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B fz 1z is analytic at every no zero point 2 z c find

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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. Examples 2.7 (A) Polynomials functions in z are entire. (B) f(z) = 1/z is analytic at every no-zero 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 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! Cauchy-Riemann Equations If f: DÆC , write f(z) = f(x, y) as u(x, y) + iv(x, y) I Theorem 2.8 (Cauchy-Riemann Equations) ∂u ∂u ∂v ∂v ,,, all exist, and satisfy ∂x ∂y ∂x ∂y ∂u ∂v...
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