Wave Propagation in Linear Media | re-examined

7.3 Implementation of OscInt and related functions
],
koffs = N[Floor[zerof[offs]/Pi],2 wp];
Re[Table[x/.FindRoot[zerof[x] == (-k+koffs-cfoffs) Pi,
fx,offs,offs+1g],
fk,irel,irel+n-1g]]
Remark (Finding the limit) The test for a de nite limit of (x) was implemented
directly with the built-in function Limit. This works well enough for any kind of polynomial
and rational functions, but usually gives problems with trancendental functions,
which, however, are not supposed to be part of (x) anyhow. The outcome of Limit is
then converted to a machine number and examined whether it is a number or not to lter
out the value Infinity.
Whether (x) is increasing or decreasing can be identi ed with the sign of 0 (x) for x>o.
Accordingly, the table of subdivision points is computed with FindRoot, which is again
straightforward because the respective equations have only one solution above the starting
points o and o +1. Although these solutions must be real-valued, there are often spurious
imaginary parts left, which are eliminated by taking only the real part of the resulting
table.
7.3.5 PartInt
This function actually computes the integral R 1
f(x) dx based on a set of n subdivision points
a
fx0;x1;::: ;xn,1g; xi a by means of extrapolation. Its syntax is
PartInt[f, zerotab, a, opts],
where f is the integrand function (de ned either as pure function or like f[x]), zerotab
is the list of partition points, and a is the lower integration limit. The essential part of the
function body is listed below without most of the option assignments. The rst statement isa
test that should always fail under normal conditions, because the functions that generate the
subdivision points already make sure that the rst point x0 a. If this condition, however, is
not satis ed, then all members greater than a are ltered out for further use, and a warning
message is issued to inform the user that only fewer sequence members than intended will be
computed.
Remark (Selecting members of lists) The ltering of the partition points is accomplished
with the function Select[list, crit], which tests each element of a list with a
given criterion and picks out those for which the test yields True.
The value of the integral consists of two parts. The rst is the de nite integral R x0
f(x) dx
a
(if, of course, x0 6= a) that is computed with the double exponential rule according to the
results of chapter 6.3. The second contribution is the extrapolation result that is obtained
in the well-known manner by computing a sequence of partial integrals R xi+1
f(x) dx and
xi
nding the numerical limit of this series. If the rst partition point equals Infinity, then
this extrapolation is skipped. This feature is imperative in order to include all special cases
of parameter-dependent integrands that lose their in nite oscillations under certain circumstances.
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