pset5

# pset5 - Physics(PHZ 3113 Mathematical Physics Florida...

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Physics (PHZ) 3113 Mathematical Physics Florida Atlantic University Fall, 2010 Problem Set V Due: Tuesday, 5 October 2010 1. (Problems 2.9.12, 18 and 23, pp. 63–64) Express each of the following complex numbers in the Cartesian form z = x + i y . z (a) := 4 e - 8 π i / 3 , z (b) := 1 + i 1 - i 4 and z (c) := (1 + i) 48 ( 3 - i) 25 . 2. (Problems 2.10.10, 19 and 23, pp. 66–67) Find all complex values of each of the following roots. z (a) := 5 32 , z (b) := 3 i and z (c) := 4 q 8i 3 - 8 . 3. (Problems 2.11.4, 7 and 10, p. 69) Find all complex values of each of the following functions. z (a) := e 3 ln 2 - i π , z (b) := tan(i ln 2) and z (c) := sin(i ln i) . 4. (Problems 2.12.2, 3, 8, 12 and 16, p. 71) Verify each of the following identities using the definitions of the standard trigonometric and hyperbolic functions of a complex variable z = x + i y . a. cos z = cos x cosh y - i sin x sinh y b. sinh z = sinh x cos y + i cosh x sin y c. cosh 2 z = cosh 2 z + sinh 2 z d. cos 4 z + sin 4 z = 1 - 1 2 sin 2 2 z e. tan i z = i tanh z 5. (Problems 2.11.27, 29 and 36, p. 71) Evaluate each of the following functions in the Cartesian form z = x + i y z (a) := sin(4 + 3i) , z (b) := cosh 2 π i and z (c) := sinh 1 + i π 2 . 6. (Problems 2.14.7, 9 and 19, p. 74) Find all complex values of each of the following functions. z (a) := ln

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