Heaviside step 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.
Note that a block ("boxcar") function BΔ 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/dx 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.
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.