Wave Propagation in Linear Media | re-examined

7 Mathematica implementation of a quadrature function
fint is the integrand function (x) u( (x)), which can be de ned in two ways. One is
sin x
to use a so-called pure function, e. g. Sin[#]/#& for x (see also the opening remark
in section 7.5). If the integrand is declared explicitly, then it must be of the form f[x]
or, if there are several other parameters, f[a,b,c][x]. In this case, only the name of
the function together with the additional parameters (f[a,b,c]) has to be passed to
OscInt.
fzero is the function being used to determine the partition points fx0;x1;x2;:::g. Depending
on the option FunctionType it is understood either as an explicit enumeration
of the partition points x (n); n 0 or as the argument function (x) (or an approximation
(x)) of the oscillating factor of the integrand, which is then used to compute
the subdivision points xn numerically. In the latter case, the function must be di erentiable.
In the former case, it is the user's responsibility to ensure that fzero returns
points in ascending order that lie exclusively to the right of the rightmost extremum or
in exion point of the argument function. It is, however, not necessary to see to it that
x (0) > a, as the routine will seek the rst x (k) > a itself. Like fint, the function
fzero can be given either as a pure function or in explicit notation.
a is the lower integration limit.
opts are options the user can set to take in uence on the computation. They are
explained below.
Normally, the partition points are chosen such that they lie in a region where (x) and its
derivative 0 (x) are monotonic, in order to avoid the problems discussed in section 6.2. If an
enumeration function is passed to OscInt, this function must be written accordingly. If the
original (x) is used as fzero, then z0 is selected to be larger than the largest real part of the
solutions of 0 (x) =0. In some cases, however, it might be more convenient tochoose the rst
subdivision point according to a di erent criterion | particularly if we use an approximation
(x). Thus there is a second way to invoke OscInt with an explicit speci cation of a lower
subdivision limit a0:
OscInt[fint, fzero, fa, a0g, opts].
Apart from the new parameter, the meanings of the other input values remain unchanged.
The partition points are now chosen such that x (k) >a0; k K(a0), or x0 >a0, depending
on the interpretation of fzero.
There is a variety of options indispensable to control the evaluation of OscInt. Some of them
are standard Mathematica options that are used also for other numerical functions, others
have been newly introduced.
WynnDegree is the standard option for the computation of the limit of the sequence of
partial sums. Its default value is Infinity.
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