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Chapter3

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

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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 = 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

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CHAPTER 3. EXAMPLES OF FUNCTIONS 24 which can be rearranged to ( ad bc )( z 1 z 2 ) = 0 . Since ad bc = 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 = 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. Proof. Translations and dilations certainly map circles and lines into circles and lines, so by the last proposition, we only have to prove the theorem for the inversion f ( z ) = 1 z . Before going on we find a standard form for the equation of a straight line. Starting with ax + by = c (where z = x + iy ), let α = a + bi . Then α z = ax + by + i ( ay bx ) so α z + α z = α z + α z = 2 Re( α z ) = 2 ax + 2 by . Hence our standard equation for a line becomes α z + α z = 2 c, or Re( α z ) = c. (3.1) Circle case : Given a circle centered at z 0 with radius r
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Chapter3 - Chapter 3 Examples of Functions Obvious is the...

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