Wave Propagation in Linear Media | re-examined

7.5 Test of the quadrature routine
The following integrals can be found in the tables of Grobner and Hofreiter [149]. Some of
them have complex-valued integrands to show that the quadrature routine is able to treat
complex functions as well.
Example 7.5.15
Z 1
0
x ,
e 10 sin p xdx = 10
2
p 10 e , 10 4
In[35]:= f[x_] := Exp[-x/10] Sin[Sqrt[x]];
OscInt[f,Sqrt[#]&,0] - N[5 Sqrt[10 Pi] Exp[-10/4]]
Out[35]= -11
2.87295 10
Example 7.5.16
Z 1
,1
e j(x2 +x) dx = p e j( 4 , 1 4 )
In order to be able to compute this integral, we divide it at the origin and transform the left half
onto the positive real axis. We thereby obtain two integrals in our standard notation R 1
0 ej(x2 +x) dx +
R 1
0 ej(x2 ,x) dx. The result is accurate only to nine digits if we leave the options at their default values.
A larger number of terms, however, signi cantly improves the result.
In[36]:= f1[x_] := Exp[I (x^2 + x)];
f2[x_] := Exp[I (x^2 - x)];
OscInt[f1,(#^2+#)&,0,NSumExtraTerms->30] +
OscInt[f2,(#^2-#)&,0,NSumExtraTerms->30] -
N[Sqrt[Pi] Exp[I (Pi/4 - 1/4)],20]
Out[36]= -13 -13
1.42331 10 - 1.4494 10 I
Example 7.5.17
Z 1
0
1 j(x,
e x ) dx
r
p =(1+j)
x 2 e,2
Since the integrand in this case, too, has an in nite number of zeros as it approaches the lower
integration limit, we again apply the successful strategy of dividing the integration interval at the
point x = 1. We map the interval [0; 1] to the semi-in nite range [1; 1[ by the transformation
x ! 1=x and obtain the two integrals R 1
0 ej(x, 1 x ) dx
px and R 1
0 ej( 1 x ,x) p x
x 2 dx that are amenable to our
quadrature routine.
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