Heaviside Step Function

Origin

Named for Oliver Heaviside.

Definition

$$H(x)=H_u(x):=\{\begin{array}{l}0,x<0\\ 1,x\ge 0\end{array}$$

The function is often written as $H_u(x)$ to denote both the names ‘Heaviside Step Function’ and ‘Unit Step Function’ at once. $H(0)=1$ if one is concerned with right-continuity. Alternately, and to be useful for Fourier analysis, it can instead be defined as

$$H(x):=\{\begin{array}{l}0,x<0\\ \frac{1}{2},x=0\\ 1,x>0\end{array}$$

It is also acceptable to simply leave $H(0)$ undefined:

$$H(x):=\{\begin{array}{l}0,x<0\\ 1,x>0\end{array}$$

The Discrete Form

This function can also be represented as a discrete series instead of a continuous function,

$$H[n]=\{\begin{array}{l}0,n<0\\ 1,n>0\end{array}$$

with the same caveats for defining $H[0]$ as 0, $1/2$, or 1 as in the continuous case.

Properties

The Dirac Delta Function

The Heaviside Step Function is the integral of the Dirac Delta Function, and the Delta Function is the derivative of the Step Function:

$$H(x)={\int }_{-\infty }^x\delta (s)ds\frac{d}{dx}H(x)=\delta (x)$$

Frequency Domain

The Fourier Transform of the Heaviside Step Function is the following:

Further Reading

1. The Fourier Transform of the Heaviside Function: A Tragedy
2. The Fourier Transform of the Heaviside Step Function by Junekey Jeon