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3.03.UseExpansions

3.03.UseExpansions - p Approximation Authors Bill Davis...

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Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.03 Using Expansions BASICS B.1) Expansions in powers of H x - b L and approximations based on them · B.1.a) When you want to study behavior near a point x = b other than x = 0, you can use expansions in powers of H x - b L . What is the expansion of a function f @ x D in powers of H x - b L ? · Answer: Given a function f @ x D , the expansion of f @ x D in powers of H x - b L is a @ 0 D + a @ 1 D H x - b L + a @ 2 D H x - b L 2 + a @ 3 D H x - b L 3 + . . . + a @ k D H x - b L k + a @ k + 1 D H x - b L k + 1 + . . . + . . . where the numbers a @ 0 D , a @ 1 D , a @ 2 D , . . ., a @ k D , a @ k + 1 D , . . . are set so that for every positive integer m, the function f @ x D and the polynomial a @ 0 D + a @ 1 D H x - b L + a @ 2 D H x - b L 2 + a @ 3 D H x - b L 3 + . . . + a @ m D H x - b L m have order of contact m at x = b. · B.1.b.i) Come up with the expansion of f @ x D = 1 1 + x in powers of H x - 2 L . · Answer: Look at: b = 2; Clear @ f, expan3, x D ; f @ x _ D = 1 1 + x ; expan3 @ x _ D = Normal @ Series @ f @ x D , 8 x, b, 3 <DD 1 3 + 2 - x 9 + 1 27 H - 2 + x L 2 - 1 81 H - 2 + x L 3 The polynomial you see above has order of contact 3 with f @ x D at x = 2. You can check that: Clear @ k D ; Table @ 8 D @ f @ x D , 8 x, k <D , D @ expan3 @ x D , 8 x, k <D< , 8 k, 0, 4 <D ê .x -> 2 :: 1 3 , 1 3 > , : - 1 9 , - 1 9 > , : 2 27 , 2 27 > , : - 2 27 , - 2 27 > , : 8 81 , 0 >> Look at this: Normal @ Series @ f @ x D , 8 x, b, 6 <DD 1 3 + 2 - x 9 + 1 27 H - 2 + x L 2 - 1 81 H - 2 + x L 3 + 1 243 H - 2 + x L 4 - 1 729 H - 2 + x L 5 + H - 2 + x L 6 2187 This polynomial has order of contact 6 with f @ x D at x = 2. The denominators are powers of 3 and the signs alternate. The expansion of f @ x D = 1 1 + x in powers of H x - 2 L is 1 3 - x - 2 3 2 + H x - 2 L 2 3 3 - H x - 2 L 3 3 4 + . . . + H - 1 L k H x - 2 L k 3 k + 1 + . . . This isn't worth memorizing. · B.1.b.ii) Come up with the expansion of f @ x D = Cos @ x D in powers of I x - p 2 M . · Answer: Look at: Clear @ f, x D ; f @ x _ D = Cos @ x D ; Normal B Series B f @ x D , : x, p 2 , 3 >FF p 2 - x + 1 6 J - p 2 + x N 3 This polynomial has order of contact 3 with f @ x D at x = p 2 . Now look at: Normal B Series B f @ x D , : x, p 2 , 12 >FF p 2 - x + 1 6 J - p 2 + x N 3 - 1 120 J - p 2 + x N 5 + I - p 2 + x M 7 5040 - I - p 2 + x M 9 362880 + I - p 2 + x M 11 39916800 This polynomial has order of contact 12 with f @ x D at x = p 2 . The expansion of f @ x D = Cos @ x D in powers of I x - p 2 M is: - I x - p 2 M + I x - p 2 M 3 3 ! - I x - p 2 M 5 5 ! + . . . + H - 1 L k + 1 I x - p 2 M 2 k + 1 H 2 k + 1 L ! + . . . which just happens to be the expansion of - Sin A x - p 2 E in powers of I x - p 2 M : Series B - Sin B x - p 2 F , : x, p 2 , 7 >F - J x - p 2 N + 1 6 J x - p 2 N 3 - 1 120 J x - p 2 N 5 + I x - p 2 M 7 5040 + O B x - p 2 F 8 Because: Cos @ x D == - Sin B x - p 2 F True Again, none of these expansions are worth memorizing. · B.1.c) When you expand in powers of H x - 0 L , you get the expansion in powers of x: b = 0; Clear @ f, approx, x D ; f @ x _ D = E x ; approx @ x _ D = Normal @ Series @ f @ x D , 8 x, b, 6 <DD 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 + x 6 720 Plot @8 f @ x D , approx @ x D< , 8 x, b - 1, b + 1 < , PlotStyle Ø 88 Thickness @ 0.02 D , Blue < , 8 Thickness @ 0.01 D , Red << , AxesLabel Ø 8 "x" , "" < , PlotLabel Ø "Powers of H x - 0 L centered at x = 0" , Epilog Ø 88 Blue, PointSize @ 0.04 D , Point @8 b, 0 <D< , Text @ "expansion point" , 8 b, 0 < , 8 0, - 2 <D<D - 1.0 - 0.5 0.5 1.0 x 1.0 1.5 2.0 2.5 Powers of H x - 0 L centered at x = 0 Here's what happens when do the same thing except this time expand in powers of H x - b L for b = 1: b = 1; Clear @ f, approx, x D ; f @ x _ D = E x ; approx @ x _ D

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3.03.UseExpansions - p Approximation Authors Bill Davis...

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