Worksheet: Möbius Transformations

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

Q1:

Consider the MΓΆbius transformations 𝑇(𝑧)=𝑧+3π‘–π‘–π‘§βˆ’2 and 𝑇(𝑧)=𝑖𝑧+12π‘§οŠ¨, where 𝑧≠2𝑖 or 0.

Write an expression for the composition π‘‡βˆ˜π‘‡(𝑧).

  • Aπ‘‡βˆ˜π‘‡(𝑧)=2π‘–π‘§βˆ’52𝑧+6π‘–οŠ§οŠ¨
  • Bπ‘‡βˆ˜π‘‡(𝑧)=7𝑖𝑧+1(βˆ’4+𝑖)𝑧+1
  • Cπ‘‡βˆ˜π‘‡(𝑧)=(1+𝑖)𝑧+1+3𝑖(2+𝑖)π‘§βˆ’2
  • Dπ‘‡βˆ˜π‘‡(𝑧)=7𝑖𝑧+1βˆ’5𝑧+π‘–οŠ§οŠ¨
  • Eπ‘‡βˆ˜π‘‡(𝑧)=(1+𝑖)π‘§βˆ’2+3𝑖2𝑧+6π‘–οŠ§οŠ¨

Q2:

A transformation that maps the 𝑧-plane to the 𝑀-plane is defined by π‘‡βˆΆπ‘§β†¦1𝑧, where 𝑧≠0.

Find an equation for the image of |𝑧|=2 under the transformation.

  • A|𝑀|=1
  • B|𝑀|=12
  • C𝑀=2
  • D𝑀=12
  • E|𝑀|=2

Find an equation for the image of arg(𝑧)=3πœ‹4.

  • Aarg(𝑀)=πœ‹4
  • Barg(𝑀)=βˆ’3πœ‹4
  • Carg(𝑀)=43πœ‹
  • Darg(𝑀)=βˆ’πœ‹4
  • Earg(𝑀)=3πœ‹4

Find a Cartesian equation for the image of Im(𝑧)=2.

  • A𝑒+𝑣+14=116
  • B𝑒+𝑣+12=14
  • C2𝑒+ο€Ό2𝑣+14=116
  • D𝑣=0
  • E𝑣=12

Find a Cartesian equation for the image of |π‘§βˆ’π‘–|=12.

  • A𝑒+𝑣+43=49
  • B𝑒+𝑣+83=0
  • C𝑒+𝑣+43=0
  • D𝑒+𝑣=0
  • E𝑒+𝑣+43=49

Q3:

A transformation that maps the 𝑧-plane to the 𝑀-plane is defined by π‘‡βˆΆπ‘§β†¦13π‘§βˆ’6𝑖.

Find the Cartesian equation for the image of |𝑧+2|=3.

  • Aο€Όπ‘’βˆ’23+ο€Όπ‘£βˆ’23=1
  • Bο€Όπ‘’βˆ’23+𝑣+23=1
  • Cπ‘’βˆ’43𝑒+π‘£βˆ’43π‘£βˆ’16π‘’π‘£βˆ’19=0
  • Dο€Όπ‘’βˆ’23+𝑣+23=0
  • Eο€Όπ‘’βˆ’221+𝑣+221=5147

Find the Cartesian equation for the image of Re(𝑧)=5.

  • Aο€Όπ‘’βˆ’130οˆβˆ’π‘£=1900
  • Bο€Όπ‘’βˆ’115+𝑣=1225
  • Cο€Όπ‘’βˆ’130+𝑣=1900
  • Dο€Όπ‘’βˆ’115οˆβˆ’π‘£=1225
  • E𝑒+𝑣=5

Q4:

A transformation, 𝑇, that maps the 𝑧-plane to the 𝑀-plane is given by 𝑇(𝑧)=(2+𝑖)𝑧+4π‘§βˆ’π‘–, where 𝑧≠𝑖.

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

  • A(π‘’βˆ’1)+𝑣+52=134
  • B𝑣=4βˆ’32𝑒
  • C𝑣=32π‘’βˆ’4
  • D𝑣=32𝑒+4
  • E(π‘’βˆ’1)+𝑣+52=12

Hence, find the image of the region Im(𝑧)>0 under the transformation 𝑇.

  • A𝑣>4βˆ’32𝑒
  • B(π‘’βˆ’1)+𝑣+52>134
  • C𝑣<4βˆ’32𝑒
  • D𝑣>32π‘’βˆ’4
  • E(π‘’βˆ’1)+𝑣+52<134

Q5:

Which of the following MΓΆbius transformations maps |𝑧|=1 to the real axis?

