Wave Propagation in Linear Media | re-examined

7 Mathematica implementation of a quadrature function
Remark (Implementation) To check whether the exponential expansion comprises
only terms with integer exponents, we simply form the derivative of the list of coe cients.
Since terms with non-integer exponents are listed together with the constant terms, the
derivative insuch cases is di erent from zero. Note that to check this, we must test the
two expressions for symbolic identity with the relational operator =!=, because a simple
test for equality with != will not be evaluated if the variables in the symbolic expression
D[f, x] are not assigned any values.
Since it is not possible to numerically solve equations that involve functions like Abs,
NSolve must be invoked twice with both the positive and negative possibility for the
variable goal. The results are ltered such that only the real and positive solutions are
retained. As NSolve returns a list of replacement rules fx->x1, x->x2 ...g, wemust
apply the replacement operator /. to the elements to enable testing for a vanishing
imaginary part or the like. The operation of the Select command has been described in
section 7.3.5 .
Example 7.4.2 The next two examples show the error checking mechanisms. The function in the
rst one cannot be expressed as apower series with integer exponents, whereas the second one has
multiple solutions. Note that the functions are speci ed as pure functions that need no separate
declaration.
In[10]:= ApproxLimGeneric[Sqrt[#^3]&,10]
ApproxLimGeneric::nopoly:
Function has no polynomial expansion with integer exponents.
In[11]:= ApproxLimGeneric[(#^2-20 Sqrt[#^2+1])&,20]
ApproxLimGeneric::nounique:
There are 3 real solutions f3.98475, 8.68827, 10.8449g,
here is the largest.
Out[11]= 10.8449
There are also two special cases of this module included in the package: ApproxLimQuad
for parabola-like functions ax 2 + b p x 2 + c + d and ApproxLimHyp for functions of the form
ax + b p x 2 + c + d. They can be found together with the code listing of the package in the
appendix.
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