118Homework11

118Homework11 - and are related by the equation = m + arg f...

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Homework 11 Due Monday, Dec. 6 We saw in class that if a function f is conformal at a point z 0 (i.e., f is analytic at z 0 and f 0 ( z 0 ) 6 = 0) then f preserves the angle between curves passing through z 0 . The following problems consider what happens when z 0 is a zero of f 0 . Suppose that f is analytic at z 0 and that f 0 ( z 0 ) = f 00 ( z 0 ) = . . . = f ( m - 1) ( z 0 ) = 0 , and f ( m ) ( z 0 ) 6 = 0 for some positive integer m 1. Also write w 0 = f ( z 0 ). 1. Use the Taylor series for f about the point z 0 to show that there is a neighborhood of z 0 in which the difference f ( z ) - w 0 can be written f ( z ) - w 0 = ( z - z 0 ) m f ( m ) ( z 0 ) m ! [1 + g ( z )] , where g ( z ) is continuous at z 0 and g ( z 0 ) = 0. 2. Let C be a smooth arc and let Γ be the image of C under the transfor- mation w = f ( z ). Let θ 0 be the angle of inclination of a tangent line to C at z 0 , and let φ 0 be the angle of inclination of a tangent line to Γ at w 0 . Note that θ 0 is the limit of arg( z - z 0 ) as z approaches z 0 along C . Also note that φ 0 is the limit of arg( f ( z ) - w 0 ) as z approaches z 0 along C . Use these facts, together with the result of Problem 1 to show that
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Unformatted text preview: and are related by the equation = m + arg f ( m ) ( z ) . 3. Let C 1 and C 2 be two smooth arcs passing through the point z . Also let 1 and 2 be the image of C 1 and C 2 , respectively, under the transforma-tion w = f ( z ). In addition, let be the angle between C 1 and C 2 as the pass through z . Show how the relation obtained in Problem 2 implies that the corresponding angle between the curves 1 and 2 as they pass through the point w = f ( z ) is equal to m . (Note that the transforma-tion is conformal at z when m = 1 and that z is a critical point when m 2.)...
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