[SOLVED] Amath 567, homework 7

1. Problem 1:
(a) Construct the bilinear transformation
w(z) = az + b
cz + d
that maps the region between the two circles |z −
1
4
| =
1
4
and |z −
1
2
| =
1
2
into
an infinite strip bounded by the vertical lines u = ℜ{w} = 0 and u = ℜ{w} = 1.
To avoid ambiguity, suppose that the outer circle is mapped to u = 1.(b) Upon finding the appropriate transformation w, carefully show that the image of
the inner circle under w is the vertical line u = 0, and similarly for the outer circle.2. Problem 2: Use the result of Problem 1 to find the steady state temperature T(x, y)
in the region bounded by the two circles, where the inner circle is maintained at
T = 0°C and the outer circle at T = 100°C. Assume T satisfies the two-dimensional
Laplace equation