MATH253 Chapter Notes - Chapter 1-24: Unit Circle, Asteroid Family, 3I

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turquoiseminnow67

18 Jun 2015

School

University of Alberta

Department

Mathematics

Course

MATH253

Professor

Souvik Goswami

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From james lewis"s lecture notes, with permission from the author. Let [a, b] r be an interval. [here it is always assumed that a < b. ] We consider a nice" curve z(t) : [a, b] c, where we write z(t) = x(t) + iy(t). Fix p c and r > 0. The curve z(t) := reit + p : [0, 2 ] c, traces out a circle of radius r centered at p. The derivative of a curve is given by: z(cid:48)(t) := d dt z(t) := lim. T 0 z(t + t) z(t) x(t + t) x(t) Example 0. 2. z(t) = cos t + it2 z(cid:48)(t) = sin t + i2t. Standard rules: (sum, product, quotient, chain) let z(t), z1(t), z2(t) : (a, b) . Also let c c and let f (z) be analytic on z(a, b). = z(cid:48) (i)(cid:0)z1 z2(t)(cid:1)(cid:48) (ii)(cid:0)cz(t)(cid:1)(cid:48) (iii)(cid:0)z1(t)z2(t)(cid:1)(cid:48) (iv) if z2(t) (cid:54)= 0, then:(cid:18) z1(t) (v)(cid:0)f (z(t))(cid:1)(cid:48)

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MATH253 Chapter Notes - Chapter 1-24: Unit Circle, Asteroid Family, 3I

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