Lecture23

# Lecture23 - 17.2 Linear Fractional Transformations An...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 z-plane is mapped onto the unit circle | w | = R = 1 in the w-plane (oriented in the opposite direction). We look at points ( x,y ) in the z-plane 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 z-plane are mapped to circles and straight lines in the w-plane 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:...
View Full Document

## This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.

### Page1 / 6

Lecture23 - 17.2 Linear Fractional Transformations An...

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

View Full Document
Ask a homework question - tutors are online