Wave Propagation in Linear Media | re-examined

Out[14]= 0.537450389063711
7 Mathematica implementation of a quadrature function
The result is correct to 13 digits, although Mathematica initially considers it less reliable. The di erence
in the last two digits is due to roundo errors. We can still improve the result by using a higher
working precision and including more terms in the extrapolation.
In[15]:= OscInt[f,#&,0,WorkingPrecision->30,NSumExtraTerms->25]
Out[15]= 0.53745038906373283
It is interesting to notice that the last digit of the `exact' value given by Ehrenmark seems to be wrong,
for if we let Mathematica compute the integral analytically, we obtain a slightly di erent result.
In[16]:= Integrate[f[x],fx,0,Infinityg]
Out[16]= Pi BesselI[0, 2] 3 3
---------------- - 2 HypergeometricPFQ[f1g, f-, -g, 1]
2 2 2
In[17]:= N[%,20]
Out[17]= 0.5374503890637328029
Example 7.5.2
Z 1
In[18]:= f[x_] := Sin[x]/Sqrt[x];
OscInt[f,#&,0] - N[Sqrt[Pi/2],20]
Out[18]= -12
5.05929 10
0
r
sin x
p dx =
x 2
Again this result could be improved with a higher working precision, albeit at the expense of an
increased computing time.
In[19]:= OscInt[f,#&,0,NSumTerms->30,NSumExtraTerms->20,
WorkingPrecision->34] - N[Sqrt[Pi/2],30]
Out[19]= -19
0. 10
The next integrals have been used by Espelid and Overholt as benchmarks [123]. Since their
algorithm was tailored to regularly oscillating integrands, the arguments of the sin-functions
are at least asymptotically linear.
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