Singularity function

Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory.

The functions are defined as: where: δ(x) is the Dirac delta function, also called the unit impulse.

The first derivative of δ(x) is also called the unit doublet.

is the Heaviside step function: H(x) = 0 for x < 0 and H(x) = 1 for x > 0.

The value of H(0) will depend upon the particular convention chosen for the Heaviside step function.

Note that this will only be an issue for n = 0 since the functions contain a multiplicative factor of x − a for n > 0.

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.

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 downward forces and sagging bending moments being positive.

Load distribution: Shear force: Bending moment: Slope: Deflection: The boundary condition u = 0 at x = 4 m allows us to solve for c = −7 Nm2