5.5 The Mapping \(w = z^{1/2}\)

If we square both sides of \(\, w = z^{1/2}\,\) get \(\, w^2 = z.\,\) Setting \(\, w = u + iv\,, \, z = x+ iy,\,\) we get \[\boxed {x = u^2 - v^2 \quad \text {and}\quad y = 2uv}\] This shows that the line \(\, x = a\, (a> 0)\,\) in the \(z-\)plane gets mapped to the parabola \(\, u^2 - v^2 = a\,, \,\) while the line \(\, y = b\, (b> 0)\,\) is mapped to the parabola \(\, b = 2uv\).