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1-dim of Fourier Transform of Heaviside is subtle enough problem, and I am not sure how to solve for 2-dim Fourier Transform of Heaviside.

Precisely, I am looking for the answer for

$\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \exp(-ik_x x-ik_y y) \theta(k_x^2+k_y^2 - a^2)$.

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Hint: Let $a\in\mathbb{R}.$ Imagine a plane with coordinates $(x_1,x_2)$. Assign a value to each point to that plane by evaluating $\theta(x_1^2+x_2^2-a^2)$. You see, if $x_1^2+x_2^2$ is greater than $a^2$ then $\theta=1$ and otherwise $\theta=0$. On your plane this means that for the points that satisfy $$x_1^2+x_2^2>a^2,$$ $\theta$ is 1. This is the complement of a ball around the origin with radius $\sqrt{x_1^2+x_2^2}$ in $\mathbb{R^2}$. Lets call this area $A$.

Now split up your integral:

$$\int_{\mathbb{R^2}}dx_1 dx_2 e^{-ik\cdot x } \theta(x^2 - a^2)=\int_{\mathbb{R^2}}dx_1 dx_2e^{-ik\cdot x }-\int_{A}dx_1 dx_2e^{-ik\cdot x }.$$

The first integral, you can solve just like you solve for the 1D case. The second integral is best to be taken in polar coordinates.

EDIT: Sorry I used other variables. To clarify: $x=(x_1,x_2)$ and $k=(k_1,k_2)$

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