​ If (x + iy)^3 = (u + iv) then prove that (u/x+v/y) = 4(x^2 – y^2). ​

Question

If (x + iy)³ = (u + iv) then prove that \((\frac{u}{x}+\frac{v}{y})\) = 4 (x² – y²).

Answer 1

Given that, (x + iy)³ = (u + iv)

⇒ x³ + (iy)³ + 3x² iy + 3xi²y² = u + iv

⇒ x³ - iy³ + 3x² iy - 3xy² = u + iv

⇒ x³ - 3xy² + i(3x²y - y³) = u + iv

On equating real and imaginary parts, we get

U = x³ - 3xy² and v = 3x²y - y³

= x² - 3y² + 3x² - y²

= 4x² - 4y²

= 4(x² - y²)