Wave Propagation in Linear Media | re-examined

In[37]:= f1[x_] := Exp[I (x - 1/x)]/Sqrt[x];
f2[x_] := Exp[I (1/x - x)] Sqrt[x]/x^2;
OscInt[f1,(#-1/#)&,1] + OscInt[f2,(1/#-#)&,1] -
N[(1+I) Sqrt[Pi/2] Exp[-2]]
Out[37]= -13 -13
9.02889 10 + 2.14218 10 I
7 Mathematica implementation of a quadrature function
Instead of using the exact argument functions x , 1=x and 1=x , x to determine the partition points
we could as well take their polynomial parts x and ,x, respectively, which yields even slightly better
results.
In[38]:= OscInt[f1,#&,1] + OscInt[f2,-#&,1] -
N[(1+I) Sqrt[Pi/2] Exp[-2]]
Out[38]= -15 -14
2.52576 10 + 3.71925 10 I
Example 7.5.18
Z 1
0
sin 3 x
x 2
In[39]:= f[x_] := Sin[x]^3/x^2;
OscInt[f,#&,0] - N[3/4 Log[3]]
Out[39]= -14
-7.39409 10
Example 7.5.19
Z 1
0
sin x , x cos x
x 3
In[40]:= f[x_] := (Sin[x] - x Cos[x])/x^3;
OscInt[f,#&,0] - N[Pi/4]
Out[40]= -13
-2.498 10
Example 7.5.20
Z 1
0
sin x 2 , 1
x 2
dx = 3
ln 3
4
dx = 4
dx = 1
r
2 2 e,2
Dividing the range of integration and with the change of variable x2 ! 1=x2 , we obtain the two
2 dx
integrals R 1
1 sin , x 2 , 1
x 2
dx and R 1
1 sin , 1
x 2 , x
188
x 2 , respectively.