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Question

If α and β are different complex numbers with |β|=1 then find βα1¯αβ
Here α and β are different complex numbers such that |β|=1

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Solution

βα1¯αβ2

=βα1¯αβ[¯¯¯¯¯¯¯¯¯βα¯¯¯¯¯¯¯¯¯¯¯¯1¯αβ] [ |z|2=z¯z]

=[βα1¯αβ][¯β¯α1αβ]

=β¯ββ¯αα¯β+α¯α1¯αβα¯β+α¯αβ¯β

=|β|2¯αβα¯β+|α|21¯αβα¯β+|α|2|β|2

=1¯αβα¯β+|α|21¯αβ¯αβ+|α|2=1

βα1αβ=1


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