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Unformatted text preview: z0]. That is,
Arg[f'(z)] ‡ Arg[∆f] - Arg[∆z]
Therefore, the argument of f'(z0) gives the direction in which f is rotating near z0. In fact,
we shall see later that f preserves angles at a point if the derivative is non-zero there.
Question What if f'(z0) = 0?
Answer Then the magnitude is zero, so, locally, f “squishes’ everything to a point.
(A) Polynomials functions in z are entire.
(B) f(z) = 1/z is analytic at every no-zero point.
(C) Find f'(z) if f(z) =
(D) Show that f(z) = Re(z) is nowhere differentiable! Indeed: think of it geometrically as
projection onto the x-axis. Choosing ∆z as a real number gives the difference quotient † Evidently not worth mentioning by the textbook 7 equal to 1, whereas choosing it to be imaginary gives a zero difference quotient.
Therefore, the limit cannot exist!
If f: DÆC , write f(z) = f(x, y) as u(x, y) + iv(x, y)
Theorem 2.8 (Cauchy-Riemann Equations)
∂u ∂u ∂v ∂v
all exist, and satisfy
∂x ∂y ∂x ∂y
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