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

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Consider the triangle formed by these 3 points: z1, z3 and z3. If z1 is one vertex, then z2 is obtained by rotating the position vector of z1 by 2π/3 in anti-clockwise sense. Likewise, z3 is obtained by rotating the position vector of z2 by the same amount. Hence, z1, z2 and z3 are the vertices of an equilateral triangle inscribed on the unit circle: |z| = 1. 26
Symmetry with respect to a circle Given a point α in the complex plane, we would like to construct the symmetry point of α with respect to the circle C R : |z| = R. The symmetry point of α is defined to be β = R 2 /α. Conversely, since we may write α = R 2 /β, we can as well consider α to be the symmetry point of β. The two points α and β are said to be symmetric with respect to the circle C R. Construction of a pair of symmetric points with respect to the circle C R. 27
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: Equating the respective real and im
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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