# Lesson Worksheet: Möbius Transformations Mathematics

In this worksheet, we will practice interpreting möbius transformation in the complex plane.

Q1:

Consider the Möbius transformations and , where or 0.

Write an expression for the composition .

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Q2:

A transformation that maps the -plane to the -plane is defined by , where .

Find an equation for the image of under the transformation.

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Find an equation for the image of .

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Find a Cartesian equation for the image of .

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Find a Cartesian equation for the image of .

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Q3:

A transformation that maps the -plane to the -plane is defined by .

Find the Cartesian equation for the image of .

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Find the Cartesian equation for the image of .

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Q4:

Which of the following Möbius transformations maps to the real axis?

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Q5:

A transformation, , that maps the -plane to the -plane is given by , where .

Find a Cartesian equation for the image of the imaginary axis under the transformation .

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Hence, find the image of the region under the transformation .

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Q6:

Suppose is a linear transformation with . What is the value of ?

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Q7:

Suppose is a linear transformation. What is the value of ?

Q8:

Which of the following Möbius transformations, , maps to 0, 0 to , and has ?

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Q9:

Find the equation for the image of under the transformation , , which maps the -plane to the -plane.

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Q10:

A transformation, , that maps the -plane to the -plane is given by , where . Find the image of the region under the transformation .

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