232sheet2 - B U Department of Mathematics Math 232...

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B U Department of Mathematics Math 232 Introduction To Complex Analysis Spring 2008 Exercise Sheet 2 1 During this session, our main concern was to have a better understanding of the stereographic projection, the Riemann Sphere Σ = { ( ξ 1 2 3 ) R 3 | ξ 1 2 + ξ 2 2 + ξ 3 2 - ξ 3 = 0 } , and the extended complex plane C = C ∪ {∞} . We discussed about the map Σ -→ C and came up with the following task: 1. We know that the image z = x + iy of a point ξ = ( ξ 1 2 3 ) on the Riemann sphere under the stereographic projection map S : Σ \ { N } -→ C is z = ± ξ 1 1 - ξ 3 ² + i ± ξ 2 1 - ξ 3 , ² h ± ξ 1 1 - ξ 3 , ξ 2 1 - ξ 3 , 0 ² Find the inverse map S - 1 : C -→ Σ \ { N } . We defined the sets C k of points on Σ by C k = { ξ = ( ξ 1 2 3 ) Σ | ξ 3 = k } where k 6 = 1 . As k ranges in (0 , 1) , these are nothing but the parallels on the Riemann Sphere. Here is another exercise. 2.
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232sheet2 - B U Department of Mathematics Math 232...

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