Wave Propagation in Linear Media | re-examined

A.1 Numerical quadrature
AsymptoticExpand::usage =
"AsymptoticExpand[f,x] returns the polynomial part of the
asymptotic expansion of f regarding the variable x at Infinity. If f
is a pure function or is defined as f[...][x] then x need not be
specified."
ApproxLimGeneric::usage =
"ApproxLimGeneric[f,goal] is used to give a figure of merit for the quality
of a polynomial approximation to an argument f of a circular function. It returns
the abscissa value where the ratio of the circular velocity of the approximation
polynomial and the approximation error, respectively, reaches a given goal (the
larger the better). This is equivalent to the number of zeros the approximating
function has within one half period of the frequency defined by the difference
between f and its approximation."
ApproxLimQuad::usage =
"ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the
function a x^2 + b Sqrt[x^2 + c] + d and its approximation a x^2 + b x + d."
ApproxLimLinear::usage =
"ApproxLimHyp[a,b,c,goal] is a special case of ApproxLimGeneric for the
function a x + b Sqrt[x^2 + c] + d and its approximation (a + b) x + d."
QuadZero::usage =
"QuadZero[a,b,c,offs][k] returns the k-th root of the quadratic equation
a x^2 + b x + c == (k+offs)*Pi for ascending x (the branch to the right of the
extremum). The parameter offs has to be determined such that the 0-th solution is
already larger than the abscissa value of the extremum (see QuadOffset)."
QuadOffset::usage =
"QuadOffset[a,b,c] returns the offset value used in QuadZero, which is
essentially the ordinate value of the extremum divided by Pi."
HypZero::usage =
"HypZero[a,b,c,d,offs][k] returns the k-th root of the quadratic equation
(a+b) x + d == (k+offs)*Pi, which is the polynomial approximation of
a x + b Sqrt[x^2 + c] + d at Infinity for ascending x (the branch to the right of the
extremum). The parameter offs has to be determined such that the 0-th solution is
already larger than the abscissa value of the extremum (see HypOffset)."
HypOffset::usage =
"HypOffset[a,b,c,d] returns the offset value used in HypZero, which is
essentially the ordinate value of the extremum or bending point divided by Pi."
HypZeroExact::usage =
"HypZeroExact[a,b,c,d,offs][k] returns the k-th root of the equation
a x + b Sqrt[x^2 + c] + d == (k+offs)*Pi for ascending x (the branch to the right of the
extremum). The parameter offs has to be determined such that the 0-th solution is
already larger than the abscissa value of the extremum (see HypOffsetExact)."
HypOffsetExact::usage =
"HypOffsetExact[a,b,c,d] returns the offset value used in HypZeroExact, which is in
principle the ordinate value of the extremum or bending point divided by Pi."
ZerosInBetween::usage =
"ZerosInBetween[f,a,b] returns the number of solutions of the function
g[x] == k*Pi for x in (a,b). The function f[k] is required to give the k-th
solution of g[x] == k*Pi. If f[0] lies in the interval (a,b) only the part
to the right of f[0] is regarded. With this function, the parameter NSumTerms
of NSum can be determined."
HypApproxError::usage =
"HypApproxError[b,c,goal] returns the abscissa value where the error between
the function b*Sqrt[x^2 + c] and its approximation b*x reaches a given value."
PartInt::usage =
"PartInt[f,partpoints,a,(opts)] computes the indefinite integral of a function
f over the range (a,Infinity) using a partition-extrapolation method when the
partition points partpoints are explicitly given. The first interval (a,p1) is
obtained using the double-exponential method, the others by a Gauss-Kronrod formula.
The partial integrals between the partition points are summed up and passed to
SequenceLimit to determine the value of the integral. If the first member of the
list of partition points in Infinity, the no extrapolation is performed, and the
integral is entirely calculated with the double-exponential rule. Note that the
partition points must be of ascending order and chosen such that extrapolation
is possible (in case of an oscillating integrand, they must be to the right of
the rightmost extremum or saddle point of the argument function)."
PartitionTable::usage =
"PartitionTable[f,a,n,(opts)] returns a list of partition points suitable for
PartInt. Depending on the option FunctionType the function f is interpreted either
as argument function of Sin[f[x]] or as a function giving the k-th zero of Sin[arg[x]].
In the first case, the first n solutions of f[x] == K*Pi to the right of the lower
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