Re: Convolve : Different Looking Results
- To: mathgroup at smc.vnet.net
- Subject: [mg130649] Re: Convolve : Different Looking Results
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Sun, 28 Apr 2013 05:17:35 -0400 (EDT)
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Am 27.04.2013 05:08, schrieb Polar Coords:
> Dear Mathgroup,
>
> May be someone could please explain the following behaviour :
>
>
> In[15]:= Convolve[Exp[-x] UnitStep[x], Sin[2 x] UnitStep[x], x, y,
> Assumptions -> y > 0]
>
> Out[15]= -(2/5) (Cos[2 y] - Cosh[y] - Cos[y] Sin[y] + Sinh[y])
>
>
The two expressions are equivalent.
In[15]:=
{1/5 (2 E^-t - 2 Cos[2 t] + Sin[2 t]),
-(2/5) (Cos[2 v] - Cosh[v] - Cos[v] Sin[v] + Sinh[v])} //
TrigToExp // ExpandAll
Out[15]=
{(2 E^-t)/5 - (1/5 - I/10) E^(-2 I t) - (1/5 + I/10) E^(2 I t),
(2 E^-v)/5 - (1/5 - I/10) E^(-2 I v) - (1/5 + I/10) E^(2 I v)}
The different output form of that expression seem to be a consequence of
internal lexicographic expression ordering, a problem, one encounters
frequently.
Here, the simplification process has a discontinuity at the
lexicographic order of the dummy summation variable "uu" and the dummy
output expression variable.
Convolve[Exp[-uu] UnitStep[uu], Sin[2 uu] UnitStep[uu], uu, #,
Assumptions -> (# > 0)] & /@ {s, t, u, v, w, x, y, z}
{1/5 (2 E^-s - 2 Cos[2 s] + Sin[2 s]),
1/5 (2 E^-t - 2 Cos[2 t] + Sin[2 t]),
1/5 (2 E^-u - 2 Cos[2 u] + Sin[2 u]),
-(2/5) (Cos[2 v] - Cosh[v] - Cos[v] Sin[v] + Sinh[v]),
-(2/5) (Cos[2 w] - Cosh[w] - Cos[w] Sin[w] + Sinh[w]),
-(2/5) (Cos[2 x] - Cosh[x] - Cos[x] Sin[x] + Sinh[x]),
-(2/5) (Cos[2 y] - Cosh[y] - Cos[y] Sin[y] + Sinh[y]),
-(2/5) (Cos[2 z] - Cosh[z] - Cos[z] Sin[z] + Sinh[z])}
--
Roland Franzius