From: John Baron <johnb@nova.stanford.edu>
Newsgroups: sci.math.num-analysis
Subject: Re: quadrature method for cos(kcos(theta))
Date: 10 Sep 1998 19:46:57 -0700

bin hu <binhu@students.uiuc.edu> writes:

>Hi

>the largest k is around 2000. the integral range is from 0~2\pi. actually
>I am trying do the integration like 
>e^{i(x*sin(theta)+y*cos(theta))}.

>do you know some efficient quadrature method to deal with that problem?

>thanks
>Bin

Can't you use the method of stationary phase for this?  The basic idea
is that for an integral of the form

I = int{g(z) exp(i * k * f(z) dz},

where g(z) and f(z) are real functions and k is a large parameter,
the oscillations due to the exponential term will essentially cancel
except where f'(z) = 0.  Expanding f(z) in a Taylor series about
the stationary point z_0, the asymptotic form of the integral is

I ~ g(z_0) exp(-j * pi / 4) * sqrt(2 * pi / (k |f''(z_0)|)) *
		exp(j * k * f(z_0))

In your case, you may have two stationary points in your interval.


On second thought, forget all this.  Just use the identities

J_0(x) = 1 / (2 * pi) * int{0, 2 * pi, cos(x * cos th) dth},

where J_0 is the zero-order Bessel function of the first kind,

and

int{0, 2 * pi, sin(x * cos th) dth} = 0 

since sin(x * cos th) is odd over the interval {0, 2 * pi}.


John
-- 
__________________________________________________________________
John Baron         johnb@nova.stanford.edu          (650) 723-3669
Center for Radar Astronomy        http://nova.stanford.edu/~johnb/