MHB Fourier Transform: Calculate $\hat{g}(\omega)$

evinda
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Hello! (Wave)

I want to calculate the Fourier transform of $g(x)=|x|$.

I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$

Is it right so far?

How can we calculate $\lim_{x \to +\infty} \frac{x \sin{(\omega x)}}{\omega}$ ?
 
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What's your definition of the Fourier Transform (including constants)?
 
Ackbach said:
What's your definition of the Fourier Transform (including constants)?

$$\hat{g}(\omega)=\int_{-\infty}^{+\infty} g(x) e^{-i x \omega} dx$$
 
evinda said:
Hello! (Wave)

I want to calculate the Fourier transform of $g(x)=|x|$.

I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$

Is it right so far?

How can we calculate $\lim_{x \to +\infty} \frac{x \sin{(\omega x)}}{\omega}$ ?
For a function to have a Fourier transform, it has to satisfy some integrability condition. The function $|x|$ is not at all integrable, in fact it tends to infinity as $x\to\infty$. I don't think there is any sense in which this function can have a Fourier transform.
 
Opalg said:
For a function to have a Fourier transform, it has to satisfy some integrability condition. The function $|x|$ is not at all integrable, in fact it tends to infinity as $x\to\infty$. I don't think there is any sense in which this function can have a Fourier transform.

Neither using distribution theory?
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...

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