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# 11 exercise set 3 p 817 1 3 5 11 13 17 21 2331 odd

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Unformatted text preview: ; (e) ln(-1) = πi + 2nπi; (f) ln(3-4i) = ln5 + i arg(3-4i) + 2nπi = ln5 + i arctan(4/3) + 2nπi; Ln 1 = 0; Ln 4 = 1.386... Ln r = ln r Ln i = πi/2; Ln (-1) = πi; Ln(3-4i) = ln5 + i arctan(4/3). More Properties 1. ln(z w) = lnz + lnw; ln(z/w) = ln(z) - ln(w). This doesn't work for Ln; eg., z = w = -1 gives Ln z + Ln w = πi + πi = 2πi, but Ln(zw) = Ln(1) = 0. 2. Ln z jumps every time you cross the negative x -axis, but is continuous everywhere else (except zero of course). If you want it to remain continuous, you must switch to another branch of the logarithm. (Lnz is called the principal branch of the logarithm.) 3. eln z = z, and ln(ez) = z + 2nπi; eLn z = z, and Ln(ez) = z + 2nπi; (For example, z = 3πi gives ez = -1, and Ln(ez) = πi ≠ z.) 11 Exercise Set 3 p. 817 #1, 3, 5, 11, 13, 17, 21, 23–31 odd, 35, 37, 45 p. 821 #1, 5, 7, 11, 13, 15, 23 Hand In 1. Find functions f that do the following: (a) Map the region {z | 0 ≤ arg(z) ≤ π/2} onto the whole plane (b) Map th...
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## This document was uploaded on 03/20/2014 for the course MATH 144 at Hofstra University.

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