# 10.8: Multi-Dimensional Fourier Transforms


When studying problems such as wave propagation, we often deal with Fourier transforms of several variables. This is conceptually straightforward. For a function $$f(x_1, x_2, \dots, x_d)$$ which depends on $$d$$ independent spatial coordinates $$x_1, x_2, \dots x_d$$, we can Fourier transform each coordinate individually: $F(k_1, k_2, \dots, k_d) = \int_{-\infty}^\infty dx_1\; e^{-ik_1x_1}\; \int_{-\infty}^\infty dx_2\; e^{-ik_2x_2}\,\cdots\, \int_{-\infty}^\infty dx_d\; e^{-ik_d x_d}\, f(x_1,x_2, \dots,x_N)$ Each coordinate gets Fourier-transformed into its own independent $$k$$ variable, so the result is also a function of $$d$$ independent variables.

We can express the multi-dimensional Fourier transform more compactly using vector notation. If $$\vec{x}$$ is a $$d$$-dimensional coordinate vector, the Fourier-transformed coordinates can be written as $$\vec{k}$$, and the Fourier transform is $F(\vec{k}) = \int d^d x \; \exp\left(-i\,\vec{k}\cdot\vec{x}\right) \, f\big(\vec{x}\big),$ where $$\int d^d x$$ denotes an integral over the entire $$d$$-dimensional space, and $$\vec{k}\cdot\vec{x}$$ is the usual dot product of two vectors. The inverse Fourier transform is $f(\vec{x}) = \int \frac{d^dk}{(2\pi)^d}\; \exp\left(i\,\vec{k}\cdot\vec{x}\right)\, F\big(\vec{k}\big).$ The delta function, which we introduced in Section 10.7, can also be defined in $$d$$-dimensional space, as the Fourier transform of a plane wave: $\delta^d(\vec{x}-\vec{x}') = \int \frac{d^dk}{(2\pi)^d} \, \exp\left[i\vec{k} \cdot \left(\vec{x}-\vec{x}'\right)\right].$ Note that $$\delta^d$$ has the dimensions of $$[x]^{-d}$$. The multi-dimensional delta function has a “filtering” property similar to the one-dimensional delta function. For any $$f(x_1,\dots,x_d)$$, $\int d^dx \; \delta^d(\vec{x}-\vec{x}') \, f(\vec{x}) = f(\vec{x}').$

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