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Re: convolution involving UnitStep

  • To: mathgroup at smc.vnet.net
  • Subject: [mg126126] Re: convolution involving UnitStep
  • From: Bob Hanlon <hanlonr357 at gmail.com>
  • Date: Fri, 20 Apr 2012 07:42:53 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201204190754.DAA04280@smc.vnet.net>

Boole also works

h[t_] = Sin[t];
g1[t_] = Piecewise[{{0, t < 0}, {2, 0 <= t < 1}}, 1];
g2[t_] = 2 Boole[0 <= t < 1] + Boole[1 <= t];

y1[t_] = Assuming[{Element[t, Reals]},
   Integrate[h[t - s] g1[s], {s, 0, t}] // Simplify];

y2[t_] = Assuming[{Element[t, Reals]},
   Integrate[h[t - s] g2[s], {s, 0, t}] // Simplify];

y1[t] === y2[t]

True


Bob Hanlon


On Thu, Apr 19, 2012 at 9:31 AM, John Davis <texasautiger at gmail.com> wrote:
> I agree this works, but am perplexed as to why my straightforward
> computation (unexpectedly) gives Mathematica trouble.
>
> Thanks for your help,
> John
>
>
>
> On Thu, Apr 19, 2012 at 8:15 AM, Bob Hanlon <hanlonr357 at gmail.com> wrote:
>>
>> h[t_] = Sin[t];
>>
>> g[t_] = Piecewise[{{0, t < 0}, {2, 0 <= t < 1}}, 1];
>>
>> y[t_] = Assuming[{Element[{s, t}, Reals]},
>>  Integrate[h[t - s] g[s], {s, 0, t}] // Simplify]
>>
>> Piecewise[{{1 + Cos[1 - t] - 2*Cos[t], t > 1},
>>     {2 - 2*Cos[t], Inequality[0, Less, t, LessEqual,
>>         1]}}, 0]
>>
>>
>> Bob Hanlon
>>
>>
>> On Thu, Apr 19, 2012 at 3:54 AM, J Davis <texasautiger at gmail.com> wrote:
>> > h[t_] = Sin[t];
>> > g[t_] = 2 UnitStep[t] - UnitStep[t - 1];
>> > y[t_] = Integrate[h[t-s]g[s],{s,0,t}]
>> >
>> > results in a conditional expression requiring t>1, but I want to
>> > evaluate and plot t values from [0,1] as well as t>1.
>> >
>> > I tried HeavisideTheta as well as := in the definition of y to no
>> > avail. Thanks for any help...
>> >



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