570 Chapter^8
Theorem 8.24 (Identity Principle for Analytic Functions). If two func-
tions f and g are analytic in a region R and assume the same values
in a subset A C R having at least an accumulation point c E R, then
f(z) = g(z) for all z E R.
Proof Consider the function F(z) = f(z)-g(z), and let {an} be a sequence
of distinct points in A such that an~ c. Then F(an) = f(an)-g(an) = 0
and F( c) = limn-.oo F( an) = 0. Thus c E R is a zero of F( z) but c is not
isolated. Hence F(z) = 0 in R, or J(z) = g(z) for all z E R.
Corollary 8.11 If two functions f and g are analytic in a region R, and
if J(z) = g(z) for all z in some subregion R' CR, or if f(z) = g(z) for all
z in some arc 'Y* CR (not a point arc), then J(z) = g(z) for all z ER.
Examples 1. The functions f ( z) = sin 2z and g( z) = 2 sin z cos z are both
analytic in C. On the real line (i.e., for z = x) we have sin 2x = 2 sin x cos x
for all x, as known from trigonometry. Hence sin 2z = 2 sin z cos z for all
complex values of z.
In a similar manner, the fact that all the well-known trigonometric iden-
tities are also valid in the complex case is a consequence of the identity
principle. Although not all the circular functions are analytic in C, they
are analytic in the region obtained by deleting from C the set of isolated
points where they cease to be analytic. For instance, in the case of tan z
this is the set {1M2k + l)7r: k = O, ±1, ±2, ... }.
2. If f is analytic in a region R and f'(z) = 0 for all z in R, then
f(z) = 0 (a constant) in R. We have already considered this property
in Theorem 6.6. To show that it is also a consequence of the identity
principle, let z 0 E R. The we have
f'(zo) f(n)(zo) n
f(z) = f(zo) + -
11
-(z -zo) + · · · + n! (z - zo) + · · ·valid for lz -zo I < r. From the assumption f' ( z) = 0 in R, it follows that
f'(zo) = f"(zo) = · · · = J(n)(zo) = · · · = 0
Hence
f(z) = f(zo) = C for lz -zol < r
Let g(z) = C for all z ER. Then we have f(z) = g(z) for lz -zol < r. By
the identity principle we conclude that f(z) = g(z) for all z in R.
8.10 Zeros of Polynomials
Most of the theorems discussed in the preceding section for analytic func-
tions apply equally well to the case of a polynomial function. However, in