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Question
If (x + iy)3 = u + iv, then show that `u/x + v/y =4(x^2 - y^2)`
Solution
`(x + iy)^3 = u + iv`
or `u + iv = x^3 + 3x^2 .iy + 3.(iy)^2 x + (iy)^3`
= `x^3 + 3x^2 yi + 3xy^2 i^2 + i^3 y^3` `[∵ i^2 = - 1]`
= `(x^3 - 3xy^2 ) + (3x^2y - y^3)i`
⇒ `x^3 - 3xy^2 = u/x`
and `3x^2y - y^3 = v`
or `3x^2 - y^2 = v/y`
On adding equations, (1) and (2)
`4x^2 - 4y^2 = u/x + v/y`
⇒ `u/x + v/y = 4 (x^2 - y^2)`
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