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    Questions on Linear Fractional Transformation (LFT)

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    ** Please see the attached file for the full problem description **

    I want someone to show me the calculations and explain where necessary. Thanks in advance.

    1) (i) Write down the equation of the line in xy-coordinates defined by
    | z - (i + 1) | = | z + 2i |.
    (ii) The LFT w = 1/z takes the line in (i) to a circle | w - P | = R.
    Determine what P and R is.

    2) Let T(z) = iz - 1/5z + 4i, L(z) = (1 + i) z + i/ iz + 1.
    (i) Compute the inverse LFT T^-1(z).
    (ii) Compute the composite T(L(z)) (as a LFT).

    3) (i) Find a LFT T such that
    T(1) = 0, T(i) = 1, T (-1) = infinity
    (ii) Describe the region T(|z| =< 1).

    4) (i) Find a LFT T such that
    T(0) = i, T(1 +i) = 2i - 1, T(-i) = -i/2,
    (ii) Using your answer T(z) in (i), verify by evaluating T(0), T(1 + i), T(-i).

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    Solution Preview

    (i) To write a equation, given in terms of a complex number, in xy-coordinates, we suppose z=x+iy
    Plugging this in the given equation, we get .
    We use the definition of modulus , we have

    The square terms get cancelled and we get ie., which represents a line
    (ii) we have
    Using the suggested transformation, we have where some complex number, we have
    This gives
    This can further be written as where the 'bar' over a complex number ...

    Solution Summary

    This solution explains how to calculate questions on Linear Fractional Transformation. More specifically it includes, finding the inverse of a Linear Fractional Transformation, composite Linear Fractional Transformation, conversion of equations using Linear Fractional Transformation, describing the region given by an inequality involving Linear Fractional Transformation, and finding a Linear Fractional Transformation as per the given conditions on it.