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

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Corollary A set C is closed if and only if its complement D = {z : z ∈ C} is open. To see the claim, we observe that the boundary of a set coincides exactly with the boundary of the complement of that set (as a direct consequence of the definition of boundary point). Recall that a closed set contains all its boundary points. Its complement shares the same boundary, but these boundary points are not contained in the complement, so the complement is open. Remark There are sets that are neither open nor closed since they contain part, but not all, of their boundary. For example, is neither open nor closed. S = {z : 1 < |z| ≤ 2} 38
Theorem A set S is closed if and only if S contains all its limit points. Proof Write N(z; ǫ) as the deleted ǫ-neighborhood of z, and S ′ as the complement of S. Note that for z ∈ S, N(z; ǫ) ∩ S = N(z; ǫ) ∩ S. S is closed ⇔ S ′ is open ⇔ given z ∈ S, there exists ǫ > 0 such that N(z; ǫ) ⊂ S ′ ⇔ given z ∈ S, there exists ǫ > 0 such that N(z; ǫ) ∩ S = φ ⇔ no point of S ′ is a limit point of S 39
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: Theorem A set D is open if and only
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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