Complex Analysis - Maths KU

APP-2 Appendix I Proof of Theorem 2.1
7.1 Appendix I Proof of Theorem 2.1
Appendix I Proof of Theorem 2.1
The following theorem was presented in Section 2.6 as a practical method
for computing complex limits. In this appendix we give the full epsilon-delta
proof of this theorem.
Theorem A.1 Real and Imaginary Parts of a Limit
Suppose that f(z) =u(x, y)+iv(x, y), z0 = x0 + iy0, and L = u0 + iv0.
Then lim f(z) =L if and only if
z→z0
lim
(x,y)→(x0,y0) u(x, y) =u0 and lim v(x, y) =v0.
(x,y)→(x0,y0)
Proof Theorem A.1 states that
if and only if
lim f(z) =L (1)
z→z0
lim
(x,y)→(x0,y0) u(x, y) =u0 and lim
(x,y)→(x0,y0) v(x, y) =v0. (2)
Because Theorem A.1 involves an “if and only if” statement, we must prove
two things:
(i) that (1) implies (2), and
(ii) that (2) implies (1).
We begin with the former.
(i) If we assume that lim f(z) =L, then by Definition 2.8 of Section 2.6 we
z→z0
have:
For every ε>0 there exists a δ>0 such that |f(z) − L|