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