Fourier transform of distributions.

  • #1

Main Question or Discussion Point

Is there any way to calculate the Fourier transform of the functions

[tex] \frac{d\pi}{dx}-1/log(x) [/tex] and [tex] \frac{d\Psi}{dx}-1 [/tex]

(both are understood in the sense of distributions)

i believe that these integrals (even with singularities) exist either in Cauchy P.V or Hadamard finite part sense but if possble i would need a help, thanks

EDIT:= 'pi(x)' here is the prime counting function and 'Psi (x) ' is the Tchebycheff function.
 

Answers and Replies

  • #2
mathwonk
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what problems have you encountered just using the definitions?
 
  • #3
The problem mathwork is that as you can see the integral is 'sngular' i was looking for a method to give a FINITE value for every frequency, for example using the Cauchy P.V however i think this method (Cacuhy's) to extract a finite value does not work.. perhaps Hadamrd finite part ??, but i don't know how to apply it
 
  • #4
The derivative of the pi function, in the distributional sense, will be an infinite series of delta functions, as there are an infinite number of primes.
[tex]\frac{d\pi}{dx}= \delta(x-2) + \delta(x-3) + \delta(x-5) + \cdots[/tex]
So its fourier transform for a particular frequency would be
[tex]\hat{f}(\omega)=e^{-i 2 \omega }+e^{-i 3 \omega }+e^{-i 5 \omega }+ \cdots [/tex]

I feel pretty sure that this would be a divergent sum, as the pi function is not L2 integrable itself. It seems highly improbable that the sum would converge for all frequencies.
 

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