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Thread: Transformation

  1. #1
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    Transformation

    The transformation $\displaystyle T$ from the $\displaystyle z$-plane, where$\displaystyle z = x + iy$, to the w-plane, where...

    $\displaystyle w=\frac{z+i}{z}, \ z \ne 0$

    (a) The transformation $\displaystyle T$ maps the points on the line with equation $\displaystyle y=x$ in the $\displaystyle z$-plane, other than $\displaystyle (0, 0)$, to points on a line $\displaystyle l$ in the $\displaystyle w$-plane. Find a cartesian equation of $\displaystyle l$.

    (b) Show that the image, under $\displaystyle T$, of the line with equation $\displaystyle x+y+1=0$ in the $\displaystyle z$-plane is a circle $\displaystyle C$ in the $\displaystyle w$-plane, where $\displaystyle C$ has cartesian equation $\displaystyle u^2+v^2-u+v=0$.

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  2. #2
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    Quote Originally Posted by Air View Post
    The transformation $\displaystyle T$ from the $\displaystyle z$-plane, where$\displaystyle z = x + iy$, to the w-plane, where...

    $\displaystyle w=\frac{z+i}{z}, \ z \ne 0$

    (a) The transformation $\displaystyle T$ maps the points on the line with equation $\displaystyle y=x$ in the $\displaystyle z$-plane, other than $\displaystyle (0, 0)$, to points on a line $\displaystyle l$ in the $\displaystyle w$-plane. Find a cartesian equation of $\displaystyle l$.
    The complex number $\displaystyle z$ has an argument of $\displaystyle \frac{ \pi}{4}$ for $\displaystyle \Im (z) > 0$ and $\displaystyle - \frac{ 3 \pi }{4}$ for $\displaystyle \Im (z) < 0$.

    Consider $\displaystyle w - 1$ which is $\displaystyle \frac{i}{z}$ therefore $\displaystyle \arg(w -1) = \arg \left( \frac{i}{z} \right) $

    $\displaystyle \Rightarrow \arg(w -1) = \arg(i) - \arg(z) $
    $\displaystyle \Rightarrow \arg(w -1) = \frac{\pi}{2} - \arg(z) $

    for $\displaystyle \Im (z) > 0$ $\displaystyle \arg(w -1) = \frac{\pi}{2} - \frac{\pi}{4} \ \ \Rightarrow \ \ \arg(w -1) = \frac{\pi}{4} $

    for $\displaystyle \Im (z) < 0$ $\displaystyle \arg(w -1) = \frac{\pi}{2} + \frac{3 \pi}{4} \ \ \Rightarrow \ \ \arg(w -1) = \frac{5 \pi}{4} $

    You should be able to pull out an equation form that.

    Bobak
    Last edited by bobak; Jun 10th 2008 at 01:32 PM.
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