202 Functions of a Complex Variableroot. Hint: if the polynomial P = P(z) has no roots, then l/P(z) is a bounded
entire function, hence constant — a contradiction.
The Liouville theorem is due to the same Joseph Liouville who discov-
ered transcendental numbers; he has one more famous theorem, related
to Hamiltonian mechanics. The first proof of the fundamental theorem of
algebra appeared in 1799 in Gauss's Doctoral dissertation; as with other
important theorems, there had been numerous unsuccessful attempts at the
proof prior to that.
EXERCISE 4.2.21."^4 Let us say that a function f = f{z) is analytic at the
point z = oo if and only if h(z) — f(l/z) is analytic at z = 0. Prove that
if an entire function is analytic at z = oo, then the function is everywhere
constant.
4.2.4 Conformal Mappings
No discussion of analytic functions is complete without mentioning confor-
mal mappings.
As the name suggests, a conformal mapping is a mapping that preserves
(local) form, or, more precisely, angles. We will see that an analytic function
with non-zero derivative defines a conformal mapping. Before giving the
precise definitions, let us recall how to compute the angle between two
curves in M^2.
As a set of points, a curve in M^2 is defined in one of the two ways: (a) by
a vector-valued function r(t) = x(t) i + y(t) j; (b) as a level set of a function
F = F(x,y), that is, a set {(x,y) : F(x,y) = const.}. If r\ = ri(t) and
**2 = T"2(«) define two smooth curves and ri(£ 0 ) = rz(so), then the angle 9
between the curves at the point of intersection is defined by cos 8 =\u\ (to) •
U2{sa)\: it is either the angle between the unit tangent vectors or n minus
that angle, whichever is smaller. If the differentiable functions F = F(x, y)
and G = G(x,y) define two curves so that F(x 0 ,yo) — G(xo,yo) and the
point (xo,j/o) is n°t critical for F and G, then the angle 0 between the
curves at the point of intersection satisfies
a \VF{xo,yo)-VG{x 0 ,yo)\\VF(x 0 ,y 0 )\\VG(x 0 ,yo)W
EXERCISE 4.2.22? Show that conjugate harmonic functions have orthogonal
level sets. In other words, let u, v be two differentiable functions satisfying
the Cauchy-Riemann equations (4-2.2) oft page 192, dndCiif CVf two curves