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Hi all.

We have a function u(x,t), wherexcan go from (-infinity;infinity) andt>0. In my book it says:

"For fixedt, the functionu(x,t)becomes a function of the spatial variablex, and as such, we can take its Fourier transform with respect to thex-variable. We denote this transform by [tex]\widehat{u}(x,t)[/tex]:

[tex]

\widehat{u}(\omega,t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}{u(x,t)e^{-i\omega x}} dx

[/tex]

****************

Questions:

1:First of all, [tex]\widehat{u}(x,t)[/tex] isonlya function of [tex]\omega[/tex], since we have fixedt, correct?

2:The author says that we have the following:

[tex]

F(\frac{d}{dt}u(x,t))(\omega) = \frac{d}{dt}\widehat{u}(\omega,t),

[/tex]

where the large F denotes the Fourier transform of u(x,t) with respect to x. Since we have fixedt, then why are we differentiating with respect tot? Doesn't this give zero?

Thanks in advance.

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# Homework Help: Fourier Transforms: Solving PDE's

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