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Where in all these formulas epsilon represents a sufficiently small real number. Some possible alternative representations for the Dirac(x) delta function are ∫ a b Dirac x − x0 f x &DifferentialD x = f x0 if x0 belongs to the open interval ]a,b[ f x0 2 if x0=a, or x0=b, and a įunctionAdvisor(integral_form, Dirac(x)) ĭirac x = ∫ − ∞ ∞ &ExponentialE I _k1 x &DifferentialD _k1 2 π, with no restrictions on x More generally, and for arbitrary real integration limits a and b , ∫ − ∞ ∞ Dirac x f x &DifferentialD x = f 0īy a sufficiently regular function, it is meant one that does not vary significantly over an interval of infinitesimal length epsilon around x = 0, or, more strictly, a function that is infinitely differentiable in some open neighborhood of 0. The Dirac function is then defined through In the one-dimensional case, the Dirac 'function' represents a distribution, that is, an object defined not by establishing the relation between the values of the function and those of its argument, but by establishing the rule for integrating its product with sufficiently regular functions. To obtain the representations on the left or right -hand sides respectively use the combine and expand commands. Where the arguments of the Dirac functions on the right-hand side are understood to be real. The Dirac command allows for a compact representation of an n-dimensional Dirac delta function, defined in terms of the product of one-dimensional Dirac functions byĭirac x1, x2. List of algebraic expressions having for domain the non-negative integers, used when representing a multidimensional Dirac List of algebraic expressions, used when representing a multidimensional Dirac xk of that k-dimensional function.Īlgebraic expression understood to be real
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The above represents: the one-dimensional Dirac delta function, the nth derivative of that Dirac function, the k-dimensional Dirac function in Cartesian coordinates, and the partial derivative of order n1 w.r.t.