Chapter3

Chapter3 - Chapter 3 Examples of Functions Obvious is the...

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Unformatted text preview: Chapter 3 Examples of Functions Obvious is the most dangerous word in mathematics. E. T. Bell 3.1 M¨obius Transformations The first class of functions that we will discuss in some detail are built from linear polynomials. Definition 3.1. A linear fractional transformation is a function of the form f ( z ) = az + b cz + d , where a,b,c,d ∈ C . If ad- bc 6 = 0 then f is called a M¨obius 1 transformation . Exercise 11 of the previous chapter states that any polynomial (in z ) is an entire function. From this fact we can conclude that a linear fractional transformation f ( z ) = az + b cz + d is holomorphic in C \ - d c (unless c = 0, in which case f is entire). One property of M¨ obius transformations, which is quite special for complex functions, is the following. Lemma 3.2. M¨obius transformations are bijections. In fact, if f ( z ) = az + b cz + d then the inverse function of f is given by f- 1 ( z ) = dz- b- cz + a . Remark. Notice that the inverse of a M¨ obius transformation is another M¨ obius transformation. Proof. Note that f : C \ {- d c } → C \ { a c } . Suppose f ( z 1 ) = f ( z 2 ), that is, az 1 + b cz 1 + d = az 2 + b cz 2 + d . As the denominators are nonzero, this is equivalent to ( az 1 + b )( cz 2 + d ) = ( az 2 + b )( cz 1 + d ) , 1 Named after August Ferdinand M¨ obius (1790–1868). For more information about M¨ obius, see http://www-groups.dcs.st-and.ac.uk/ ∼ history/Biographies/Mobius.html . 23 CHAPTER 3. EXAMPLES OF FUNCTIONS 24 which can be rearranged to ( ad- bc )( z 1- z 2 ) = 0 . Since ad- bc 6 = 0 this implies that z 1 = z 2 , which means that f is one-to-one. The formula for f- 1 : C \{ a c } → C \{- d c } can be checked easily. Just like f , f- 1 is one-to-one, which implies that f is onto. Aside from being prime examples of one-to-one functions, M¨ obius transformations possess fas- cinating geometric properties. En route to an example of such, we introduce some terminology. Special cases of M¨ obius transformations are translations f ( z ) = z + b , dilations f ( z ) = az , and in- versions f ( z ) = 1 z . The next result says that if we understand those three special transformations, we understand them all. Proposition 3.3. Suppose f ( z ) = az + b cz + d is a linear fractional transformation. If c = 0 then f ( z ) = a d z + b d , if c 6 = 0 then f ( z ) = bc- ad c 2 1 z + d c + a c . In particular, every linear fractional transformation is a composition of translations, dilations, and inversions. Proof. Simplify. With the last result at hand, we can tackle the promised theorem about the following geometric property of M¨ obius transformations. Theorem 3.4. M¨obius transformations map circles and lines into circles and lines....
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This note was uploaded on 06/18/2011 for the course MATH 375 taught by Professor Marchesi during the Spring '10 term at Binghamton.

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Chapter3 - Chapter 3 Examples of Functions Obvious is the...

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