Heaviside step function: Difference between revisions

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==Discrete form==

An alternative form of the unit step, defined instead as a function <math>H: \mathbb{Z} \to \mathbb{R}</math> (i.e.that is, taking in a discrete variable {{mvar|n}}), is:

<math display="block">H[n]=\begin{cases} 0, & n < 0, \\ 1, & n \ge 0, \end{cases} </math>

or using the half-maximum convention: <ref>{{cite book |last=Bracewell |first=Ronald Newbold |date=2000 |title=The Fourier transform and its applications |language=en |___location=New York |publisher=McGraw-Hill |isbn=0-07-303938-1 |page=61 |edition=3rd}}</ref>

<math display="block">H[n]=\begin{cases} 0, & n < 0, \\ \tfrac12, & n = 0,\\ 1, & n > 0, \end{cases} </math>

where {{mvar|n}} is an [[integer]]. If {{mvar|n}} is an integer, then {{mvar|n}} < 0 must imply that {{mvar|n}} <=≤ -1−1, whilstwhile {{mvar|n}} > 0 must imply that the function attains unity at {{mvar|n}} = 1. Therefore the "step function" exhibits ramp-like behaviourbehavior over the ___domain of [-1−1,1], and can notcannot authentically be a step function, using the half-maximum convention.

Unlike the continuous case, the definition of {{math|''H''[0]}} is significant.