Singularity function

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Singularity functions or singularity brackets are a notation used to describe discontinuous functions.

$<x-a>^n = \begin{cases} \delta'(x-a) : & n=-2\\ \delta(x-a) : & n=-1\\ 0 : & n\ge0, x<a\\ (x-a)^n : & n\ge0, x\ge a \end{cases}$

δ'(x) is the first derivative of δ(x), also called the unit doublet.
δ(x) is the Dirac delta function, also called the unit impulse.

[edit] Integration

Integrating $< x - a > n$ can be done in a convenient way in which the constant of integration is automatically included so the result will be 0 at x=a.

$\int<x-a>^n dx = \begin{cases} <x-a>^{n+1}, & n<0 \\ \frac{<x-a>^{n+1}}{n+1}, & n \ge 0 \end{cases}$

[edit] 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 downwards forces and sagging bending moments being positive.