Heaviside step function

The Heaviside step function is the function $H:ℝ\to ℝ$ defined as

$H(x)$$=$

Here, there are many conventions for the value at $x=0$. Themotivation for setting $H(0)=1/2$ is that we can then write$H$ as a function of the signum function (seethis page (http://planetmath.org/SignumFunction)). In applications, such asthe Laplace transform, where the Heaviside function is used extensively,the value of $H(0)$ is irrelevant.The Fourier transform of heaviside function is

$${ℱ}_{0}H(t)=\frac{1}{2}\left(\delta (t)-\frac{i}{\pi t}\right)$$

where $\delta$ denotes the Dirac delta centered at $0$.The function is named after Oliver Heaviside (1850-1925)[1]. However, the function was already used byCauchy[2], who defined the function as

$$u(t)=\frac{1}{2}\left(1+t/\sqrt{t^2}\right)$$

and called it a coefficient limitateur [3].