Real and Complex Analysis (Rudin)

CONFORMAL MAPPING 297
(d) Let ex be a complex number. Show that to every cp E A which has ex for its unique fixed point
there corresponds a p such that
1 1
cp(z) - ex z - ex
--=-+p.
Let G. be the set of all these cp, plus the identity transformation. Prove that G. is a subgroup of A and
that G. is isomorphic to the additive group of all complex numbers.
(e) Let ex and p be distinct complex numbers, and let G •. , be the set of all cp E A which have ex
and p as fixed points. Show that every cp E G •. , is given by
cp(z) - ex z - ex
cp(z) - p = y . z - p'
where y is a complex number. Show th~t G •. , is a subgroup of A which is isomorphic to the multiplicative
group of all nonzero complex numbers.
(f) If cp is as in (d) or (e), for which circles C is it true that CP(C) = C? The answer should be in
terms of the parameters ex, p, and y.
32 For z E 0, Z2 #' 1, define
J(z) = exp {i log 1 + z},
1-z
choosing the branch oflog that has log 1 = o. DescribeJ(E) if E is
(a) U,
(b) the upper half of T,
(c) the lower half of T,
(d) any circular arc (in U) from -1 to 1,
(e) the radius [0, 1),
(f) any disc {z: Iz - rl < 1- r},O< r < 1.
(g) any cUrve in U tending to 1.
33 If CP. is as in Definition 12.3, show that
(a) ~ r 1 cp~ 12 dm = 1,
11: Ju
1 i 1 -lexl 2 1
(b) - Icp~1 dm=--log--.
11: u lexl 2 1-lex1 2
Here m denotes Lebesgue measure in R2.