The Heaviside step function is usually defined as

$$ \theta(x)=\left\{\begin{array}{ll}0&\text{if }x<0\\\tfrac12&\text{if }x=0\\1&\text{if }x>0.\\\end{array}\right. $$

It is remarkably simple and doesn't take a lot of work to define, though if used properly it does play an important role in a number of fields, including notably the study of distributions. It is nevertheless named after Oliver Heaviside.

What was his involvement with this function, and why does it merit this honour? In particular, I'm looking for well-documented sources which show his use of the concept and the context in which he used it. Ideally, I would also like references to the initial works which associate his name with the function, and their reasons for doing so.

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    $\begingroup$ He invented it. $\endgroup$ – Jiminion Feb 17 '15 at 18:38
  • $\begingroup$ Check out the biography by Paul Nahin. $\endgroup$ – Jiminion Feb 18 '15 at 12:48
  • $\begingroup$ @Jiminion Apologies, but that hardly counts as "references". Could you be more specific? $\endgroup$ – Emilio Pisanty Feb 18 '15 at 12:52
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    $\begingroup$ "Electromagnetic induction and its propagation". 1885-1887. It was in his earlier works. $\endgroup$ – Jiminion Feb 18 '15 at 13:02
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    $\begingroup$ myreckonings.com/wordpress/2007/12/07/… for some links to original papers by Heaviside $\endgroup$ – Tom Copeland Mar 5 '15 at 3:05

Oliver Heaviside invented what is called "symbolic calculus" which was a mathematically non-rigorous (at that time) but very effective way of solving differential equations that occur in physics and engineering. Later it was justified using Laplace transform and distributions theory. Heaviside function plays an important role in his formalism, analogous to the delta-function in Dirac's formalism. (Derivative of the Heaviside function in the sense of distributions is the delta-function. It has no derivative in the usual, "high-school" sense). Of course his contribution cannot be described as "inventing the Heaviside function" :-)

Reference: any book which has "Laplace transform" or "Operational calculus" in the title.

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    $\begingroup$ I disagree. He invented the step function because he was analyzing telegraph transmission circuits. The function was needed because it modeled a switch suddenly turning off. $\endgroup$ – Jiminion Feb 18 '15 at 5:50

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