(^196) Functions of a Complex Variable
of |/(.z)| on the curve C. The result is the following inequality for the
integral (4.2.4):
L
f(z)dz < Lc(a,b) max\f(z). (4.2.8)
Note that maxzec \f(z)\ = maxa<t<b \f(z(t))\; since the functions / = f(z)
and z = z(t) are both continuous and the interval [a, b] is closed and
bounded, a theorem from the one-variable calculus ensures that the maxi-
mal value indeed exists. If you remember that the curve C is only piece-wise
smooth, and insist on complete rigor, apply the above argument to each
smooth piece of the curve separately and then add the results.
We will now look more closely at the line integrals along closed curves.
Recall (page 25) that a curve C, defined by a vector-valued function r =
r(t), a < t < b, is called simple closed if the equality r(ti) = rfa) holds for
h = a, i 2 = b and for no other £1,^2 € [a, b. By default, the orientation of
such a curve is counterclockwise: as you walk along the curve, the domain
enclosed by the curve stays on your left.
EXERCISE 4.2.14? Let f be a function, analytic in a domain G, and letC be
a simple, closed, piece-wise smooth curve in G so that the domain enclosed
by C lies entirely in G. Assuming that the derivative /' of f is continuous
in G, show that §c f(z)dz = 0. Hint: use Green's formula to evaluate the line
integrals in (4-2.5), then use the Cauchy-Riemann equations (4-2.2).As with line integrals of real-valued vector functions, we say that the
function / = f(z) has the path independence property in a domain G
of the complex plane if / is continuous in G and §c f(z)dz = 0 for every
simple closed curve C in G (recall that we always assume that C is piece-wise
smooth).
EXERCISE 4.2.15.c Show that if the function f has the path independence
property in G and C\, C2 are two curves in G with a common starting point
and with a common ending point, then L f(z)dz — Jc f(z)dz. Hint: make
a closed curve by combining C\ and C2.
EXERCISE 4.2.16.C Verify that the function f(z) = 1 has the path indepen-
dence property in every domain, and therefore Jc,z z ,dz = Z2 — z\, where
C(.z\iz2) is a curve that starts at z\ and ends at Z2- Hint: use (4-2.5) and
the result about path independence from vector analysis.
We will show next that a continuous function with the path indepen-