University of Tasmania

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Generalised product integration

posted on 2023-05-27, 17:37 authored by Paget, David Frederick
This thesis is concerned with the construction of quadrature rules for the numerical evaluation of integrals of the form ‚Äöv†¬¥ba w(x)f(x)K(x; ˜í¬™)dx, where the functions w, f and K are required to a possess certain distinct characteristics. Initially, we describe the construction and implementation of a quadrature rule when K(x; ˜í¬™) = (x - ˜í¬™) -1 , a < ˜í¬™ < b, and the above integral is taken to be a Cauchy principal value integral. Convergence of this quadrature rule is investigated when f is a Holder continuous function and when f is an analytic function. In the former case sufficient conditions for convergence of the rule are established, and in the latter case the remainder is expressed as a contour integral from which asymptotic estimates of the remainder may be derived. Particular attention is given to the case when w is a Jacobi weight function. The generalised product integration rule for arbitrary K is developed in Chapter 5. Sufficient conditions for convergence of the generalised rule are established for continuous functions f and K. The implementation and convergence of the rule is further investigated for three functions K of considerable interest, namely: K(x; ˜í¬™) = exp(ix˜í¬™), ˜í¬™ real, I˜í¬™I large; K(x; ˜í¬™) = lnIx - ˜í¬™I,-1 < ˜í¬™ < 1; K(x; ˜í¬™) = Ix - ˜í¬™Is, s > -I, -I < ˜í¬™ < 1. In each case we obtain conditions sufficient to ensure convergence of the rule when w is a Jacobi weight function.


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Copyright 1976 the author - The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s). Thesis (Ph.D.)--University of Tasmania, 1977. Bibliography: l. 96-99

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