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

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No other points are boundary points • Since points inside the circle are interior points, they cannot be boundary points. • Given a point outside the circle, we can enclose it in a disk that does not intersect the disk Br(z0). Hence, such a point is not boundary point. In this example, none of the boundary points of Br(z0) belong to Br(z0). 34
Open and closed sets • A set which consists only of interior points is called an open set. • Every point of an open set has a neighborhood contained completely in the set. • A set is closed if it contains all its boundary points. • The closure S of a set S is the closed set consisting of all points in S together with the boundary of S. We may think of any two-dimensional set without a boundary as an open set. For example, the set S = {z : |z| < 1} is an open set. The closure S is the set {z : |z| ≤ 1}. 35
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: Example Show that the boundary of i
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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