# Question:"Recognizing" a Fourier Transform in an expression

October 07 2013
Maple

0

Hi:

I would like to ask maple to "recognize" a Fourier transform in a (possibly complicated) expression.

Maple defines a Fourier transform in this way:
convert(fourier(f(x), x, k), Int);

So if f(x)=1 we should get something that is proportional to Dirac(k), which we do:
fourier(1, x, k);

...but given an expression
exp1:=Int(exp(-I*x*k), x = -infinity .. infinity);

... how would tell Maple to write exp1 in terms of Dirac(k)??
I was hoping that something like say:
convert(exp1, Dirac);
or this
simplify(exp1, Dirac);
would work, but these don't.

Taking the invfourier also does not work in getting rid of the integral.

In my particular work, I am using the Physics package do deal with large expressions of non-commuting operators which I want to simplify. So my expressions consist of many terms that may look like this

restart:
with(Physics):
Setup(noncommutativeprefix={u});

exp1:=Int(-m* mp * exp(-I*(m*x+mp*x-omega[m]*t-omega[mp]*t))*u(m)*u(mp), x=-infinity..infinity);

By hand, I can "recognize" the Fourier transform (or alternatively "do the integral" with respect to x), and can rewrite this as simply something like

constant *(-m)* mp * Dirac(m + mp) * exp(-I*(-omega[m]*t-omega[mp]*t))*u(m)*u(mp);

In practice I would also integrate the above expression with respect to m and that's where the Dirac(mp+m) would come in.

So is there an easy way to have Maple help me with these?

thanks!

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