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

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• When any closed curve is constructed in a simply connected domain, every point inside the curve lies in the domain. • It is always possible to construct some closed curve inside a multiply connected domain in such a way that one or more points inside the curve do not belong to the domain. 44
Complex infinity • Unlike the real number system, which has +∞ and −∞, there is only one infinity in the complex number system. This is because the complex number field C is not an ordered field ∗ . Recall that in the real number system, +∞ (−∞) is an upper (lower) bound of every subset of the extended real number system (set of all real numbers augmented with +∞ and −∞). ∗ C is not an ordered field can be argued as follows. Suppose C is an ordered field, then for any non-zero x ∈ C, we must have either x or −x being positive, and x 2 = (−x) 2 being positive. Consider i and −i, we have i 2 = (−i) 2 = −1, which is negative. A contradiction is encountered. 45
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 and 16: To prove the other half of the tria
Page 17 and 18: Example For a non-zero z and −π
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: Simply and multiply connected domai
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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