scholarly journals An Inversion Formula for the Generalised Laplace Transform

1949 ◽  
Vol 8 (3) ◽  
pp. 126-127 ◽  
Author(s):  
R. S. Varma
Keyword(s):  
Laplace Transform ◽  
Inversion Formula ◽  
Whittaker Functions ◽  
Laplace Integral ◽  
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Recently I have given a generalisation of the Laplace integralin the formwhere Wk, m (x) stands for Whittaker Functions.

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 ◽  
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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

An inversion formula for the Varma transform

1966 ◽  
Vol 62 (3) ◽  
pp. 467-471 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Bessel Function ◽  
Inversion Formula ◽  
Fractional Integration ◽  
Integration Theory ◽  
Modified Bessel Function ◽  
Laplace Integral ◽  
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AbstractRecently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equationcan be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.

An inversion formula for the kernel Kv(x)

1965 ◽  
Vol 61 (2) ◽  
pp. 457-467 ◽  
Author(s):  
Charles Fox
Keyword(s):  
Integral Equation ◽  
Laplace Transform ◽  
Inversion Formula ◽  
Bessel Functions ◽  
Fractional Integration ◽  
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AbstractThe problem discussed is that of solving the integral equationwhere g(x) is given, Kv(z) is associated with Bessel functions of purely imaginary argument and f(x) is to be determined.I prove that, by means of fractional integration, it is possible to reduce this equation to the form of a Laplace transform which can be solved by known methods.

scholarly journals A Relation between Laplace and Hankel Transforms

1962 ◽  
Vol 5 (3) ◽  
pp. 114-115 ◽  
Author(s):  
B. R. Bhonsle
Keyword(s):  
Laplace Transform ◽  
Hankel Transform ◽  
Laplace And Hankel Transforms ◽  
Hankel Transforms ◽  
The Laplace Transform ◽  
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The Laplace transform of a function f(t) ∈ L(0, ∞) is defined by the equationand its Hankel transform of order v is defined by the equationThe object of this note is to obtain a relation between the Laplace transform of tμf(t) and the Hankel transform of f(t), when ℛ(μ) > − 1. The result is stated in the form of a theorem which is then illustrated by an example.

On Some Properties of Functions Analytic in a Half-Plane

1959 ◽  
Vol 11 ◽  
pp. 432-439 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Laplace Transform ◽  
Half Plane ◽  
The Laplace Transform ◽  
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The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where1.1Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,then f ∈ , where1.2and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
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Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

An Inversion Formula for the Weierstrass Transform

1961 ◽  
Vol 13 ◽  
pp. 593-601 ◽  
Author(s):  
G. G. Bilodeau
Keyword(s):  
Inversion Formula ◽  
Basic Properties ◽  
Weierstrass Transform ◽  
Gauss Transform ◽  
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The Weierstrass transform f(x) of a function ϕ(y) is defined by1.1wherewhenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operatorwill invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definitionfor f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).

A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform

1994 ◽  
Vol 124 (5) ◽  
pp. 909-916 ◽  
Author(s):  
O. Arino ◽  
M. A. El Attar
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
General Expression ◽  
Exponential Growth ◽  
Mixed Type ◽  
Deviating Arguments ◽  
Neutral Differential Equations ◽  
Equations Of Mixed Type ◽  
The Laplace Transform ◽  
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Consider the general expression of such equations in the formwhere Ai, Bj, ∊ ℝ, δo = 0 dn/ 0, dn are n-derivatives, n ≧ l, the σj'S and δj,'s respectively, are ordered as an increasing family with possibly positive and negative terms. These are the deviating arguments. In this paper, we provide a proof of this result based on the use of the Laplace transform. Our method involves new results regarding the exponential growth of positive solutions for such equations.

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