# Questions on Linear Fractional Transformation (LFT)

** 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).

https://brainmass.com/math/complex-analysis/questions-linear-fractional-transformation-lft-570425

#### 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.