1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

464 CHAPTER 11 • APPLICATIONS OF H ARMONIC FUNCTIONS

3. Find the electrostatic potential (x, y) in the crescent-shaped region that lies inside
   the disk lz - 21 < 2 a nd outside the circle !z -I I = 1 that satisfies the following
   boundary values (shown in Figure 11.41). (x, y) = 100 , for lz - 21 = 2, z f O;
   (x, y) = 50, for lz - l l = 1, z f 0.
   y
   

x

Figure 11.41

4. Find the electrostatic potential</> (x, y) in the semi-infinite strip -i~ < x < ~. y > 0
   that has the boundary values shown in Figme 11.42:G.v) = o,
   (x, 0) = 50,
   ( ~1r' y) = 100,
   

Figure 11.42

for y > O;

-1r 1r

for 2 < x < zi

for y > 0.

5. Find the electrostatic potential (x, y) in the domain Din the half-plane Re (z) > 0
   tha.t lies to the left of the hyperbole. 2x^2 - 2y^2 = 1 and satisfies the following
   boundary values (shown in Figure 1 1. 43 ).(0,y)=50, forally;
   (x, y) = 100 when 2x^2 - 2y^2 = 1.
   

y

Figure 11.43