  • A𝑇(𝑧)=𝑖𝑧+𝑖𝑧+1
  • B𝑇(𝑧)=π‘–π‘§βˆ’2𝑖𝑧+2
  • C𝑇(𝑧)=π‘–π‘§βˆ’π‘–π‘§+1
  • D𝑇(𝑧)=π‘§βˆ’1𝑧+1
  • E𝑇(𝑧)=π‘§βˆ’2𝑧+2

Q6:

Suppose πΏβˆΆβ„β†’β„ is a linear transformation with 𝐿(2)=7. What is the value of 𝐿(1)?

  • A72
  • BThis depends on the definition of 𝐿.
  • C1
  • D27
  • E7

Q7:

Suppose πΏβˆΆβ„β†’β„ is a linear transformation. What is the value of 𝐿(0)?

Q8:

Which of the following MΓΆbius transformations, 𝑇, maps βˆ’2 to 0, 0 to βˆ’3, and has limο™β†’βˆžπ‘‡(𝑧)=βˆ’3𝑖?

  • A𝑇(𝑧)=π‘–π‘§βˆ’33𝑧+6
  • B𝑇(𝑧)=𝑧+2π‘–βˆ’3π‘–π‘§βˆ’6𝑖
  • C𝑇(𝑧)=βˆ’3π‘–π‘§βˆ’6𝑖𝑧+2𝑖
  • D𝑇(𝑧)=3𝑧+6π‘–π‘§βˆ’3
  • E𝑇(𝑧)=βˆ’βˆ’π‘§+οŠ§οŠ©οƒο™οŠ§οŠ¬οƒοŠ§οŠ¨οƒ

Q9:

Find the equation for the image of arg(𝑧)=πœ‹2 under the transformation 𝑀=𝑧7π‘§βˆ’2𝑖, 𝑧≠2𝑖7, which maps the 𝑧-plane to the 𝑀-plane.

  • Aarg(𝑀)=πœ‹2
  • Barg(𝑀)=βˆ’πœ‹2
  • Carg(𝑀)=0
  • Darg(𝑀)=2πœ‹3
  • Earg(𝑀)=πœ‹6

Q10:

A transformation, 𝑇, that maps the 𝑧-plane to the 𝑀-plane is given by 𝑇(𝑧)=𝑖𝑧+2(2βˆ’π‘–)𝑧+4𝑖, where π‘§β‰ βˆ’4𝑖2βˆ’π‘–. Find the image of the region |𝑧+2𝑖|≀|π‘§βˆ’3βˆ’π‘–| under the transformation 𝑇.

  • A𝑒+109+ο€Όπ‘£βˆ’19β‰₯89
  • B𝑒+109+ο€Όπ‘£βˆ’19οˆβ‰€89
  • C𝑒+109+ο€Όπ‘£βˆ’19β‰₯7481
  • Dο€Όπ‘’βˆ’49+ο€Όπ‘£βˆ’13β‰₯3481
  • E𝑒+109+𝑣+19οˆβ‰€89

Q11:

Given that π‘Žπ‘‘βˆ’π‘π‘=0, simplify the expression π‘Žπ‘§+𝑏𝑐𝑧+𝑑, where π‘§β‰ βˆ’π‘‘π‘.

  • Aπ‘Ž+𝑏𝑐+𝑑
  • Bπ‘Žπ‘‘
  • C𝑏𝑐
  • Dπ‘π‘Ž
  • E𝑏𝑑

Q12:

A transformation which maps the 𝑧-plane to the 𝑀-plane is given by 𝑀=𝑖𝑧+6π‘§βˆ’3𝑖, where 𝑧≠3𝑖.

Find the Cartesian equation of the image of |𝑧|=3 under the transformation.

  • A𝑒+ο€Όπ‘£βˆ’12=34
  • B𝑒+𝑣=34
  • C𝑣=32
  • D𝑀=𝑖+21βˆ’π‘–
  • E𝑣=34

Find the Cartesian equation of the image of |𝑧|=1 under the transformation.

  • A𝑒+ο€Όπ‘£βˆ’178=94
  • B𝑒+𝑣+178=964
  • C𝑒+ο€Όπ‘£βˆ’178=964
  • D𝑒+ο€Όπ‘£βˆ’52=βˆ’3
  • E𝑒+ο€Όπ‘£βˆ’52=34

Q13:

Consider the two MΓΆbius transformations 𝑇(𝑧)=π‘Žπ‘§+𝑏𝑐𝑧+π‘‘οŠ§ and 𝑇(𝑧)=𝛼𝑧+𝛽𝛾𝑧+π›ΏοŠ¨. Write an expression for the composition π‘‡βˆ˜π‘‡οŠ§οŠ¨ in terms of π‘Ž, 𝑏, 𝑐, 𝑑, 𝛼, 𝛽, 𝛾, and 𝛿 and determine whether the resulting transformation is a MΓΆbius transformation.

