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

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Take |z| > 1 as an illustration Geometric interpretation: Consider the triangle whose vertices are 1, z and |z|, by the Triangle Inequality, we have |z − 1| ≤ |z| − 1 + z − |z| . The chord joining |z| and z is always shorter than the circular arc joining the same two points. Note that the arc length is given by |z| |Arg z|. 18
Geometric applications How to find the equation of the perpendicular bisector of the line segment joining the two points z1 and z2? Since any point z on the bisector will be equidistant from z1 and z2, the equation of the bisector can be represented by |z − z1| = |z − z2| . For a given equation f(x, y) = 0 of a geometric curve, if we set x = (z + z)/2 and y = (z − z)/2i, the equation can be expressed in terms of the pair of conjugate complex variables z and z as z + z z − z f(x, y) = f , = F(z, z) = 0 . 2 2i For example, the unit circle centered at the origin as represented by the equation x2 + y2 = 1 can be expressed as zz = 1. 19
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: Example For a non-zero z and −π
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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