The 3D Fourier Transform of 1/r3: A Comprehensive Analysis

The 3D Fourier transform of a function plays a crucial role in many areas of physics and engineering, especially when dealing with radial functions. In this article, we will explore the 3D Fourier transform of the function (frac{1}{r^3}), where (r) is the radial distance in three-dimensional space.

Setting Up the Integral

To find the 3D Fourier transform of the function (f(mathbf{r}) frac{1}{r^3}), we will use the Fourier transform formula for three dimensions:

[mathcal{F}{f(mathbf{r})}(mathbf{k}) int_{mathbb{R}^3} f(mathbf{r}) , e^{-i mathbf{k} cdot mathbf{r}} , d^3r]

We will express (f(mathbf{r})) in spherical coordinates, where (d^3r r^2 sintheta , dr , dtheta , dphi). Therefore, the integral becomes:

[mathcal{F}{f(mathbf{r})}(mathbf{k}) int_0^{infty} int_0^{pi} int_0^{2pi} frac{1}{r^3} , e^{-i k r costheta} , r^2 sintheta , dphi , dtheta , dr]

Here, (mathbf{k} cdot mathbf{r} k r costheta) with (k |mathbf{k}|).

Simplifying the Integral

Integrating over (phi) gives a factor of (2pi):

[mathcal{F}{f(mathbf{r})}(mathbf{k}) 2pi int_0^{infty} frac{1}{r^3} , r^2 , sintheta , dr int_0^{pi} e^{-i k r costheta} , dtheta]

Evaluating the Angular Integral

The angular integral can be evaluated using the formula for the integral of the spherical Bessel function:

[int_0^{pi} e^{-i k r costheta} , sintheta , dtheta frac{2 sin(kr)}{kr}]

Thus, the integral simplifies to:

[int_0^{pi} e^{-i k r costheta} , sintheta , dtheta frac{2 sin(kr)}{kr}]

Substituting and Evaluating the Radial Integral

Substituting this result back into the equation, we get:

[mathcal{F}{f(mathbf{r})}(mathbf{k}) 2pi int_0^{infty} frac{1}{r^3} , r^2 cdot frac{2 sin(kr)}{kr} , dr]

This simplifies to:

[mathcal{F}{f(mathbf{r})}(mathbf{k}) frac{4pi}{k} int_0^{infty} frac{sin(kr)}{r} , dr]

Evaluating the Final Integral

The integral (int_0^{infty} frac{sin(kr)}{r} , dr) is known to be:

[int_0^{infty} frac{sin(kr)}{r} , dr frac{pi}{2}]

Therefore, the final result is:

[mathcal{F}left{frac{1}{r^3}right}(mathbf{k}) frac{4pi}{k} cdot frac{pi}{2} frac{2pi^2}{k}]

Final Result and Discussion

The 3D Fourier transform of (frac{1}{r^3}) is:

[mathcal{F}left{frac{1}{r^3}right}(mathbf{k}) frac{2pi^2}{k}]

where (k |mathbf{k}|).

It is important to note that this result is only valid away from the origin. At the origin, the integral involves the term (sintheta ln r), which, when integrated from zero to infinity, is unbounded. Therefore, no solution exists at the origin.

Overall, the 3D Fourier transform of (frac{1}{r^3}) is a valuable tool for analyzing radial functions in three-dimensional space.