Wave Propagation in Linear Media | re-examined

Example 7.5.12
Z 1
0
sin x 2
7 Mathematica implementation of a quadrature function
cos x2
4
dx
x2 = e, , 1
4 p 2
This integral, taken from an earlier work of Sidi [126], is di erent from the others examined so far
in that the integrand has an in nite number of oscillations both for x ! 1 and x ! 0. Thus it
cannot be treated in a straightforward manner like the integrals before and we split the integration
interval at the point x =1. The right part I1 = R 1
sin 0 x2 cos x2 dx
4 x2 remains unchanged, whereas we
transform the left part by achange of variable x ! 1=x to obtain a second integral in our standard
form I2 = R 1
sin( x 1 2 ) cos 4x2 dx.
In[32]:= f1[x_] := Sin[Pi/x^2] Cos[Pi x^2/4]/x^2;
f2[x_] := Sin[Pi x^2] Cos[Pi/(x^2 4)];
OscInt[f1,Pi/4 #^2&,1] + OscInt[f2,Pi #^2&,1] -
N[(Exp[-Pi] - 1)/(4 Sqrt[2])]
Out[32]= -14
4.54636 10
The next integrals are taken from the tables of Abramowitz and Stegun [148] and involve
Bessel functions that are known to behave like cos x for large values of x. Thus the partition
points may be computed by a linear function.
Example 7.5.13
In[33]:= f[x_] := BesselJ[0,x];
OscInt[f,#&,0] - 1
Out[33]= -14
-4.74065 10
Example 7.5.14
Z 1
0
Z 1
0
J0(x) dx =1
cos xK0(x)dx =1
In[34]:= f[x_] := Cos[x] BesselK[0,x];
OscInt[f,#&,0] - N[Pi/(2 Sqrt[2])]
Out[34]= -14
-2.84217 10
186