Chap 8 Mapping by Elementary Functions

Transcript

1 Chap 8 Mapping b Elementar Fntions 68. Linear Transformations A, where A is a nonero onstant and iα iθ Let A= ae, = re i( α+ θ) ( ar) e rotate b α = arg A. epand or ontrat radis b a = A + B Let + iv = + i B= b+ ib then ( v, ) = ( + b, + b) translation The general linear transformation A+ B ( A ) is a omposition of Z = A and w = Z + B E: ( + i) + i 4 = e + v B A

2 69. The Transformation mapping between w = nonero points of and w planes. Sine =, is the omposite of Z =, Z a refletion in the real ais An inversion with respet to nit irle Z = arg Z = arg Bease lim = and lim = Z w Z 3 To define a one to one transformation T( ) from the etended plane onto the etended w plane b writing T() =, T( ) = and T( ) = for, T is ontions throghot the etended plane. Sine w = + iv is the image of = + i nder w = i w = = + =, v= (3) + + -v Similarl, =, = = w + v + v 4

3 Let,,, be real nmbers, and + > 4 A BCD B C AD The eqation ( ) (5) A + + B+ C+ D= represents an arbitrar irle or line, where A for a irle and A= for a line A B C D = A A A B C 4AD B C A A 4A + ( + ) + ( + ) + = B C B + C 4AD ( + ) + ( + ) = ( ) A A A a irle when B + C 4AD> 5 A= B + C > B C, whih means and are not both ero. B + C + D = is a line. v sbstitting b, b in (5), + v + v D + v + B Cv+ A= we get ( ) represents a irle or line The mapping w = transforms irles and lines into irles and lines. (a) A, D, A irle not passing throgh the origin = is tranformed to a irle not passing throgh the origin w =. T (b) A, D=, A irle thr = line not passing. A= = T (), D, a line not passing a irle throgh. A= D= = T (d),, a line thr a line thr. 6

4 E. v vertial line =, w = ( ) + v = ( ) a point = (, ) b eq.(3) ( v, ) = (, ) + +, + + C < C C < C > C > C C > C < 7 E. = w = + ( v+ ) + ( ) E3. ( ) + v ( ) C C C C 8

5 7. Linear Frational Transformation () a + b ( ad b ), a, b,, d, omple onstants + d is alled a linear frational transformation or Mobis transformation. Eq.() w + dw a b = ( b ( ad) ) (~ A w+ B+ Cw+ D=, AD- BC ) bilinear transformation when =, the ondition beomes ad. a b () + a linear fntion d d linear in linear in w bilinear in and w 9 when, a ad ( + d) + b () w = + d a b ad = + + d is a omposition of a b ad Z = + d, W =, + W Z a linear frational transformation alwas transforms irles and lines into irles and lines. ( a + b, )

6 solving () for, dw + b = ( ad - b ) w a d b If =, = w one-to-one mapping a a a If, & w has one-to-one mapping eept at. Denominator= define a linear frational tranformation T on the etended plane sh that a. in the image of = when. a + b T( ) = + d ( ad b ) (5) T( ) = if = a T( ) = if d T( ) = if This makes T ontinos on the etended plane (E, se4). We enlarge the domain of definition, (5) is a one-to-one mapping of the etended plane onto the etended w plane.

7 i.e., T( ) T( ) whenever and for eah point w in w-plane, there is a point in the -plane, sh that T( ) = w. There is an inverse transformation T T - T w T w T T T ( ) = iff ( ) = dw + b ( w) = w a ( ) = if = a ( ) = if d ( ) = - ( ad b ) A linear frational transformation 3 There is alwas a linear frational transformation that maps three given distint points,, and 3 onto three speified distint points w, w and w3. a + b E. find w = that + d map = = = 3 onte w = i w = w = i 3 a + b b ( b( a ) ) + b a a+ b a+ b i = i= + b + b i ib = a + b i + ib = a + b i = b, = -ib a= ib ( b ) ib+ b i+ i w = = = ib+ b i+ i+ b = d b = d 4

8 E: =, =, 3 = w = i, w =, w = 3 d = a + b ( b ) a+ b a+ b i = = i = a + b = a + b i ( i ) = b, b= i i+ a = i b= ( i ) = ( i+ ) + ( i ) w = 5 7. An Impliit Form The eqation ( w w)( w w3) ( )( 3) = ( w w )( w w ) ( )( ) 3 3 defines (impliitl) a linear frational transformation that maps distint points,, onto distint w, w, w, respetivel. Rewrite () as () ( )( w w )( )( w w ) () = ( )( w w )( )( w w ) If =, right-hand side= w If =, left-hand side= w 3 6

9 If = ( w w )( w w ) = ( w w )( w w ) 3 3 w Epanding () get A w+ B+ Cw+ D= a linear frational transformation. E. = w = i = w = = w = i 3 3 ( w+ i)( i) ( + )( ) = ( w i)( + i) ( )( + ) ( w+ i)( )( i) = ( w i)( + )( + i) ( w + i w i)( i) = ( w i + w i)( + i) ( w + i w i iw + + iw ) = ( w + i w + i iw iw ) i- w + iw = (- + i) w( + i) = ( i - ) w = i+ 7 eqation () an be modified for point at infinit. sppose = replae b, and let ( )( 3) ( )( 3) lim = lim ( )( ) ( 3)( ) The desired eqation is = ( w w )( w w ) = ( w w )( w w )

