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MathGroup Archive 2002

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Re: L2 inner product. Integrate and Conjugate?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36345] Re: L2 inner product. Integrate and Conjugate?
  • From: Tom Burton <tburton at brahea.com>
  • Date: Mon, 2 Sep 2002 04:08:58 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

On 8/30/02 10:25 PM, in article akpk8a$s2s$1 at smc.vnet.net, "Andreas
Dietrich" <andreas.dietrich at physik.uni-marburg.de> wrote:

> I am trying to implement the inner product in the space of
> complex-valued, square integrable functions over [-1/2,1/2], which can
> be expressed in Mathematica code as
> 
> inner[f_Function,g_function]:=Integrate[Conjugate[f[x]]*g[x],{x,-1/2,1/2}]
> ...

Mathematica cannot Integrate through Conjugate in general. I suspect,
however, that most or all specific cases where you cannot expand Conjugate
are not integrable in closed form anyway. So I suggest that you attempt to
expand the Conjugate before integrating:

inner[f_Function, g_Function] :=
    Integrate[ComplexExpand[f[x] Conjugate[g[x]]], {x, -1/2, 1/2}]

Although the above function works for the few simple cases I tried just now,
my experience tells me that the following, more cumbersome form might be
more robust:
    
inner[f_Function, g_Function] := Integrate[Simplify[ComplexExpand[
  f[x] Conjugate[g[x]], TargetFunctions -> {Re, Im}]], {x, -1/2, 1/2}]

Hope this helps,

Tom Burton




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