 1 y

</>(x, y) = 1 +- In - +-=-Arctan--

2n (x+1)^2 + y2 7r x-1

• x +^1 Arctan-Y-.
  7f x + 1
  Figure 11. 15 shows the graph of (x,y).