1.3. SINGULAR OPERATORS

Singular operators are widely used in the analysis of two-dimensional systems, especially systems in which discretization of continuous functions is performed. Dirac’s 2D delta function is a singular operator with the following properties:

(1.3.1a)

(1.3.1b)

(1.3.1b)

(1.3.1g)

Magnitude here denotes an infinitesimal limit of integration.

The two-dimensional delta function can be represented as the product of two one-dimensional delta functions of orthogonal coordinates :

(1.3.2)

where one-dimensional delta functions satisfy one-dimensional relations similar to (1.3.1). The delta function can also be defined as the limit of some functions [1, p. 275], for example, a rectangular function

(1.3.3a)

circular function

(1.3.3b)

Gaussian function

(1.3.3b)

sinc functions

(1.3.3g)

Bessel function

(1.3.3d)

Where

(1.3.4a)

(1.3.4b)

(1.3.4в)

Another useful definition of the delta function is the following identity [2, p. 99]:

(1.3.5)

Where .