Fourier Expansion in 3D: Expansion of V(r)

• Logarythmic
In summary, the conversation discusses the use of Fourier expansions in one dimension, with basis functions being products of 1-D functions in x, y, and z. The question then arises on how to expand a function in three dimensions, and whether r or x should be used as the variable. The expert suggests using a 1-D expansion in r for functions that are only dependent on r, but notes that there are other orthogonal functions that can be used in 3-D in cylindrical or spherical coordinates.
Logarythmic
I'm used to use

$$\tilde{f} (x) = a_n|e_n>$$

where

$$|e_n> = e^{2 \pi inx / L}$$

and

$$a_n = \frac{1}{L}<e_n|f>$$

for my Fourier expansions.

How do I expand a function in 3 dimensions, for example

$$V(\vec{r}) = \frac{e^{-\lambda r}}{r}$$

?

There is still just one variable in there, no?

---I deleted this, it's mostly nonsense and it does't apply to the problem.---

But in my opinion, mathematically, it makes no difference if you have an r or an x in there; just do the Fourier expansion btw r_0 and r_1 as you would a fct of x.

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Logarythmic said:
I'm used to use

$$\tilde{f} (x) = a_n|e_n>$$

where

$$|e_n> = e^{2 \pi inx / L}$$

and

$$a_n = \frac{1}{L}<e_n|f>$$

for my Fourier expansions.

How do I expand a function in 3 dimensions, for example

$$V(\vec{r}) = \frac{e^{-\lambda r}}{r}$$

?
The basis functions are products whose factors are the 1-D functions in x, y, and z.

So $$|e_n> = e^{2 \pi in \vec{r} /L} = e^{2 \pi in x /L} e^{2 \pi in y /L} e^{2 \pi in z /L}$$ and $$r = \sqrt{x^2 + y^2 + z^2}$$?

Yes for the last product on the right and the r; the Ls could be different for each dimension.

$$|e_n> = e^{2 \pi in x /L_x} e^{2 \pi in y /L_y} e^{2 \pi in z /L_z}$$

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So to expand $$V(\vec{r})$$ I have to rewrite it in terms of x, y ,z or does quasar987 have a point there?

Logarythmic said:
So to expand $$V(\vec{r})$$ I have to rewrite it in terms of x, y ,z or does quasar987 have a point there?
I thought you wanted the expansion for any function in 3-D. If the function is only a function of r, then you could do a 1-D expansion in r. There are other orthogonal functions that are often used in 3-D in cylindrical or spherical coordinates.

Yes, first of all I want to solve this problem but I also want to learn something from it. ;)

1. What is Fourier Expansion in 3D?

Fourier Expansion in 3D is a mathematical technique used to represent a function in three-dimensional space as a sum of sinusoidal functions. It is based on the concept of Fourier series, which states that any periodic function can be expressed as a sum of sine and cosine functions of different frequencies.

2. How is Fourier Expansion in 3D used in scientific research?

Fourier Expansion in 3D is widely used in scientific research to analyze and model various physical phenomena, such as electromagnetic fields, heat transfer, and fluid dynamics. It allows researchers to break down complex functions into simpler components, making it easier to understand and manipulate data.

3. What is the purpose of expanding V(r) using Fourier Expansion in 3D?

The purpose of expanding V(r) using Fourier Expansion in 3D is to obtain a more accurate representation of the function in three-dimensional space. This allows for better analysis and understanding of the behavior of V(r) and its interactions with other variables.

4. What are the key components of Fourier Expansion in 3D?

The key components of Fourier Expansion in 3D are the Fourier coefficients, which represent the amplitude and phase of each sinusoidal function, and the Fourier basis functions, which are the sinusoidal functions themselves. These components work together to form the expanded function in 3D space.

5. Can Fourier Expansion in 3D be applied to non-periodic functions?

Yes, Fourier Expansion in 3D can be applied to non-periodic functions by using a technique called the Fourier transform. This allows for the expansion of non-periodic functions in terms of complex exponential functions instead of sine and cosine functions, making it useful for a wider range of applications in scientific research.

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