Unformatted text preview: f is indeed diFerentiable at 0 and ± i only. ±rom the CauchyRiemann equations we can deduce: Theorem 3.5. [S&T4.7] If f is diFerentiable in a domain D and f ′ ( z ) = 0 for all z ∈ D then f is constant in D . Proof. We have f ′ ( z ) = ∂u ∂x + i ∂v ∂x = ∂v ∂y − i ∂u ∂y , so f ′ ( z ) = 0 implies all the partial derivatives of u and v are zero. Two points in D are joined by a step path. On each horizontal and vertical segment, u and v have zero derivatives and so are constant. Therefore f takes the same values at the two endpoints of the path and is hence constant on D . s 1...
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 Spring '14
 Sidorov
 Equations, Derivative, Partial differential equation

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