Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2007
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2007

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Evaluating a convolution integral in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80263] Re: Evaluating a convolution integral in Mathematica
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Thu, 16 Aug 2007 04:43:00 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f9ud4q$a5q$1@smc.vnet.net>

rdbeer at indiana.edu wrote:
> I am having trouble evaluating a convolution integral in Mathematica.
> Define B[x,{min,max}] to be a function that takes on the value 1 when
> x is between min and max and 0 otherwise.  I need to find the
> convolution of x^n B[x, {min1, max1}] with x^m B[x, {min2, max2}],
> where x is Real, n and m are nonnegative integers, and the mins and
> maxs are Real with the constraints that min1<=max1 and min2<=max2.
> 
> The resulting convolution integeral is Integrate[t^n B[t, {min1,
> max1}] (x-t)^m B[x-t, {min2, max2}], {t, -Infinity, Infinity}].
> 
> Mathematica 6.0.1 has no problem evaluating this integral when
> constant values are substituted for n, m, and the mins and maxs.
> However, I need the general value of this integral.  Mathematica also
> claims to be able to evaluate this general integral, returning a
> complicated peicewise expression involving gamma functions and
> hypergeometric functions.  However, when specific values for n, m and
> the mins and maxs are then substituted into this general expression,
> it always returns either 0 or Indeterminate.
> 
> Any help with evaluating this integral in general would be greatly
> appreciated.

It is not clear, at least for me, how you coded your function B? Did you 
use Piecewise, If, Which, Boole, ... ? Here is a simple example with Boole,

B[x_, {min_, max_}] := Boole[min <= x <= max]

Also, it is not clear whether you have tried to set up your assumptions 
in Integrate itself. For instance,

Integrate[
  t^n B[t, {min1, max1}] (x - t)^m B[
    x - t, {min2, max2}], {t, -Infinity, Infinity},
  Assumptions -> {{x, min1, mmax1, min2, max2} \[Element]
     Reals, {n, m} \[Element] Integers, n >= 0, m >= 0, min1 <= max1,
    min2 <= max2}]

returns a long piecewise function with some hypergeometric functions (I 
  did not try to simplify nor to check the validity of the expression 
since I did not know what the function B really was).

Also, it may be worthwhile to try Maxim Rytin's "Integration of 
Piecewise Functions with Applications" package available at

http://library.wolfram.com/infocenter/MathSource/5117/

Especially, you should check the function *PiecewiseIntegrate*. From the 
notebook piecewise.nb, we can read,

"PiecewiseIntegrate[f,{x,xmin,xmax},{y,ymin,ymax},\[Ellipsis]] gives the 
definite integral of function f. It is intended for integrating 
piecewise continuous functions, and also generalized functions. It 
handles integrands and integration bounds involving the following 
expressions:

. UnitStep, Sign, Abs, Min, Max
. Floor, Ceiling, Round, IntegerPart, FractionalPart, Quotient, Mod
. DiracDelta and its derivatives, DiscreteDelta, KroneckerDelta
. If, Which, Element, NotElement
. Piecewise, Boole, Clip"

For instance,

PiecewiseIntegrate[
  t^n B[t, {min1, max1}] (x - t)^m B[
    x - t, {min2, max2}], {t, -Infinity, Infinity},
  Assumptions -> {{x, min1, mmax1, min2, max2} \[Element]
     Reals, {n, m} \[Element] Integers, n >= 0, m >= 0, min1 <= max1,
    min2 <= max2}]

returns a long answer made of nested If constructs that return some 
simple expressions such as

If[m == 0 && min1 < max1 && min2 == -max1 + max2 + min1 &&
   x == max1 + min2,
  If[min1 > 0, (-min1^(1 + n) + max1^n (-min2 + x))/(1 + n), [...]

(Again, I have not try to simplify nor to check the validity of the 
expression.)

HTH
-- 
Jean-Marc



  • Prev by Date: Re: framed plots with two y axes scales
  • Next by Date: Re: Can model parameters be global?
  • Previous by thread: Evaluating a convolution integral in Mathematica
  • Next by thread: Re: Evaluating a convolution integral in Mathematica