1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

458 CHAPTER 11 • APPLICATIONS OF HARMONIC F UNCTIONS

in the upper half-disk lzl < 1, Im(z) > 0, show that the isothermals T(x,y) =a

are portions of circles that pass through the points + 1 and - 1, as illustrated in

Figure 11.33.

y

Figure 11.33

14. For the temperature function

T(x, y) =^3 00R.e(Arcsinz)

1T

in the upper half-plane Im (z) > 0, show that the isothermals T (x, y) = ar are

portions of hyperbolas that have foci at the points ±1, as illustrated in Figure

11.34.

T = 150

F igure 11. 34

1 5. Find the t emperature function in the portion of the upper half-plane Im (z) > 0
that lies inside the ellipse
x2 y2
--+--=l
cosh^2 2 sinh^2 2
and satisfies the following boundary conditions (shown in Figure 11.35). Hint: Use
w = Arcsin z.

T (x, y) = 80 , for (x, y) on the ellipse;

T(x,O) = 40 , for - l<x<l;

fJT

8n = Ty (x, 0) = 0 when 1 < lxl < cosh 2.