Heaviside step function: Difference between revisions

Revision as of 02:44, 13 December 2021

Revision as of 22:02, 19 January 2022

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<math display="block">\hat{H}(s) = \lim_{N\to\infty}\int^N_{-N} e^{-2\pi i x s} H(x)\,dx = \frac{1}{2} \left( \delta(s) - \frac{i}{\pi} \operatorname{p.v.}\frac{1}{s} \right).</math>

Here {{math|p.v.{{sfrac|1|''s''}}}} is the [[distribution (mathematics)|distribution]] that takes a test function {{mvar|φ}} to the [[Cauchy principal value]] of {{<math|>\textstyle\int_{{intmath|int|−∞|∞}-\infty}^\infty \frac{{sfrac|''φ''\varphi(''s'')|''}{s''}} ''\, ds''}}</math>. The limit appearing in the integral is also taken in the sense of (tempered) distributions.

== Unilateral Laplace transform ==