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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 06/18/2011 for the course MATH 375 taught by Professor Marchesi during the Spring '10 term at Binghamton.

Page1 / 17

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online