- #1
stunner5000pt
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Show that the stereographic projections of the points z and [itex] 1/\overline{z} [/itex] are reflections of each other in teh equatorial plane of the Reimann sphere
ok so let z = x + iy
then [tex] \frac{1}{\overline{z}} = \frac{x - iy}{x^2 + y^2} [/tex]
so the magnitude of [tex] \frac{1}{\overline{z}} [/tex] is [tex] \frac{1}{x^2 + y^2} [/tex]
the stereogrpahic projection of z is
[tex] x_{1} = \frac{2x}{x^2 + y^2 +1} [/tex]
[tex] y_{1} = \frac{2y}{x^2 + y^2 +1} [/tex]
[tex] z_{1} = \frac{x^2 + y^2 -1}{x^2 + y^2 +1} [/tex]
for 1/ z bar is
[tex] x_{2} = \frac{2x(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
[tex] y_{2} = \frac{2y(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
[tex] z_{2} = \frac{1-(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
i fail to see how these are reflections of each other, then shouldn't the x1 = - x2?? and so on??
please help!
ok so let z = x + iy
then [tex] \frac{1}{\overline{z}} = \frac{x - iy}{x^2 + y^2} [/tex]
so the magnitude of [tex] \frac{1}{\overline{z}} [/tex] is [tex] \frac{1}{x^2 + y^2} [/tex]
the stereogrpahic projection of z is
[tex] x_{1} = \frac{2x}{x^2 + y^2 +1} [/tex]
[tex] y_{1} = \frac{2y}{x^2 + y^2 +1} [/tex]
[tex] z_{1} = \frac{x^2 + y^2 -1}{x^2 + y^2 +1} [/tex]
for 1/ z bar is
[tex] x_{2} = \frac{2x(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
[tex] y_{2} = \frac{2y(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
[tex] z_{2} = \frac{1-(x^2 + y^2)^2}{1 + (x^2 + y^2)^2} [/tex]
i fail to see how these are reflections of each other, then shouldn't the x1 = - x2?? and so on??
please help!
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