Signals and Systems, Alexander D. Poularikas, University of Alabama in Huntsville Fourier Transforms, Kenneth B. Howell, University of Alabama in Huntsville Sine and Cosine Transforms, Pat Yip, McMaster University, Ontario The Hartley Transform, Kraig J. Olejniczak, University of Arkansas, Fayetteville Laplace Transforms, Samuel Seely (deceased), Westbrook, Connecticut The Z-Transform, Alexander D. Poularikas, University of Alabama in Huntsville Hilbert Transforms, Stefan L. Hahn, Warsaw University of Technology Radon and Abel Transforms, Stanley Deans, University of South Florida The Hankel Transform, Robert Piessens , Katholieke Universieit Leuven, Belgium Wavelet Transform, Yunlong Sheng, Laval University, Quebec The Mellin Transform, Jacqueline Bertrand, Pierre Bertrand, Universite de Paris VII, and Jean-Philippe Ovarlez, ONERA/DES, France Mixed Time-Frequency Signal Transformations, G. Faye Boudreaux-Bartels, University of Rhode Island Fractional Fourier Transforms, Dr. Mustafa Abushagur, University of Alabama in Huntsvill, Dr. Ahmed M. Almanasrah, Photronix, Malaysia The Lapped Transforms, Ricardo L. de Queiroz, Xerox Corporation The Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Discrete Time and the Discrete Transforms, Alexander D. Poularikas, University of Alabama in Huntsville Appendices, Alexander D. Poularikas, University of Alabama in Huntsville Index

The transforms and applications handbook

We introduce and investigate a framework for constructing algorithms to invert Laplace transforms numerically. Given a Laplace transform

 \hat{f} 

of a complex-valued function of a nonnegative real-variable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula

 f(t) \approx f_n (t) \equiv \frac{1}{t} \sum_{k = 0}^{n}\omega_{k}\hat{f}\biggl(\frac{\alpha_{k}}{t}\biggr),\quad 0 

where the weights ωk and nodes αk are complex numbers, which depend on n, but do not depend on the transform

 \hat{f} 

or the time argument t. Many different algorithms can be put into this framework, because it remains to specify the weights and nodes. We examine three one-dimensional inversion routines in this framework: the Gaver-Stehfest algorithm, a version of the Fourier-series method with Euler summation, and a version of the Talbot algorithm, which is based on deforming the contour in the Bromwich inversion integral. We show that these three building blocks can be combined to produce different algorithms for numerically inverting two-dimensional Laplace transforms, again all depending on the single parameter n. We show that it can be advantageous to use different one-dimensional algorithms in the inner and outer loops.

/pdf/a-unified-framework-for-numerically-inverting-laplace-3e82rtbzln.pdf

A Unified Framework for Numerically Inverting Laplace Transforms

SUMMARY

High-quality high-resolution transmission and reflection images produced using a scanning optical microscope and the split-detector technique are presented. These images exhibit differential phase contrast, the method avoiding some drawbacks of the usual Nomarski DIC arrangement. Imaging is treated theoretically and compared with the Nomarski method.

Differential phase contrast in scanning optical microscopy

In this paper, we study the properties of Sumudu transform and relationship between Laplace and Sumudu transforms Further, we also provide an example of the double Sumudu transform in order to solve the wave equation in one dimension which is having singularity at initial conditions

/pdf/on-the-applications-of-laplace-and-sumudu-transforms-3zh71m2h2h.pdf

On the applications of Laplace and Sumudu transforms

A note on solutions of wave, Laplace's and heat equations with convolution terms by using a double Laplace transform

Operational Calculus in Two Variables and Its Applications.

Operational Calculus in Two Variables and Its Applications

Операционное исчисление по двум переменным и его приложения

Integral Transforms and Operational Calculus.

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Papers

Operational Calculus in Two Variables and Its Applications.

Operational Calculus in Two Variables and Its Applications

Операционное исчисление по двум переменным и его приложения

Integral Transforms and Operational Calculus.