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In general all these circles pass through the origin

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Unformatted text preview: age of such a point is the zero vector, and so we can say nothing about the direction of the path at that point—anything can happen. Examples 6.3 (A) f(z) = z + b, or w = z + b Translation by b. (B) f(z) = az, or w = az Expansion/Contraction + Rotation iø If a = r is real, we get expansion or contraction. If a = e we get rotation by ø. Therefore, in general, we get a composite of the two. (C) f(z) = az + b or w = az + b Affine: A combination of all 3 This is the stuff of geometry. Note that, in geometry, two objects in the plane are congruent iff one can be obtained from the other using an affine transformation. z (D) f: C ÆC ; f(z) = e . Here is a better illustration than that pathetic one in the book: II Vertical lines Æ circles Horizontal lines Æ rays (E) What about the inverse mapping, Ln(z)? Recall that Ln: {z | z ≠ 0 and arg(z) ≠ π} Æ C I 20 Think of it as the above map in reverse: The above picture on the right shows the top half the domain, and we get: (F) f: C ÆC ; f(z) = sin z II For this it is useful to remember that f(x + iy) = sinx c...
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