18 - Möbius transformatios, part I

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welcome to the third lecture in the

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fourth week of our course analysis of a

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complex kind today we'll start learning

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about Mobius transformations which are

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special conformal mappings this is a two

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lecture part series so we're also

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learning about Mobius transformations in

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lecture 4 let's start with a definition

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a Mobius transformation is also

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sometimes called a fractional linear

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transformation and it is simply a

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function of the form f of Z equals AZ

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plus B over C Z plus D with these

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numbers a b c and d are complex numbers

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such that ad minus bc is not equal to 0

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so for example you could say something

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like f of z equals 3 z minus 4 divided

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by z plus 15i in this case a d which is

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3 times 59 minus bc wishes for is not

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equal to 0 and so this is a Mobius

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transformation in other words you have a

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linear function at the top and a linear

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function at the bottom what happens to

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su led Z go toward infinity what the

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term a Z becomes more and more important

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than term B because more and more

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unimportant and the same thing happens

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in the denominator so in that limit f of

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Z it's something like a z / CZ because

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the b and d have lost their impo