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Unformatted text preview: 17.2 Linear Fractional Transformations An important class of mappings are linear fraction transformations (or M¨obius transformations), which are of the form w = f ( z ) = az + b cz + d , (990) a,b,c,d ∈ C , ad bc negationslash = 0. Then we have f ′ ( z ) = a ( cz + d ) ( az + b ) c ( cz + d ) 2 = ad bc ( cz + d ) 2 negationslash = 0 , z ∈ C , (991) and by Thm. 34, the mapping (990) is conformal everywhere in C . Special cases of (990) include • a = d = 1, c = 0: w = z + b (translations) •  a  = d = 1, b = c = 0: w = az ,  a  = 1 (rotations) • c = 0, d = 1: w = az + b , a negationslash = 0 (linear transformations) • a = d = 0, b = c = 1: w = 1 /z (inversion in the unit circle) Example: The inversion w = 1 /z . We use polar forms z = re iϑ , w = Re iϕ to obtain Re iϕ = 1 re iϑ = 1 r e iϑ ⇒ R = 1 r , ϕ = ϑ (mod 2 π ) . (992) Hence the unit circle  z  = r = 1 in the zplane is mapped onto the unit circle  w  = R = 1 in the wplane (oriented in the opposite direction). We look at points ( x,y ) in the zplane which satisfy the equation A ( x 2 + y 2 ) + Bx + Cy + D = 0 , A,B,C,D ∈ R . (993) For A = 0, the points lie on a straight line ( Bx + Cy = D ), whereas for A negationslash = 0, the points lie on a circle: parenleftbigg x + B 2 A parenrightbigg 2 + parenleftbigg y + C 2 A parenrightbigg 2 = B 2 + C 2 4 AD 4 A 2 (994) With z = x + iy (993) becomes Az z + B 2 ( z + z ) + C 2 i ( z z ) + D = 0 . (995) 178 Replacing z = 1 /w and multiplying by w w we have A + B 2 ( w + w ) C 2 i ( w w ) + Dw w = 0 . (996) With w = u + iv , we obtain D ( u 2 + v 2 ) + Bu Cv + A = 0 , (997) which is of the same form as (993). We conclude that circles and straight lines in the zplane are mapped to circles and straight lines in the wplane by the mapping w = 1 /z . This is not only true for the inversion, but for every linear fractional transformation. Before we prove that, it is useful to notice that every linear fractional transformation can be written as a composition of inversions and linear transformations:...
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.
 Spring '08
 Staff
 Math, Transformations

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