By C. W. Celia, A. T. F. Nice, K. F. Elliott

ISBN-10: 0333348273

ISBN-13: 9780333348277

ISBN-10: 1349067113

ISBN-13: 9781349067114

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Extra info for Advanced Mathematics 3

Sample text

In Fig. 1, the point A represents the complex number a, with lal < 1 and arg a ~ Jr. The points P and Q represent the two square roots of a. o< Example Find the square roots of 21 + 20i. Method 1 = = Let 21 + 20i = R(cos a + i sin a) R2 = 2 F + 202 = 841 R = 29. The two square roots will be ± j29[cos (ta) By equating real parts, + i sin (ta)]. r A p 1 Q Fig. 1 26 Advanced mathematics 3 Square roots x 29 cos a = 21 cos a = 21/29 . = By equating imaginary parts, 29 sin a = 20 sin a = 20/29. = Since cos a and sin a are both positive, 0 < a < lr/2 and cos be both positive.

21 (b) On AB, let z = I + z it, where 0 :::; t :::; I. L 4' This is the equation of the circle in the w-plane with centre given by w = t and with radius 1As t increases from 0 to I, u decreases from I to i and v decreases from 0 to - 1. 2' Hence as z moves from A to B, w moves along the arc AIB I shown in Fig . 21. (c) On BC, let z =i+ t, where 0 :::; t :::; I. t - i I This gives w=--=--- => u=--- i + ( t I + (2 ' I + (2 - I v=--I + /2 Complex numbers 55 1 =---= 1 + t2 => => u2 + (v + t)2 = -v t· This is the equation of the circle in the w-plane with centre given by w = - i/2 and with radius t .

10 The sum of two roots of the equation + Z4 2z 3 + 2z 2 + 0 is 2. Solve the equation. 6 The exponential function The function e", or exp z, is defined with domain C by the series e= = I + z Z2 + 2! + Z3 3! + . + z" n! + .... It can be shown that this series converges for all values of z. Let z == x + iy. When y == 0, Z == x and the series above becomes the Maclaurin series for e . When x == 0, z = iy and series for e' becomes e iy = 1 + . + (iy)2 + (iy)3 + 2! ly 3! (iy)" +-+ n! The real part of the series for eiy is y2 y4 y6 2!