Fourier transform of distributions.

1. Jul 11, 2007

Klaus_Hoffmann

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

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

(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.

2. Jul 11, 2007

mathwonk

what problems have you encountered just using the definitions?

3. Jul 12, 2007

Klaus_Hoffmann

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. Jul 12, 2007

ObsessiveMathsFreak

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.
$$\frac{d\pi}{dx}= \delta(x-2) + \delta(x-3) + \delta(x-5) + \cdots$$
So its fourier transform for a particular frequency would be
$$\hat{f}(\omega)=e^{-i 2 \omega }+e^{-i 3 \omega }+e^{-i 5 \omega }+ \cdots$$

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.