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 nonzero there.
Question What if f'(z0) = 0?
Answer Then the magnitude is zero, so, locally, f “squishes’ everything to a point.
Examples 2.7
(A) Polynomials functions in z are entire.
(B) f(z) = 1/z is analytic at every nozero point.
2
z
(C) Find f'(z) if f(z) =
2
(z+1)
(D) Show that f(z) = Re(z) is nowhere differentiable! Indeed: think of it geometrically as
projection onto the xaxis. 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!
CauchyRiemann Equations
If f: DÆC , write f(z) = f(x, y) as u(x, y) + iv(x, y)
I
Theorem 2.8 (CauchyRiemann Equations)
∂u ∂u ∂v ∂v
,,,
all exist, and satisfy
∂x ∂y ∂x ∂y
∂u
∂v...
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This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.
 Fall '03
 StefanWaner
 Math, Algebra, Geometry, Complex Numbers

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