Top 10 similar words or synonyms for dirac_delta_function
heaviside_step_function 0.873016
levi_civita_symbol 0.866060
scalar_curvature 0.862146
dirac_delta 0.857237
laplace_transform 0.854955
laplacian 0.854447
hilbert_transform 0.852191
kronecker_delta 0.850120
integrand 0.847220
laplace_operator 0.843609
Top 30 analogous words or synonyms for dirac_delta_function
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Dirac delta function | In science and mathematics, the Dirac delta function, or function, is a generalized function, or distribution, on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line. The delta function is sometimes thought of as a hypothetical function whose graph is an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents the density of an idealized point mass or point charge. It was introduced by theoretical physicist Paul Dirac. |
Dirac delta function | The graph of the delta function is usually thought of as following the whole "x"-axis and the positive "y"-axis. Despite its name, the delta function is not truly a function, at least not a usual one with range in real numbers. For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if "f" and "g" are functions such that almost everywhere, then "f" is integrable if and only if "g" is integrable and the integrals of "f" and "g" are identical. Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions. |
Dirac delta function | Joseph Fourier presented what is now called the Fourier integral theorem in his treatise "Théorie analytique de la chaleur" in the form: |
Dirac delta function | The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, |
Dirac delta function | The notion of a Dirac measure makes sense on any set. Thus if "X" is a set, is a marked point, and Σ is any sigma algebra of subsets of "X", then the measure defined on sets by |
Dirac delta function | is the delta measure or unit mass concentrated at "x". |
Dirac delta function | Then "δ" is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure dω of for "ξ" in the unit sphere "S": |
Dirac delta function | A fundamental result of elementary Fourier series states that the Dirichlet kernel tends to the a multiple of the delta function as . This is interpreted in the distribution sense, that |
Dirac delta function | In spite of this, the result does not hold for all compactly supported "continuous" functions: that is "D" does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of summability methods in order to produce convergence. The method of Cesàro summation leads to the Fejér kernel |
Dirac delta function | for all holomorphic functions "f" in "D" that are continuous on the closure of "D". As a result, the delta function "δ" is represented on this class of holomorphic functions by the Cauchy integral: |