Convolution and Integration of Top Hat Functions
- To: mathgroup at smc.vnet.net
- Subject: [mg33196] Convolution and Integration of Top Hat Functions
- From: Roger Jones <rmj at SLAC.Stanford.EDU>
- Date: Thu, 7 Mar 2002 02:24:08 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
The first convolution of two Top Hat functions is the triangle function. This does not come out explicitly in Mathematica 4.1 as can be seen from the below example. The first part of the IF statement is indeed the correct solution but the second option to the IF is not required. Obviously this can be coded to force only the first part of the IF condition to be displayed but I would have expected this to be automatic. I also, note that in Mathematica 3.0 this was not the case as only the first part of the if conditional was given as the solution. Thanks! -Roger Jones Clear[II]; (* Define "Top Hat" function *) Clear[II]; II[xx_] = UnitStep[1\/2 + xx] - UnitStep[xx - 1\/2]; (* Convolve *) f[yy_]=Integrate[II[xx] II[yy-xx],{xx,-Infinity,Infinity}] 1 1 If[- - yy > -Infinity && yy < -(-) && 2 2 1 yy < -, (-1 + yy) UnitStep[-1 + yy] - 2 2 yy UnitStep[yy] + (1 + yy) UnitStep[1 + yy], 1 Integrate[(-UnitStep[-(-) + xx] + 2 1 UnitStep[- + xx]) 2 1 (-UnitStep[-(-) - xx + yy] + 2 1 UnitStep[- - xx + yy]), 2 {xx, -Infinity, Infinity}]]