# Heaviside step function

In mathematics, physics, and engineering the **Heaviside step function** is the following function,

The function is undetermined for *x* = 0, sometimes one defines .

From the definition it follows immediately that

The function is named after the English mathematician Oliver Heaviside.

## Derivative

Note that a block ("boxcar") function *B*_{Δ} of width Δ and height 1/Δ can be given in terms of step functions (for positive Δ), namely

Knowing this, the derivative of *H* follows easily

where δ(*x*) is the Dirac delta function, which may be defined as the block function in the limit of zero width, see the article on the Dirac delta function.

The step function is a generalized function (a distribution).
When *H*(x) is multiplied under the integral by the derivative of an arbitrary differentiable function *f*(*x*) that vanishes for plus/minus infinity, the result of the integral is minus the function value for *x* = 0,

Here the "turnover rule" for d/d*x* is used, which may be proved by integration by parts and which holds when *f*(*x*) vanishes at the integration limits.

## H(x) as limit of arctan

In the figure it is shown that

Note here that the function arctan returns angles on the full interval 0 to 2π. In particular, if a point *x* + *iy* in the complex plane has *x* < 0 and *y* approaches zero from above, then the function arctan returns a value approaching π. Most computer languages use a two parameter function for this form of the inverse tangent.

## Fourier transform

where δ(*u*) is the Dirac delta function and P stands for the Cauchy principal value.

### Proof

Write

where we used

Now use the following relation,

and the result is proved.

In order to prove the last relation we write a complex number in polar form

Take the natural logarithm and the limit for *y* → 0

- .

Differentiation of the last expression gives gives

In fact, the functions in this expression are distributions (generalized functions) and are to be used in an integrand multiplied by a well-behaved function *f*(*x*). Since 1/*x* is singular for *x* = 0, we must take the Cauchy principal value of the integral, i.e., exclude the singularity from the integral; we make the replacement

and the result follows.