  • Aπ‘‡βˆ˜π‘‡(𝑧)=π‘Žπ›Όπ‘§+𝑏𝛽𝑐𝛾𝑧+π‘‘π›ΏοŠ§οŠ¨ is not a MΓΆbius transformation.
  • Bπ‘‡βˆ˜π‘‡(𝑧)=(π‘Žπ›Ό+𝑐𝛽)𝑧+(𝑏𝛼+𝑑𝛽)(π‘Žπ›Ύ+𝑐𝛿)𝑧+(𝑏𝛾+𝑑𝛿) is a MΓΆbius transformation.
  • Cπ‘‡βˆ˜π‘‡(𝑧)=π‘Žπ›Όπ‘§+(π‘Žπ›½+𝑏𝛼)𝑧+𝑏𝛽𝑐𝛾𝑧+(𝑐𝛿+𝑑𝛾)𝑧+π‘‘π›ΏοŠ§οŠ¨οŠ¨οŠ¨ is not a MΓΆbius transformation.
  • Dπ‘‡βˆ˜π‘‡(𝑧)=(π‘Žπ›Ό+𝑏𝛾)𝑧+(π‘Žπ›½+𝑏𝛿)𝑐𝑧+π‘‘οŠ§οŠ¨ is not a MΓΆbius transformation.
  • Eπ‘‡βˆ˜π‘‡(𝑧)=(π‘Žπ›Ό+𝑏𝛾)𝑧+(π‘Žπ›½+𝑏𝛿)(𝑐𝛼+𝑑𝛾)𝑧+(𝑐𝛽+𝑑𝛿) is a MΓΆbius transformation.

Q14:

Write an expression for the composition π‘‡βˆ˜π‘‡(𝑧), where 𝑇(𝑧) and 𝑇(𝑧) are the MΓΆbius transformations 𝑇(𝑧)=(2+3𝑖)𝑧+5βˆ’π‘§+2βˆ’π‘–οŠ§ and 𝑇(𝑧)=2π‘§βˆ’3+𝑖(βˆ’1βˆ’2𝑖)𝑧+5, where 𝑧≠2βˆ’π‘– or βˆ’5βˆ’1βˆ’2𝑖.

  • Aπ‘‡βˆ˜π‘‡(𝑧)=(7+5𝑖)𝑧+5+5𝑖(βˆ’1βˆ’7𝑖)𝑧+5βˆ’15π‘–οŠ¨οŠ§
  • Bπ‘‡βˆ˜π‘‡(𝑧)=(βˆ’1βˆ’4𝑖)𝑧+16βˆ’7𝑖(βˆ’6βˆ’3𝑖)𝑧+13βˆ’6π‘–οŠ¨οŠ§
  • Cπ‘‡βˆ˜π‘‡(𝑧)=(4+3𝑖)𝑧+2+𝑖(βˆ’1βˆ’2𝑖)𝑧+5βˆ’π‘–οŠ¨οŠ§
  • Dπ‘‡βˆ˜π‘‡(𝑧)=(βˆ’1βˆ’4𝑖)𝑧+16βˆ’7𝑖(βˆ’2βˆ’3𝑖)𝑧+7βˆ’4π‘–οŠ¨οŠ§
  • Eπ‘‡βˆ˜π‘‡(𝑧)=(5+2𝑖)𝑧+5𝑖(4βˆ’7𝑖)π‘§βˆ’10π‘–οŠ¨οŠ§

Q15:

A transformation that maps the 𝑧-plane to the 𝑀-plane is given by 𝑇(𝑧)=𝑖𝑧+6βˆ’3π‘–π‘§βˆ’3𝑖, where 𝑧≠3𝑖.

Find the Cartesian equation of the image of |𝑧|=6 under the transformation.

  • A𝑒+13+ο€Όπ‘£βˆ’23=89
  • B𝑒+13+ο€Όπ‘£βˆ’23=34
  • C𝑒+13+𝑣+23=βˆ’29
  • D(𝑒+1)+𝑣=4
  • E(𝑒+1)+𝑣=2

Q16:

A transformation, 𝑇, that maps the 𝑧-plane to the 𝑀-plane is given by 𝑇(𝑧)=𝑧+π‘–π‘§βˆ’π‘–, where 𝑧≠𝑖. Find the image of the region Re(𝑧)<4 under the transformation 𝑇.

  • A(π‘’βˆ’1)+ο€Όπ‘£βˆ’14>116
  • B(π‘’βˆ’1)+ο€Όπ‘£βˆ’14<116
  • C𝑒+ο€Όπ‘£βˆ’43<19
  • D𝑒+ο€Όπ‘£βˆ’43>19
  • E𝑒+ο€Όπ‘£βˆ’83<499

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