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Math Help - Curve Sketching: Complex domain

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    Senior Member I-Think's Avatar
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    Curve Sketching: Complex domain

    Let f(z)=z^2
    Find the images of the line Re(z)=0 and Im(z)=1 under the above mapping, w=f(z)

    Just to make sure I'm understanding this curve sketching
    Represented parametrically, Re(z)=0 is z_1(t)=it and Im(z)=1 is z_2(t)=t+i, t\in{\mathbb{R}}

    So f(z_1(t))=-t^2+0i and f(z_2(t))=t^2-1 +2ti
    x_1=-t^2, x_2=t^2-1
    y_1=0, y_2=2t

     x_2=\frac{y^2}{4}-1

    Thus the line Re(z)=0 is mapped onto the negative portion of the x-axis and the line Im(z)=1 is mapped onto this parabola: x_2=\frac{y^2}{4}-1
    Any mistakes present?
    Last edited by I-Think; May 28th 2012 at 05:29 PM.
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