Wave Propagation in Linear Media | re-examined

7.4 Auxiliary functions
Out[8]= 3
f0, -, 1g
2
The correct asymptotic expansion comprises all terms in x down to x 0 , but of course no error term.
In[9]:= AsymptoticExpand[x+Sqrt[x^4+x^3-x+1],x]
Out[9]= 1 3 x 2
-(-) + --- + x
8 2
These supporting functions are used inside a more intricate module that rst determines the
asymptotic expansion, then solves the equation given above and returns the largest real-valued
solution. There are several cases that lead to error or warning messages:
As the asymptotic expansion must be of the form (x) =xpP1ai i=0 xi , p being a nonnegative
integer, all expansions with other than integer exponents are rejected (for instance
(x) = p x3 ).
If no solution exists, an error is issued.
The existence of more than one solution is reported to the user, and the largest solution
is returned.
The function ApproxLimGeneric takes as inputs the argument function that is to be approximated
and the number of subdivision points that are sought within one wavepacket.
ApproxLimGeneric[f_,goal_] :=
Module[fx,fapprox,g,coeffs,respos,resneg,resg,
fapprox = AsymptoticExpand[f[x],x];
coeffs = CoefficientList[fapprox,x];
If[D[coeffs,x] =!= Table[0,fLength[coeffs]g],
Message[ApproxLimGeneric::nopoly],
g = D[fapprox,x] / D[fapprox - f[x],x];
respos = Select[NSolve[g == goal,x],(Im[x/.#] == 0)&];
resneg = Select[NSolve[g == -goal,x],(Im[x/.#] == 0)&];
res = Select[Join[respos,resneg],Positive[x/.#]&];
If[Length[res] == 0,
Message[ApproxLimGeneric::nosolution],
If[Length[res] > 1,
Message[ApproxLimGeneric::nounique,Length[res],
Map[x/.#&,res]]];
x/.Last[Sort[res]]]]
]
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