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Abstract

This thesis is concerned with the construction of quadrature
rules for the numerical evaluation of integrals of the form ∫ba w(x)f(x)K(x; λ)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; λ) = (x - λ) -1 , a < λ < 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; λ) = exp(ixλ), λ real, IλI large; K(x; λ) = lnIx - λI,-1 < λ < 1;
K(x; λ) = Ix - λIs, s > -I, -I < λ < 1. In each case we obtain
conditions sufficient to ensure convergence of the rule when w is a
Jacobi weight function.

Item Type:	Thesis - PhD
Authors/Creators:	Paget, David Frederick
Keywords:	Numerical integration
Copyright Holders:	The Author
Copyright Information:	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).
Additional Information:	Thesis (Ph.D.)--University of Tasmania, 1977. Bibliography: l. 96-99
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