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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 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.

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 ...

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