Complex Analysis - Maths KU

7.2 Linear Fractional Transformations 399
nonzero, then the angle θ between C1 and C2 is given by (1). Explain why
φ = arg (f ′ (0) · z ′ 2) − arg (f ′ (0) · z ′ 1) does not represent the angle between
the images C ′ 1 and C ′ 2 of C1 and C2 under the mapping w = f(z) =z 2 ,
respectively.
(b) Use Problem 20 to write down an expression involving arguments that
does represent the angle φ between C ′ 1 and C ′ 2. [Hint: C ′ 1 and C ′ 2 are
parametrized by w1(t) = f (z1(t)) = [z1(t)] 2 and w2(t) = f (z2(t)) =
[z2(t)] 2 .]
(c) Use (8) of Section 1.3 to show that your expression for φ from (b) is equal
to 2θ.
22. In this problem you will prove Theorem 7.2. Let f be an analytic function at
the point z0 such that f ′ (z0) =f ′′ (z0) =... = f (n−1) (z0) =0andf (n) (z0) �= 0
for some n>1.
(a) Explain why f can be written as
f(z) =f(z0)+ f (n) (z0)
n!
(z − z0) n (1 + g(z)) ,
where g is an analytic function at z0 and g(z0) =0.
(b) Use (a) and Problem 20 to show that the angle between two smooth curves
intersecting at z0 is increased by a factor of n by the mapping w = f(z).
7.2 Linear Fractional Transformations
In manyapplications 7.2 that involve boundary-value problems associated with Laplace’s equation,
it is necessaryto find a conformal mapping that maps a disk onto the half-plane v ≥ 0.
Such a mapping would have to map the circular boundaryof the disk to the boundaryline
of the half-plane. An important class of elementaryconformal mappings that map circles
to lines (and vice versa) are the linear fractional transformations. In this section we will
define and studythis special class of mappings.
Linear Fractional Transformations In Section 2.3 we examined
complex linear mappings w = az +b where a and b are complex constants and
a �= 0. Recall that such mappings act byrotating, magnifying, and translating
points in the complex plane. We then discussed the complex reciprocal
mapping w =1/z in Section 2.5. An important propertyof the reciprocal
mapping, when defined on the extended complex plane, is that it maps certain
lines to circles and certain circles to lines. A more general type of mapping
that has similar properties is a linear fractional transformation defined next.