Wave Propagation in Linear Media | re-examined

];
N[zerof[0]] < max,
Ceiling[i/.FindRoot[zerof[i] == max,{i,0,1}]],
True,
Message[ZerosInBetween::outside];0]
HypOffset[a_,b_,c_,d_] := Which[
(Abs[b] > Abs[a]) && (c >= 0) && (a+b) > 0,
Ceiling[((a+b) (-Sign[a b] Sqrt[a^2 c/(b^2-a^2)]) + d)/Pi],
(Abs[b] > Abs[a]) && (c >= 0) && (a+b) < 0,
Floor[((a+b) (-Sign[a b] Sqrt[a^2 c/(b^2-a^2)]) + d)/Pi],
(a+b) > 0, Ceiling[d/Pi],
(a+b) < 0, Floor[d/Pi],
True, 0];
HypZero[a_,b_,c_,d_,o_][k_] := Which[
(a+b) > 0, (( k + o) Pi - d)/(a + b),
(a+b) < 0, ((-k + o) Pi - d)/(a + b),
True, Infinity];
HypOffsetExact[a_,b_,c_,d_] := Which[
(Abs[b] > Abs[a]) && (c >= 0) && (a+b) > 0,
Ceiling[((a x + b Sqrt[x^2+c] + d)/Pi)/.
x->(-Sign[a b] Sqrt[a^2 c/(b^2-a^2)])],
(Abs[b] > Abs[a]) && (c >= 0) && (a+b) < 0,
Floor[((a x + b Sqrt[x^2+c] + d)/Pi)/.
x->(-Sign[a b] Sqrt[a^2 c/(b^2-a^2)])],
(a+b) > 0, Ceiling[(b Sqrt[c] + d)/Pi],
(a+b) < 0, Floor[(b Sqrt[c] + d)/Pi],
True, 0];
HypZeroExact[a_,b_,c_,d_,o_][k_] := Which[
(a+b) > 0 && (Abs[b] > Abs[a]),
(a (( k+o) Pi-d) - Sqrt[b^2 ((a^2-b^2) c + (( k+o) Pi-d)^2)])/(a^2-b^2),
(a+b) < 0 && (Abs[b] > Abs[a]),
(a ((-k+o) Pi-d) - Sqrt[b^2 ((a^2-b^2) c + ((-k+o) Pi-d)^2)])/(a^2-b^2),
(a+b) < 0 && (Abs[b] < Abs[a]),
(a ((-k+o) Pi-d) - Sign[a b] Sqrt[b^2 ((a^2-b^2) c + ((-k+o) Pi-d)^2)])/(a^2-b^2),
(a+b) > 0 && (Abs[b] < Abs[a]),
(a (( k+o) Pi-d) - Sign[a b] Sqrt[b^2 ((a^2-b^2) c + (( k+o) Pi-d)^2)])/(a^2-b^2),
a == b && N[(2 a ((k+o) Pi - d))] != 0,
(((k+o) Pi - d)^2 - a^2 c)/(2 a ((k+o) Pi - d)),
True, Infinity];
PartInt[f_, zerotab_List, a_, opts___Rule] :=
Module[{zeros = zerotab, seqtab,
wp = WorkingPrecision/.{opts}/.Options[OscInt],
ag = AccuracyGoal/. {opts}/.Options[OscInt],
pg = PrecisionGoal/. {opts}/.Options[OscInt],
mi = MinRecursion/. {opts}/.Options[OscInt],
ma = MaxRecursion/. {opts}/.Options[OscInt],
nt = NSumTerms/. {opts}/.Options[OscInt],
ne = NSumExtraTerms/. {opts}/.Options[OscInt],
wd = WynnDegree/. {opts}/.Options[OscInt]},
If[N[zeros[[1]]] < N[a],
zeros = Select[zeros,(N[#] >= a)&];
Message[OscInt::lowfirst,Length[zeros]]];
If[N[zeros[[1]]] == N[a], 0,
NIntegrate[f[x],{x,a,zeros[[1]]},
Method->DoubleExponential,
WorkingPrecision->wp,
AccuracyGoal->ag,
PrecisionGoal->pg,
MinRecursion->mi,
MaxRecursion->ma]] +
If[N[zeros[[1]]] == Infinity, 0,
seqtab = Table[NIntegrate[f[x],{x,zeros[[i]],zeros[[i+1]]},
Method->GaussKronrod,
WorkingPrecision->wp,
AccuracyGoal->ag,
PrecisionGoal->pg,
MinRecursion->mi,
MaxRecursion->ma],
{i,1,Length[zeros]-1}];
SequenceLimit[Take[FoldList[Plus,0,seqtab],{nt+1,nt+ne}],
WynnDegree->wd]]
];
PartitionTable[zerof_, a_, n_, opts___Rule] :=
Module[{ofs=0,x,xoffs,koffs,extrema,cfoffs=0,
wp = WorkingPrecision/.{opts}/.Options[PartitionTable],
ag = AccuracyGoal/. {opts}/.Options[PartitionTable],
ma = MaxRecursion/. {opts}/.Options[PartitionTable],
ft = FunctionType/. {opts}/.Options[PartitionTable]},
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