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Thread: Complex Analysis Question regarding mappings.

  1. #1
    Junior Member ginafara's Avatar
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    Complex Analysis Question regarding mappings.

    I am studying for an exam and I missed this problem on a homework. Can someone help me with it so that I can understand it before the exam?

    I tried an approx along the lines of Mx+b but I butchered the problem... TIA

    Problem:

    Find the linear transformation $\displaystyle w=f(z)$ that satisfies the following...

    circle $\displaystyle |x|=1$ maps onto a circle $\displaystyle |w - 3 + 2i| = 5 $ and $\displaystyle f(-i) = 3 + 3i$...

    Any help would be appreciated...
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  2. #2
    Senior Member
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    Geometry

    Think about it geometrically: you're mapping the unit circle to the circle with center 3-2i and radius 5, with the condition $\displaystyle f(-i)=3+3i$. What does $\displaystyle f(z)=Mz+b$ with complex M and Z mean geometrically? It means we scale by $\displaystyle |M|$, rotate by $\displaystyle \arg{M}$, and then translate by b. Does that help?

    --Kevin C.
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