Singularity function: Difference between revisions

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{{Short description|Class of discontinuous functions}}

'''Singularity functions''' are a class of [[discontinuous function]]s that contain [[Mathematical singularity|singularities]], i.e., they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of [[generalized functions]] and [[Distribution (mathematics)|distribution theory]].<ref>{{citation |first=A. H.|last=Zemanian|title=Distribution Theory and Transform Analysis|publisher=McGraw-Hill Book Company| year=1965}}</ref><ref>{{citation|first=R. F.|last=Hoskins|title=Generalised Functions|publisher=Halsted Press|year=1979}}</ref><ref>{{citation |first=M.J.|last=Lighthill|title=Fourier Analysis and Generalized Functions|publisher=Cambridge University Press|year=1958}}</ref> The functions are notated with brackets, as <math>\langle x-a\rangle ^n</math> where ''n'' is an [[integer]]. The "<math>\langle \rangle</math>" are often referred to as '''singularity brackets''' . The functions are defined as:

:{| class="wikitable"

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== Integration ==

Integrating <math>\langle x-a \rangle^n</math> can be done in a convenient way in which the [[constant of integration]] is automatically included so the result will be {{math|0}} at {{math|1=''x'' = ''a''}}.

<math display="block">\int\langle x-a \rangle^n dx = \begin{cases}

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== Example beam calculation ==

The deflection of a simply supported beam, as shown in the diagram, with constant cross-section and elastic modulus, can be found using [[Euler–Bernoulli beam theory]]. Here, we are using the [[sign convention]] of downwardsdownward forces and sagging bending moments being positive.

[[Image:Loaded beam.svg|center|650px]]

Load distribution:

:<math>w=-3\text{ N}\langle x-0 \rangle^{-1}\ +\ 6\text{ Nm}^{-1}\langle x-2\text{ m} \rangle^0\ -\ 9\text{ N}\langle x-4\text{ m}\rangle^{-1}\ -\ 6\text{ Nm}^{-1}\langle x-4\text{ m} \rangle^0\ </math>

Shear force:

:<math>S=\int w\, dx</math>