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Re: Complex Integral

From: Dirk Laurie
Subject: Re: Complex Integral
Date: Tue, 5 May 1998 09:43:55 +0200 (SAT)

address@hidden wrote:
> Does anyone know how to representive the following in octave:
> the integral from x to 0 of ((1-cos(t))/t dt.
It's not an elementary function.  In the "Handbook of Mathematical
Functions" by Abramowitz and Stegun (nowadays published by Dover)
it is called Cin(x).  You can calculate it by numerical
integration.  For that purpose it is better
to write it as    

     Cin(t) = 2 integral    sin(t/2)^2/t   dt

The Handbook has a table of Cin(t) for x=0:pi/10:10*pi  on p244,
which you can use to check.

I've just spent a few minutes trying out the idea.  It's a good idea
to collapse full intervals of length 2*pi into one: since sin(t/2)^2
vanishes with its first derivatives at both endpoints, numerical
integration is easier.  I've also made a tiny substitution to
make the integrand simpler (u=t/2).

Here is a test run:

» type longpart
longpart is a user-defined function:

function y = longpart (u)
  global n
  y = sin (u) ^ 2 * sum (1 ./ (u + (0:n - 1) * pi));
» type shortpart
shortpart is a user-defined function:

function y = shortpart (u)
  global n
  y = sin (u) ^ 2 / (u + n * pi);
» global n; x=3.6*pi; n=fix(x/(2*pi)), r=rem(x,2*pi)/2,
n = 1
r = 2.51327412287183
» 2*(quad('longpart',0,pi)+quad('shortpart',0,r))
ans = 3.08807509702346
» x=9.4*pi; n=fix(x/(2*pi)), r=rem(x,2*pi)/2,
n = 4
r = 2.19911485751286
» 2*(quad('longpart',0,pi)+quad('shortpart',0,r))
ans = 3.99443579621003

The Handbook gives 3.0880751 and 3.9944358 respectively.


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