scholarly journals Laplace transform of certain functions with applications

2000 ◽  
Vol 23 (2) ◽  
pp. 99-102
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Laplace Transform ◽  
Bessel Functions ◽  
Whittaker Functions ◽  
Special Cases ◽  
The Laplace Transform ◽  
Special Case

The Laplace transform of the functionstν(1+t)β,Reν>−1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity∫0∞sin (ax)/x dx=π/2is a special case of our main result.

The Inverse Laplace Transform of the Product of Two Whittaker Functions

1962 ◽  
Vol 58 (4) ◽  
pp. 580-582 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Laplace Transform ◽  
Whittaker Function ◽  
Inverse Laplace Transform ◽  
Whittaker Functions ◽  
Original Function ◽  
The Laplace Transform ◽  
Image Position

The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters

The integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

POTENTIALS OF ARBITRARY FORCES WITH FRACTIONAL DERIVATIVES

2004 ◽  
Vol 19 (17n18) ◽  
pp. 3083-3092 ◽  
Author(s):  
EQAB M. RABEI ◽  
TAREQ S. ALHALHOLY ◽  
AKRAM ROUSAN
Keyword(s):  
Laplace Transform ◽  
General Formula ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Hamiltonian Mechanics ◽  
Dissipative Effects ◽  
Special Cases ◽  
Exact Agreement ◽  
The Laplace Transform ◽  
Lagrangian And Hamiltonian Mechanics

The Laplace transform of fractional integrals and fractional derivatives is used to develop a general formula for determining the potentials of arbitrary forces: conservative and nonconservative in order to introduce dissipative effects (such as friction) into Lagrangian and Hamiltonian mechanics. The results are found to be in exact agreement with Riewe's results of special cases. Illustrative examples are given.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

scholarly journals The ruin probability in a special case

1971 ◽  
Vol 6 (1) ◽  
pp. 66-68 ◽  
Author(s):  
H. Bohman
Keyword(s):  
Numerical Analysis ◽  
Laplace Transform ◽  
Ruin Probability ◽  
Claim Amount ◽  
The Laplace Transform ◽  
Desk Calculator ◽  
Numerical Tools ◽  
The Mean ◽  
Image Position ◽  
Special Case

It is fantastic how the computer has changed our attitude to numerical problems. In the old days when our numerical tools were paper, pencil, desk calculator and logarithm tables we had to stay away from formulas and methods which led to too lengthy calculations. A consequence is that we have a tendency to think of numerical analysis in terms of the classical tools. If we go back to the results of earlier writers it seems, however, very likely that many results and formulas developed by them which had earlier a theoretical interest only could nowadays be applied successfully in numerical analysis.As an example I take the ruin probability ψ(x). The Laplace transform of ψ(x) is given by the following expressionwhere c > 1. In fact (c — 1) is equal to the “security loading”. The function p(y) is equal to the Laplace transform of the claim distribution. We assume that the mean claim amount is equal to one, i.e. p′(0) = — 1.In his book from 1955 [1] Cramer points out that this formula will be more easy to handle if the claim distribution is an exponential polynomial. In this case we havewhereCramér's results are given on pages 81-83 in his book. We reproduce them here with a slight change of notations only.

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

scholarly journals Exact Analytical Solution of theN-Dimensional Radial Schrödinger Equation with Pseudoharmonic Potential via Laplace Transform Approach

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Tapas Das ◽  
Altuğ Arda
Keyword(s):  
Schrödinger Equation ◽  
Laplace Transform ◽  
Schrodinger Equation ◽  
Exact Analytical Solution ◽  
Bound State ◽  
Pseudoharmonic Potential ◽  
Special Cases ◽  
Radial Schrödinger Equation ◽  
The Laplace Transform ◽  
Generalized Morse Potential

The second-orderN-dimensional Schrödinger equation with pseudoharmonic potential is reduced to a first-order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Some special cases are verified and variations of energy eigenvaluesEnas a function of dimensionNare furnished. To give an extra depth of this paper, the present approach is also briefly investigated for generalized Morse potential as an example.

Application of q-Bessel Functions in the Solution of Generalized Fractional Kinetic Equations

2021 ◽  
Vol 88 (1-2) ◽  
pp. 01
Author(s):  
Garima Agarwal ◽  
Sunil Joshi ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Laplace Transform ◽  
Bessel Function ◽  
Bessel Functions ◽  
Kinetic Equations ◽  
Mathematical Physics ◽  
The Laplace Transform ◽  
Fractional Kinetic Equations

The present investigation aims to extract a solution from the generalized fractional kinetic equations involving the generalized q-Bessel function by applying the Laplace transform. Methodology and results can be adopted and extended to a variety of related fractional problems in mathematical physics.

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