Complex-variable distribution theory for Laplace and z transforms

Corinthios, M.J.

doi:10.1049/ip-vis:20050999

Cited by 8 publications

(7 citation statements)

References 14 publications

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“…Using the Dirac δ function ξ (z) generalized to complex plane [25,26] and carrying out some algebra based on contour integrals, Eq. (5) transforms to the simple relation…”

Section: Relating the Fourier Coefficients D 1 (ω) And D 2 (K)mentioning

confidence: 99%

Testing the reliability of a velocity definition in a dispersive medium

Tașgın

2012

Phys. Rev. A

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We introduce a method to test if a given velocity definition corresponds to an actual physical flow in a dispersive medium. We utilize the equivalence of the pulse dynamics in the real-ω and real-k Fourier expansion approaches as a test tool. To demonstrate our method, we take the definition introduced by Peatross et al. [Phys. Rev. Lett. 84, 2370] and calculate the velocity in two different ways. We calculate (i) the mean arrival time between two positions in space, using the real-ω Fourier expansion for the fields and (ii) the mean spatial displacement between two points in time, using the Fourier expansion in real-k space. We compare the velocities calculated in the two approaches. If the velocity definition truly corresponds to an actual flow, the two velocities must be the same. However, we show that the two velocities differ significantly (3%) in the region of superluminal propagation even for the successful definition of Peatross et al.

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“…Using the Dirac δ function ξ (z) generalized to complex plane [25,26] and carrying out some algebra based on contour integrals, Eq. (5) transforms to the simple relation…”

Section: Relating the Fourier Coefficients D 1 (ω) And D 2 (K)mentioning

confidence: 99%

Testing the reliability of a velocity definition in a dispersive medium

Tașgın

2012

Phys. Rev. A

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“…The test function F(s) has derivatives of any order along straight lines in the s-plane going through the origin, and approach zero more rapidly than any power of jsj. For example, if the generalized distribution is the generalized impulse x(s) (Corinthios 2005), we may write The properties of the generalized distributions are generalizations of those of the well-known properties of the theory of distributions (Corinthios 2005). In what follows, we focus our attention on the generalized Dirac-delta impulse and its application to transforms of general two-sided and one-sided functions and sequences.…”

Section: Generalized Distributions For Laplace Domainmentioning

confidence: 99%

“…The properties of generalized distributions on the complex z-plane have been recently explored (Corinthios 2005). A distribution G(z) may be defined as the value of the integral, denoted as I G [F(z)], of its product with a test function F(z).…”

Section: Z -Transform Domainmentioning

confidence: 99%

“…Recently, a generalization of the Dirac-delta impulse (Corinthios 2003) and an extension of distribution theory to generalized functions of a complex variable (Corinthios 2005) were proposed. The objective of this paper is to apply the proposed new generalized distributions to extend further the domains of existence of Laplace and z transforms as well as generalizing an important class of related transforms, including Hilbert, Hartley and Mellin transforms.…”

Section: Introductionmentioning

confidence: 99%

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New Laplace, z and Fourier-related transforms

Corinthios

2007

Proc. R. Soc. A.

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In this paper, the author uses his recently proposed complex variable generalized distribution theory to expand the domains of existence of bilateral Laplace and z transforms, as well as a whole new class of related transforms. A vast expansion of the domains of existence of bilateral Laplace and z transforms and continuous-time and discrete-time Hilbert, Hartley and Mellin transforms, as well as transforms of multidimensional functions and sequences are obtained. It is noted that the Fourier transform and its applications have advanced by leaps and bounds during the last century, thanks to the introduction of the theory of distributions and, in particular, the concept of the Dirac-delta impulse. Meanwhile, however, the truly two-sided ‘bilateral’ Laplace and z transforms, which are more general than Fourier, remained at a standstill incapable of transforming the most basic of functions. In fact, they were reduced by half to one-sided transforms and received no more than a passing reference in the literature. It is shown that the newly proposed generalized distributions expand the domains of existence and application of Laplace and z transforms similar to and even more extensively than the expansion of the domain of Fourier transform that resulted from the introduction, nearly a century ago, of the theory of distributions and the Dirac-delta impulse. It is also shown that the new generalized distributions put an end to an anomaly that still exists today, which meant that for a large class of basic functions, the Fourier transform exists while the more general Laplace and z transforms do not. The anomaly further manifests itself in the fact that even for the one-sided causal functions, such as the Heaviside unit step function u ( t ) and the sinusoid sin βtu ( t ), the Laplace transform does not exist on the j ω -axis, and the Fourier transform which does exist cannot be deduced thereof by the substitution s =j ω in the Laplace transform, which by definition it should. The extended generalized transforms are well defined for a large class of functions ranging from the most basic to highly complex fast-rising exponential ones that have so far had no transform. Among basic applications, the solution of partial differential equations using the extended generalized transforms is provided. This paper clearly presents and articulates the significant impact of extending the domains of Laplace and z transforms on a large family of related transforms, after nearly a century during which bilateral Laplace and z transforms of even the most basic of functions were undefined, and the domains of definition of related transforms such as Hilbert, Hartley and Mellin transforms were confined to a fraction of the space they can now occupy.

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“…A more rigorous mathematical theory for the delta function has also been developed and expanded under the branch in pure mathematics called the theory of distributions by L. Schwartz (Schwartz 1945). Further development is the generalized delta impulse (Corinthios 2003), which is an extension of the Dirac delta function to that on the complex plane and is applied to theories of generalized Laplace, z , Hilbert, and Fourier-related transforms (Corinthios 2005, 2007). …”

Section: Introductionmentioning

confidence: 99%

Decomposition of multivariate function using the Heaviside step function

Chikayama

2014

SpringerPlus

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Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 to develop his theory of quantum mechanics has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given by Dirac, has been poorly studied. Following Dirac’s method, we demonstrate the decomposition of a multivariate function into a sum of integrals in which each integrand is composed of a derivative of the function and a direct product of Heaviside step functions. It is an extension of Dirac’s single-variable form to that for multiple variables.

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Complex-variable distribution theory for Laplace and z transforms

Cited by 8 publications

References 14 publications

Testing the reliability of a velocity definition in a dispersive medium

Testing the reliability of a velocity definition in a dispersive medium

New Laplace, z and Fourier-related transforms

Decomposition of multivariate function using the Heaviside step function

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