# Worksheet: Möbius Transformations

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 .

• A
• B
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• E

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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• E

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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• E

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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• E

Find the Cartesian equation for the image of .

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• E

Q4:

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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Q5:

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

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• E

Q6:

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

• A
• BThis depends on the definition of .
• C1
• D
• E7

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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Q11:

Given that , simplify the expression , where .

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Q12:

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

Find the Cartesian equation of the image of under the transformation.

• A
• B
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• E

Find the Cartesian equation of the image of under the transformation.

• A
• B
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• E

Q13:

Consider the two Möbius transformations and . Write an expression for the composition in terms of , , , , , , , and and determine whether the resulting transformation is a Möbius transformation.

• A is not a Möbius transformation.
• B is a Möbius transformation.
• C is not a Möbius transformation.
• D is not a Möbius transformation.
• E is a Möbius transformation.

Q14:

Write an expression for the composition , where and are the Möbius transformations and , where or .

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• E

Q15:

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

Find the Cartesian equation of the image of under the transformation.

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• B
• C
• D
• E

Q16:

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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• E