Fourier transform of the Heaviside step function ($\hat{H}(k)=\pi \delta (k)+\mathrm{PV}(1/(ik))$)

With the $e^{-ikx}$ convention, exponential damping on the positive half-line gives $1/(a+ik)$. Its even part tends to $\pi \delta (k)$ and its odd part to $-i\mathrm{PV}(1/k)$. Thus the Heaviside step function transform requires a Cauchy principal value, not an ordinary reciprocal function at zero.