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I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to a \left(1-\text{sgn}(w)^2\right)-\frac{i e^{i a w} \operatorname{sgn}(w)^2}{w}+\pi \delta (w)$$

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to ~PV\frac{-i}{k} +\pi\delta(k)+ a$$

(which follows from Sokhotski–Plemelj theorem)

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to a \left(1-\text{sgn}(w)^2\right)-\frac{i e^{i a w} \operatorname{sgn}(w)^2}{w}+\pi \delta (w)$$

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

For this function I want the following:

$$\frac1{x^2}\to \frac{1}{\pi \delta (0)}-\frac{\pi^2}2-\pi w \operatorname{sgn}(w)$$

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to a \left(1-\text{sgn}(w)^2\right)-\frac{i e^{i a w} \operatorname{sgn}(w)^2}{w}+\pi \delta (w)$$

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to a \left(1-\text{sgn}(w)^2\right)-\frac{i e^{i a w} \operatorname{sgn}(w)^2}{w}+\pi \delta (w)$$

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

For this function I want the following:

$$\frac1{x^2}\to \frac{1-\operatorname{sgn}(w)^2}{\pi \delta (w)}-\frac{\pi^2}2-\pi w \operatorname{sgn}(w)$$

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me.

For instance, some transforms from the tables that disturb me:

• The shift rule:

$$f (x-a) \to e^{-iaw}\hat {f}(w ) $$

It is given in the tables without exceptions but obviously it should not work for instance, for Heaviside Theta function and other functions that do not tend to zero at infinity.

Update. I want to clarify that I want the following:

$$\theta(x+a)\to a \left(1-\text{sgn}(w)^2\right)-\frac{i e^{i a w} \operatorname{sgn}(w)^2}{w}+\pi \delta (w)$$

instead of

$$\theta(x+a)\to -\frac{i e^{i a w} }{w}+\pi \delta (w)$$

which is given in the tables ($\theta(x)$ is the Heaviside Theta).

• Fourier transform for negative even powers:

$$\frac1{x^2}\to -\pi w \operatorname{sgn}(w)$$

Since this function has divergent integral over zero, the Fourier transform obviously should be infinite at $w=0$. But the table formula gives $0$!

For this function I want the following:

$$\frac1{x^2}\to \frac{1}{\pi \delta (0)}-\frac{\pi^2}2-\pi w \operatorname{sgn}(w)$$

Where can I find fair tables for Fourier transforms? Particularly, such that if $\int_{-\infty}^\infty f(x)dx=\infty$ the expression for Fourier transform at $w=0$ should take infinite values.