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Math Help - [SOLVED] Transformation

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Transformation

    Consider the transformation
    z \mapsto w = − 5z^2+3 .

    (a) Find the image of the line Re(z) = c (for c a non-zero real constant) under the
    transformation, using as usual z = x + iy and w = u + iv.

    (b) Find the equation of the line Im(z) = k (for k a real constant).

    Help please explain how to solve this.
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by ronaldo_07 View Post
    Consider the transformation
    z \mapsto w = − 5z^2+3 .

    (a) Find the image of the line Re(z) = c (for c a non-zero real constant) under the
    transformation, using as usual z = x + iy and w = u + iv.
    The equation w=3-5z^2 tells you that u+iv = 3 - 5(x+iy)^2 = 5y^2 - 10 ixy + (3-5x^2). Put x=c and equate real and imaginary parts to get u = 5y^2 + (3-5c^2) and v = -10cy. Solve the second of these for y and substitute that into the first: u = \frac{v^2}{20c^2} + 3-5c^2. You will recognise that as the equation of a parabola.

    Quote Originally Posted by ronaldo_07 View Post
    (b) Find the equation of the line Im(z) = k (for k a real constant).

    Help please explain how to solve this.
    I would have said that the equation of the line \text{Im}(z) = k is \text{Im}(z) = k (or equivalently y=k). But perhaps the question is meant to ask for the image of that line under the z→w transformation. If so, do it the same way as part (a).
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