Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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Example. Let<br />
Then f is totally differentiable, with<br />
f: C → C, z ↦→ |z| 2 .<br />
ux = 2x, uy = 2y, vx = vy = 0,<br />
noting that v = 0. The Cauchy–Riemann equations<br />
ux(z0) = vy(z0) and uy(z0) = −vx(z0)<br />
thus hold if and only if z0 = 0. By Theorem 2.1, this means that f is complex<br />
differentiable at z0 if and only if z0 = 0.<br />
Corollary 2.1.1. Let D ⊂ C be open and connected, and let f: D → C be complex<br />
differentiable. Then f is constant on D if and only if f ′ ≡ 0.<br />
Pro<strong>of</strong>. Suppose that f ′ ≡ 0. From the remark after Theorem 2.1, it follows that<br />
ux = vx = uy = vy ≡ 0.<br />
Multivariable calculus then yields that f is constant.<br />
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