10 E. = w = i = w = = w = 3 3 w w ( )( 3) = w w ( )( ) 3 3 w i ( )(+ ) = w ( + )( ) ( w i)( + ) = ( w )( ) w + i w + i = w w + w = i + i + = ( i+ ) + ( i-) ( i+ ) + ( i ) w = 9 7. Mapping of the pper Half Plane Determine all inear frational transformation T that Im > T w < Im = T w = Choose three points =,, that will be mapped to w = a + b b ( ad -b ) + d b = w = =, b = d d a = onl if ( = ) a w =, a = = b a + w = a + d

11 a b d Sine = and = a =, = (5) iα w e = w = e = iα, = or ( )( ) = ( )( ) bt if = + = + ie.. Re = Re =, or = =, (5) is a onstant transformation, = iα e when =, sine is inside w = or Im > is above the ais. if is above the -ais - - < if is below the -ais - - w = > (6) if is on the -ais = (6) is what we want Z Z Z

12 E. i i i = e has the above mapping propert i+ i E. maps > onto v> + = onto v= () real w real Sine the image of = is either a line or a irle. it mst be the real ais v =. ( -)( + ) + () v= Im Im = Im = ( + )( + ) + + >, v> <, v< also linear frational transformation is onto. QED Eponential and Logarithmi Transformations The transformation e iφ i iφ ρe = e e ρe, = + i Ths ρ = e, φ = + n, n an integer or ρ = e, φ = transformation from plane to w plane () = vertial line = = e φ = (, ), its image ρ, (, ) (,) e 4

13 () = horiontal line C C E e a b a b, d maps onto e ρ e, φ d D' φ = d C' A' φ = B' 5 E. ib i ia φ = b φ = φ = a log = ln r+ iθ ( r >, α < θ < α + ) an branh of log, maps onto a strip v i( α + ) θ α iθ iα 6

14 E3. w = log + prinipal branh is a omposition of Z = and log Z + maps pper half plane > onto ( < θ < ) pper half plane v > maps pper half plane onto the strip < v < The transformation sin Sine sin = sin osh + ios sinh sin = sin osh, v= os sinh E. sin maps, onto v (-to-) E M L A M ' L' D B E ' D ' B ' A' 8

15 A. bondar of the strip real ais () BA segment e + e =, osh = e e = osh, v= sinh = () DB segment = = sin v= (3) DE segment = -, = osh, v= 9 B. Interior of strip maps onto pper half v> of w plane line = < < = sin osh, v= os sinh (- < < ) sin v = hperbola os with foi at the points w =± sin + os =± 3

16 Consider a horiontal line segment =,, > its image is = sin osh, v= os sinh osh v + = an ellipse sinh with foi at ± osh sinh =± v C' A B C D E = C > B' A ' E' D' 3 E. D E bi C L F B A D ' E' C' L' A ' =, = sin, v= ( ) E3. os = sin( + ) Z = +, sinz E4. sinh isin( i) Z = i, W = sin Z w = iw B' osh = os( i) 3

17 75. Mapping b Branhes of are the two sqare roots of when if = rep( iθ) ( r >, < θ ) i( θ + k) then = rep ( k =,) iθ prinipal root r ep an also be written = ep( log ) The prinipal branh F ( ) of is obtained b taking the prinipal branh of log or F ( ) = ep( log ) iθ F ( ) = r ep ( >, < Arg< ) ( r >, < θ < ) 33 E C D R R B A D' v C' R ' R ' B' A' E r, θ ρ, φ 4 F (sin ) Z = sin, F ( Z) ( >, < Arg < ) D C B A sin Z D' C ' B ' A' F ( ) C '' v B '' D' ' A' ' 34

18 when < θ < and the branh log = ln r+ i( θ + ) is sed, i( θ + ) = F ( ) = r ep iθ = r ep < θ + < 3 = F ( ) other branhes of iθ fa ( ) = r ep ( r >, α < θ < α + ) n i( θ + k) n = ep( log ) = r ep, k =,,,... n n n Sqare roots of polnomials E. Branhes of ( ) is a omposition of Z = with Z Eah branh of Z ields a branh of ( ) iθ When Z = Re, branhes of Z are iθ Z = R ep ( R>, α < θ < α + ) If we write R =, Θ= Arg( - ) and θ = arg( ) two branhes of ( ) are iθ G ( ) = R ep ( R>, <Θ< ) iθ and g( ) = R ep ( R >, < θ < ) 36

19 G ( ) is defined at all points in the plane eept = and the ra Arg =. The transformation G ( ) is a one-to-one mapping of the domain - >, < Arg( ) < onto the right half Re w> of the w-plane v Z R θ R R θ θ w The transformation g ( ) maps the domain >, < arg( ) < in a ont-to-one manner onto the pper half plane Im w > E. ( ) = ( ) ( + ) ( ± ) 37