Exponential Operator

2016 ◽  
pp. 220-222
Author(s):  
Samirnath Mallik ◽  
Sourav Sarkar
Keyword(s):  
Exponential Operator

scholarly journals Exponential Function of a bounded Linear Operator on a Hilbert Space.

2014 ◽  
Vol 11 (3) ◽  
pp. 1267-1273
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Exponential Function ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Exponential Operator

In this paper, we introduce an exponential of an operator defined on a Hilbert space H, and we study its properties and find some of properties of T inherited to exponential operator, so we study the spectrum of exponential operator e^T according to the operator T.

scholarly journals A Further Extension for Ramanujan’s Beta Integral and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 118
Author(s):  
Gao-Wen Xi ◽  
Qiu-Ming Luo
Keyword(s):  
Gamma Functions ◽  
Integral Formulas ◽  
Beta Integral ◽  
Exponential Operator

In 1915, Ramanujan stated the following formula ∫ 0 ∞ t x - 1 ( - a t ; q ) ∞ ( - t ; q ) ∞ d t = π sin π x ( q 1 - x , a ; q ) ∞ ( q , a q - x ; q ) ∞ , where 0 < q < 1 , x > 0 , and 0 < a < q x . The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As some applications, we obtain some new integral formulas of Ramanujan and also show some new representation with gamma functions and q-gamma functions.

The q-exponential operator

2013 ◽  
Vol 7 ◽  
pp. 6369-6380 ◽  
Author(s):  
Husam L. Saad ◽  
Abbas A. Sukhi
Keyword(s):  
Exponential Operator

The Nonlinear Gaussian Spectrum of Log-Normal Stochastic Processes and Variables

1999 ◽  
Vol 66 (4) ◽  
pp. 964-973 ◽  
Author(s):  
R. Ghanem
Keyword(s):  
Stochastic Processes ◽  
Polynomial Chaos ◽  
Analytical Investigation ◽  
Polynomial Chaos Expansion ◽  
Stochastic Finite Element Method ◽  
Chaos Expansion ◽  
Mathematical Properties ◽  
Log Normal ◽  
Exponential Operator ◽  
Gaussian Spectrum

A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The polynomial chaos expansion results in a representation of a stochastic process in terms of multidimensional polynomials orthogonal with respect to the gaussian measure with the dimension defined through a set of independent normalized gaussian random variables. Such a representation is useful in the context of the spectral stochastic finite element method, as well as for the analytical investigation of the mathematical properties of lognormal processes.

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