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Thread: Curve Sketching: Complex domain

  1. #1
    Senior Member I-Think's Avatar
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    Curve Sketching: Complex domain

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

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

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

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

    Thus the line $\displaystyle Re(z)=0$ is mapped onto the negative portion of the x-axis and the line $\displaystyle Im(z)=1$ is mapped onto this parabola:$\displaystyle x_2=\frac{y^2}{4}-1$
    Any mistakes present?
    Last edited by I-Think; May 28th 2012 at 04:29 PM.
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