2.3. RADIAL STRETCHINGS
7
Studying quasiconformal mappings via the Beltrami equation is a particularly
valuable idea because from this point of view the mapping is the solution of an
elliptic equation and as such enjoys various nice properties not obvious from the
definition. Before getting too far into the theory here it is important to have a
concrete example in hand.
2.3. Radial Stretchings
The first example anyone usually meets in the theory is the radial stretching.
Fortunately it is also one of the most useful examples.
Let p p{t) be a piecewise differentiate function defined and valued in the
interval [0,+oo]. We assume that p is either increasing or decreasing. A mapping
h : C C of the form
(2.5) h(z)
=p{r)el\
z =
reie,
is called a radial stretching. Note that h preserves (respectively reverses) the ori-
entation if p is increasing (decreasing).
We calculate the complex partials of h at points where the derivative p(r) exists
to find
(2-6) hz = I[p(|
z
|) + -Lp(|*|)]
(2-7) h-z = 1 [p(|z|) - ijp(|*|)] £
In the orientation preserving case we find that h satisfies the Beltrami equation
h-z = p(z)hz
with the Beltrami coefficient p given by
(2-8)
M* ) = H y ' M ' l i "
\z\p(\z\) + p{\z\) z
and hence
l - K ^ O I I p(\z\) \z\p{\z\)\
The Jacobian determinant is given by
(2.io) J{z,h)
=
M-M=mm
\z\
Notice that if h preserves orientation, J{z, h) is always locally integrable. Indeed
L
J(z,h) = 2TT / p(t)p(t)dt
Jo
= 7 T /
dp2
=
np2(X)
Jo
this latter term being the area of the disk of radius p(X) of course. Finally, we
compute the operator norm of the differential matrix
(2.11) \Dh{z)\ = \hz\ + \hz\ = max{p(|z|), |*rV(l*l)}
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