From: bsmith@mickey.NoSubdomain.NoDomain (Brian Smith)
Newsgroups: comp.compression
Subject: Re: Math Question
Date: 16 Nov 1992 03:53:00 GMT
Organization: University of California, Berkeley
Distribution: world
NNTP-Posting-Host: mickey.cs.berkeley.edu

In article <24005@hacgate.SCG.HAC.COM>, abirenbo@rigel.cel.scg.hac.com (Aaron Birenboim) writes:
|> 
|> While working on an optimization problem related to compression
|> I bumped into the following nasty little problem :
|> 
|>    evaluate:
|> 
|>                 {           [(i+1)q                                      }
|>              d  {           [                                            }
|>             --- {  Integral [      (x - (i + 0.5) q)^2 * exp(-c * x) dx  }
|>              dq {           [                                            }
|>                 {           [i*q                                         }
|> 
|>   I have evaluated the integral..... and it is terribly nasty.  Even if I
|> didn't goof on the evaluation, i would surely goof on the differentiation.
|> 
|>   c and i are constants.  I'm trying to minimize error for q.
|> 
|>   I seem to remember some terribly nasty formulas for handling the
|> differentiation before evaluating any integral.
|> 
|>   I think it involved the bounds, derivitives of the bounds, the integrand
|> and the derivitave of the integrand.
|> 
|>   My memory comes from a dim recolection of some stuff covered
|> by Dr. Sawchuk at USC (southern cal) in a class on random variables
|> based on the book by papoulis.  I cannot find it mentioned in papoulis...
|> i think is was ONLY in Sawchuk's notes (which i cannot find).
|> 
|>   any advise, help, or refrences are welcome.
|> 
|>   If i knew the NAME of this theorem, I might be able to look it up.
|> 
|>      thanks in advance,
|> 
|> --
|> Aaron Birenboim         |
|> birenb@hac2arpa.hac.com |
|>                         |
|> H (303) 871-8271        |

I believe the following is correct (this is the result of about 2 minutes
of calculation, but it seems to work).  I decided to post it so other people
could check it, since I'm not perfectly confident in the answer (my calculus
was a long time ago!).

Theroem:

If f(x) is a function that is analytically integrable, then

                 {           [y         }
              d  {           [          }
             --- {  Integral [  f(x) dx }   = f(y)
              dy {           [          }
                 {           [0         }

Proof:

Let g(x) be the indefinate integral of f(x).  Then, by the fundamental theorem
of integral calculus, g'(x) = f(x).  Now, the definate intergral from 0 to y
of f(x) is g(y) - g(0).  The derivative of this expression wrt y is g'(y) which
is f(y). QED

Given this, and a few reductions, the expression

                 {           [(i+1)q                                      }
              d  {           [                                            }
             --- {  Integral [      (x - (i + 0.5) q)^2 * exp(-c * x) dx  }
              dq {           [                                            }
                 {           [i*q                                         }

evaluates to

q^2 * exp(-c * q) * [ ({(i+1) - (i+0.5)}^2) - ({i - (i+0.5)}^2) ]

which, if I didn't make any algebra errors, is zero.

-----
Brian C. Smith				arpa:  bsmith@cs.Berkeley.EDU
University of California, Berkeley	uucp: uunet!ucbvax!postgres!bsmith
Computer Sciences Department		phone: (510)642-9585
Newsgroups: comp.compression