RE: L2 inner product. Integrate and Conjugate?
- To: mathgroup at smc.vnet.net
- Subject: [mg36339] RE: [mg36308] L2 inner product. Integrate and Conjugate?
- From: "David Park" <djmp at earthlink.net>
- Date: Mon, 2 Sep 2002 04:08:48 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Andreas,
inner[f_Function, g_Function] :=
Integrate[ComplexExpand[Conjugate[f[x]]]*g[x], {x, -1/2, 1/2}]
inner[# &, # &]
1/12
I'm not too knowledgable about using complex functions in Mathematica but
sometimes I think that "ComplexExpand" should be renamed "ComplexSimplify".
One very often needs it.
David Park
djmp at earthlink.net
http://home.earthlink.net/~djmp/
From: Andreas Dietrich [mailto:andreas.dietrich at physik.uni-marburg.de]
To: mathgroup at smc.vnet.net
Hello.
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}]
This is simple enough. Problem is, Mathematica seamingly cannot
evaluate the Integral for even the simplest of functions:
In[10]:=inner[#&,#&]
Out[10]:=\!\(\[Integral]\_\(-\(1\/2\)\)\%\(1\/2\)\(x\ Conjugate[
x]\) \[DifferentialD]x\)
As you see, the Integrate returns unevaluated. It works fine if I
remove the Conjugate. Unfortunately the Conjugate is needed for
positive definiteness.
Various variants with Composition, Re and Im etc. don't work either.
This should be a So how do I get Integrate to work with Conjugate?
I use Mathematica 4.1.2.0 on Linux/i386.
Thank you,
Andreas
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