On Self-reciprocal functions for Fourier-Bessel integral transforms

1961 ◽  
Vol 57 (4) ◽  
pp. 778-781
Author(s):  
Afzal Ahmad ◽  
V. Lakshmikanth
Keyword(s):  
Bessel Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Special Cases ◽  
Image Position

Following Hardy and Titchmarsh(1) a function f(x) is said to be self-reciprocal if it satisfies the Fourier-Bessel integral transformwhere Jp(x) is a Bessel function of order P ≥ –½. This integral is denoted by Rp. The special cases P ½ and P ½, we denote by Rs and Rc, respectively.

INTEGRAL TRANSFORM OF K4-FUNCTION

2021 ◽  
Vol 21 (2) ◽  
pp. 429-436
Author(s):  
SEEMA KABRA ◽  
HARISH NAGAR
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Integral Transform ◽  
Integral Transforms ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Generalized Wright Function ◽  
Special Cases ◽  
Euler Transform

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function.

Generalized Mellin Convolutions and their Asymptotic Expansions

1984 ◽  
Vol 36 (5) ◽  
pp. 924-960 ◽  
Author(s):  
R. Wong ◽  
J. P. Mcclure
Keyword(s):  
Asymptotic Expansions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Integral Transforms ◽  
Generalized Function ◽  
Positive Parameter ◽  
Ordinary Sense ◽  
Integrable Functions ◽  
Classical Integral ◽  
Image Position

A large number of important integral transforms, such as Laplace, Fourier sine and cosine, Hankel, Stieltjes, and Riemann- Liouville fractional integral transforms, can be put in the form1.1where f(t) and the kernel, h(t), are locally integrable functions on (0,∞), and x is a positive parameter. Recently, two important techniques have been developed to give asymptotic expansions of I(x) as x → + ∞ or x → 0+. One method relies heavily on the theory of Mellin transforms [8] and the other is based on the use of distributions [24]. Here, of course, the integral I(x) is assumed to exist in some ordinary sense.If the above integral does not exist in any ordinary sense, then it may be regarded as an integral transform of a distribution (generalized function). There are mainly two approaches to extend the classical integral transforms to distributions.

scholarly journals On Integral Transforms

1956 ◽  
Vol 10 (3) ◽  
pp. 125-128
Author(s):  
B. W. Conolly
Keyword(s):  
Integral Transform ◽  
Integral Transforms ◽  
Mellin Transforms ◽  
Image Position

Titchmarsh and others [4] havo studied integral transform pairs of tho typeshowing how such pairs may be generated as a consequence of the relation which holds between the Mellin transforms of the kernels.

scholarly journals Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Transform ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Formulas ◽  
Special Cases

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

scholarly journals Geometric properties of some linear operators defined by convolution

2008 ◽  
Vol 39 (4) ◽  
pp. 325-334 ◽  
Author(s):  
R. Aghalary ◽  
A. Ebadian ◽  
S. Shams
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Integral Transform ◽  
Integral Transforms ◽  
Linear Operators ◽  
Geometric Properties ◽  
Special Cases ◽  
Classical Integral ◽  
Alpha Beta

Let $\mathcal{A}$ denote the class of normalized analytic functions in the unit disc $ U $ and $ P_{\gamma} (\alpha, \beta) $ consists of $ f \in \mathcal{A} $ so that$ \exists ~\eta \in \mathbb{R}, \quad \Re \bigg \{e^{i\eta} \bigg [(1-\gamma) \Big (\frac{f(z)}{z}\Big )^{\alpha}+ \gamma \frac{zf'(z)}{f(z)} \Big (\frac{f(z)}{z}\Big )^{\alpha} - \beta\bigg ]\bigg \} > 0. $ In the present paper we shall investigate the integral transform$ V_{\lambda, \alpha}(f)(z) = \bigg \{\int_{0}^{1} \lambda(t) \Big (\frac{f(tz)}{t}\Big )^{\alpha}dt\bigg \}^{\frac{1}{\alpha}}, $ where $ \lambda $ is a non-negative real valued function normalized by $ \int_{0}^{1}\lambda(t) dt=1 $. Actually we aim to find conditions on the parameters $ \alpha, \beta, \gamma, \beta_{1}, \gamma_{1} $ such that $ V_{\lambda, \alpha}(f) $ maps $ P_{\gamma}(\alpha, \beta) $ into $ P_{\gamma_{1}}(\alpha, \beta_{1}) $. As special cases, we study various choices of $ \lambda(t) $, related to classical integral transforms.

