       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

"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

```

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