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app2 - Poularikas A.D Appendix 2 Series and Summations The...

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Poularikas, A.D . Appendix 2 : Series and Summations .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000
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© 2000 by CRC Press LLC Appendix 2: Series and Summations Series The expression in parentheses following certain of the series indicates the region of convergence. If not otherwise indicated, it is to be understood that the series converges for all finite values of x . Binomial ( y 2 < x 2 ) ( x 2 < 1) ( x 2 < 1) ( x 2 < 1) ( x 2 < 1) Reversion of Series Let a series be represented by ( a 1 0) to find the coefficients of the series x y x nx y n n x y n n n x y n n n n n + ( ) = + + ( ) + ( ) ( ) + 1 2 2 3 3 1 2 1 2 3 ! ! L 1 1 1 2 1 2 3 2 3 ± ( ) = ± + ( ) ± ( ) ( ) + x nx n n x n n n x n ! ! L 1 1 1 2 1 2 3 2 3 ± ( ) = + + ( ) + ( ) + ( ) + x nx n n x n n n x n m m L ! ! 1 1 1 2 3 4 5 ± ( ) = + + + x x x x x x m m m L 1 1 2 3 4 5 6 2 2 3 4 5 ± ( ) = + + + x x x x x x m m m L y a x a x a x a x a x a x = + + + + + + 1 2 2 3 3 4 4 5 5 6 6 L x A y A y A y A y = + + + + 1 2 2 3 3 4 4 L
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© 2000 by CRC Press LLC Taylor 1. (Increment form) 2. 3. If f ( x ) is a function possessing derivatives of all orders throughout the interval a x b , then there is a value X , with a < X < b , such that A a A a a A a a a a A a a a a a a a A a a a a a a a a a a a a A a a a 1 1 2 2 1 3 3 1 5 2 2 1 3 4 1 7 1 2 3 1 2 4 2 3 5 1 9 1 2 2 4 1 2 3 2 2 4 1 3 5 1 2 2 3 6 1 11 1 3 1 1 2 1 5 5 1 6 3 14 21 1 7 = = = ( ) = ( ) = + + ( ) = 2 5 1 3 3 4 1 2 3 3 1 4 6 1 2 2 2 4 1 2 2 3 2 2 5 7 1 13 1 4 2 6 1 4 3 5 1 4 4 2 1 2 2 3 4 1 2 2 2 3 2 2 6 1 7 84 28 28 42 1 8 8 4 120 180 132 a a a a a a a a a a a a a a a a A a a a a a a a a a a a a a a a a a + + ( ) = + + + ( + + 5 7 1 3 2 2 5 1 3 2 3 4 1 3 3 3 1 2 4 3 36 72 12 330 a a a a a a a a a a a a a ) f x f a x a f a x a f a x a f a x a n f a n n ( ! ! ! = ( ) + ( ) ( ) + ( ) ′′ ( ) + ( ) ′′′ ( ) + + ( ) ( ) + ( ) ( ) 2 3 2 3 L L Taylor's Series f x h f x hf x h f x h f x f h xf h x f h x f h + ( ) = ( ) + ( ) + ′′ ( ) + ′′′ ( ) + = ( ) + ( ) + ′′ ( ) + ′′′ ( ) + 2 3 2 3 2 3 2 3 ! ! ! ! L L f b f a b a f a b a f a b a n f a b a n f X n n n n ( ) = ( ) + ( ) ( ) + ( ) ′′ ( ) + + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) 2 1 1 2 1 ! ! ! L
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© 2000 by CRC Press LLC b = a + h , 0 < θ < 1.
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