On a family of the incomplete H-functions and associated integral transforms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar
Keyword(s):  
Integral Transform ◽  
Special Functions ◽  
Natural Extension ◽  
Integral Transforms ◽  
Legendre Transform ◽  
Physical Sciences ◽  
Special Cases ◽  
Improper Integrals ◽  
Jacobi Transform ◽  
Math Phys

Abstract Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results.

scholarly journals A Class of Integral Transforms

1964 ◽  
Vol 14 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problems ◽  
Inversion Formula ◽  
Heat Conduction Equation ◽  
Boundary Value ◽  
Integral Transforms ◽  
Conduction Equation ◽  
New Class ◽  
Special Cases ◽  
Image Position

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))and the generalised Mehler transform pair (7)These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

On Hankel Transformable Spaces and a Cauchy Problem

1985 ◽  
Vol 37 (1) ◽  
pp. 84-106 ◽  
Author(s):  
R. S. Pathak
Keyword(s):  
Cauchy Problem ◽  
Adjoint Method ◽  
Kernel Method ◽  
Direct Method ◽  
Integral Transforms ◽  
Hankel Transform ◽  
Generalized Functions ◽  
One Dimension ◽  
Special Cases ◽  
Image Position

The classical Hankel transform of a conventional function ϕ on (0, ∞) defined formally bywas extended by Zemanian [21-23] to certain generalized functions of one dimension. Koh [9, 10] extended the work of [21] to n-dimensions, and that of [22] to arbitrary real values of μ. Motivated from the work of Gelfand and Shilov [6], Lee [11] introduced spaces of type Hμ and studied their Hankel transforms. The results of Lee [11] and Zemanian [21] are special cases of recent results obtained by the author and Pandey [14]. The aforesaid extensions are accomplished by using the so-called adjoint method of extending integral transforms to generalized functions. Dube and Pandey [2], Pathak and Pandey [15, 16] applied a more direct method, the so-called kernel method, for extending the Hankel and other related transforms.

Some self-reciprocal functions and kernels

1961 ◽  
Vol 57 (3) ◽  
pp. 690-692 ◽  
Author(s):  
V. Lakshmikanth
Keyword(s):  
Integral Equation ◽  
Bessel Function ◽  
Integral Transforms ◽  
Homogeneous Integral Equation ◽  
Image Position ◽  
Class Of Functions

The aim of this note is to find out some self-reciprocal functions and kernels for Fourier-Bessel integral transforms. Following Hardy and Titchmarsh(i), we shall denote by Rp the class of functions which satisfy the homogeneous integral equationwhere Jp(x) is a Bessel function of order p ≥ − ½. For particular values of p = ½, − ½, we write Rs and Rc irrespectively.

Fractional Integration of Generalized Bessel Function of the First Kind

2011 ◽  
Author(s):  
Pradeep Malik ◽  
Saiful R. Mondal ◽  
A. Swaminathan
Keyword(s):  
Bessel Function ◽  
Hypergeometric Functions ◽  
Fractional Integration ◽  
Integral Transforms ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Generalized Bessel Function ◽  
Hyperbolic Sine ◽  
Hyperbolic Cosine ◽  
Special Cases

Generalizing the classical Riemann-Liouville and Erde´yi-Kober fractional integral operators two integral transforms involving Gaussian hypergeometric functions in the kernel are considered. Formulas for composition of such integrals with generalized Bessel function of the first kind are obtained. Special cases involving trigonometric functions such as sine, cosine, hyperbolic sine and hyperbolic cosine are deduced. These results are established in terms of generalized Wright function and generalized hypergeometric functions.

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