1.5 Finding Real and Imaginary Parts

where $x_{2} + iy_{2} \neq 0$ and $n=\pm 1$$n = \pm 1$ and $n=\pm 2$$n = \pm 2$

$(x + iy)^n$ when $n=\pm 1$$n = \pm 1$

Let $z = (x + iy)^n$ be a complex number.

(i) For $n = 1$

We have $z = (x + iy)^1 = x + iy$. $Re(z) = x, Im(z) = y$

(ii) For $n = - 1$

We have $z = (x + iy)^{- 1}$

$\therefore Re(z)=\frac{x}{x^2+y^2},Im(z)=\frac{-y}{x^2+y^2}$$\therefore Re(z) = \frac{x}{x^2 + y^2}, Im(z) = \frac{- y}{x^2 + y^2}$