# Fourier Transform of Heaviside Function

I'm trying to find the Fourier transform of $H(k - |x|)$, where $H$ is the Heaviside step function. I've solved a few Fourier transforms recently, but this one is giving me a bit of trouble. I'd appreciate any help.

• $H(k-\lvert x\rvert)$ is the characteristic function of the ball with radius $k$, isn't it? Are we talking about $\mathbb{R}$ or $\mathbb{R}^n$? Dec 7 '14 at 15:08
• I think just $\mathbb{R}$. I know that a sine should pop up in the answer. Dec 7 '14 at 15:10
• In that case $$\int_{-k}^k e^{-2\pi i \omega x}\,dx$$ or $$\frac{1}{\sqrt{2\pi}}\int_{-k}^k e^{-i\omega x}\,dx$$ can easily be evaluated directly, I'd say. It gets more complicated (Bessel functions) in higher dimensions. Dec 7 '14 at 15:13

As you said, there is a sinus behind:

$$\int_{-k}^ke^{−i2\pi\omega x}dx=\frac{e^{−i2\pi\omega k}-e^{i2\pi\omega k}}{-i2\pi\omega}=\frac{\sin 2\pi \omega k}{2\pi\omega}$$