1. Complex numbers A complex number z is defined as an ordered ...

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Example Find an upper bound for |z 5 − 4| if |z| ≤ 1. Solution Applying the triangle inequality, we get |z 5 − 4| ≤ |z 5 | + 4 = |z| 5 + 4 ≤ 1 + 4 = 5, since |z| ≤ 1. Hence, if |z| ≤ 1, an upper bound for |z 5 − 4| is 5. In general, the triangle inequality is considered as a crude inequality, which means that it will not yield least upper bound estimate. Can we find a number smaller than 5 that is also an upper bound, or is 5 the least upper bound? Yes, we may use the Maximum Modulus Theorem (to be discussed later). The upper bound of the modulus will correspond to a complex number that lies on the boundary, where |z| = 1. 16
Example For a non-zero z and −π < Arg z ≤ π, show that Solution |z − 1| ≤ |z − 1| ≤ since (cos θ−1) 2 +sin 2 θ = and θ sin 2 ≤ |z| − 1 +|z| |Arg z|. z − |z| + (|z| − 1) ≤ = |z| |cos θ + isin θ − 1| + = |z| (cos θ − 1) 2 + sin 2 θ + θ = |z| 2sin + |z| − 1 θ 2 . 2 1 − 2sin 2 θ 2 z − |z| + |z| − 1 |z| − 1 |z| − 1 ≤ |z||Arg z| + 2 − 1 , θ = Arg z |z| − 1 , + 4sin 2 θ 2 cos2 θ 2 = 4sin2 θ 2 17
Page 1 and 2: 1. Complex numbers A complex number
Page 3 and 4: Polar coordinates Modulus of z : |z
Page 5 and 6: Complex conjugate The complex conju
Page 7 and 8: From the relation v 2 = u 2 − a,
Page 9 and 10: To prove the theorem, we consider t
Page 11 and 12: which implies qφ = pθ + 2kπ or
Page 13 and 14: In the complex plane, the n roots o
Page 15: To prove the other half of the tria
Page 19 and 20: Geometric applications How to find
Page 21 and 22: Example Find the region in the comp
Page 23 and 24: Example If z1, z2 and z3 represent
Page 25 and 26: Equating the respective real and im
Page 27 and 28: Symmetry with respect to a circle G
Page 29 and 30: (ii) When |α| = R, the inversion p
Page 31 and 32: Example Show that z = 1 is a limit
Page 33 and 34: Example Show that the boundary of i
Page 35 and 36: Open and closed sets • A set whic
Page 37 and 38: Theorem A set D is open if and only
Page 39 and 40: Theorem A set S is closed if and on
Page 41 and 42: Motivation for defining connectedne
Page 43 and 44: Simply and multiply connected domai
Page 45 and 46: Complex infinity • Unlike the rea
Page 47 and 48: Examples • The open half-plane Re
Page 49 and 50: Riemann sphere and stereographic pr

1. Complex numbers A complex number z is defined as an ordered ...

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