Graduate Thesis Or Dissertation
 

On the generalization and application of the Eular MacLaurin formula

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/nz8063291

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  • The Euler-MacLaurin sum formula has appeared in the titles of two quite recent papers whose authors were primarily interested in certain applications. In this paper a somewhat different approach to the myriad of formulas for summation, integration, differentiation, etc. , is based on the simple identity which defines the set of Bernoulli numbers. Variations of this identity are obtained by the most elementary manipulations, then application of the Laplace transformation leads to the well-known formulas, trapezoidal rule, Simpson's rule, etc., complete with an infinite series of higher derivatives. This type of formula is particularly valuable in carrying out a Frank type of inversion of a Laplace transform. In particular, the Frank method has been extended to the alternating series case. The representation of error of the approximation formula by means of an integral involving a periodic polynomial has been extended to Simpson's rule, with indication of a general method for extending the theory for more general approximation formulas.
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