Math144Notes

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online