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Generalised product integration

Paget, David Frederick 1976 , 'Generalised product integration', PhD thesis, University of Tasmania.

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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
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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
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Additional Information:

Thesis (Ph.D.)--University of Tasmania, 1977. Bibliography: l. 96-99

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