app2 - Poularikas, A.D. Appendix 2: Series and Summations....

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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 Fnite values of x . Binomial ( y 2 < x 2 ) ( x 2 < 1) ( x 2 ( x 2 ( x 2 Reversion of Series Let a series be represented by ( a 1 0) to Fnd the coefFcients of the series xy x n x y nn n n n n + () =+ + + + −− 12 2 33 1 2 3 ! ! L 11 1 2 3 23 ± =± + ± + xn x x n x n !! L 1 2 3 ± + + + + x x n x n mm L 1 2345 ± = +++ x xxxxx m L 1 1 2 3456 2 ± + + x x xxxx m L ya xa x = ++++++ 2 3 3 4 4 5 5 6 6 L xA yA y = ++++ 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 aa a A a aaa a a a A a a A a 1 1 2 2 1 3 3 1 5 2 2 13 4 1 7 123 1 2 42 3 5 1 9 1 2 24 1 2 3 2 2 4 1 3 51 2 2 3 6 1 11 1 3 11 2 1 55 1 6 3 14 21 1 7 == = () =− =+ + = 25 1 3 34 12 3 31 4 61 2 2 2 41 2 23 2 2 5 7 1 13 1 4 26 1 4 35 1 4 4 2 1 2 2 3 4 1 2 2 2 3 2 2 6 1 78 4 2 82 84 2 1 8 8 4 120 180 132 a a A a a a + +− −−− + + ( ++ 5 71 3 2 2 3 234 1 3 3 3 4 3 36 72 12 330 a aaaa −− ) fx fa x af a xa fa n n n ( !! ! = + ′′ + ′′′ + LL Taylor's Series fx h f x h fx h fh x f h x fh x + = + + + + = + + + + L L fb b af a ba n n fX 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. or where 0 < The above forms are known as Taylor’s series with the remainder term. 4. Taylor’s series for a function of two variables If ; etc., and if with the bar and subscripts means that after differentiation we are to replace x by a and y by b , Maclaurin fa h fa h f a h h n h n fah n n n n + () = + + ′′ ++ 21 1 2 1 !
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This note was uploaded on 12/03/2009 for the course EEE transforn taught by Professor Profcenk during the Spring '09 term at Dogus University.

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app2 - Poularikas, A.D. Appendix 2: Series and Summations....

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