Commutative neutrix convolution product of generalized Fresnel sine integrals and applications

Commutative neutrix convolution products of functions

The commutative neutrix convolution product of the functions xe − and xe μx + is evaluated for r, s = 0, 1, 2, . . . and all λ, μ. Further commutative neutrix convolution products are then deduced.

On the Fresnel Integrals and the Convolution

The Fresnel cosine integral C(x), the Fresnel sine integral S(x), and the associated functions C + (x), C − (x), S + (x), and S − (x) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel cosine integral and its associated functions with x r + and x r are evaluated. The Fresnel cosine integral C(x) is defined by C(x) = 2 π x 0 cos ...

On the Fresnel Sine Integral and the Convolution

The Fresnel sine integral S(x), the Fresnel cosine integral C(x), and the associated functions S + (x), S − (x), C + (x), and C − (x) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel sine integral and its associated functions with x r + , x r are evaluated. 1. Introduction. The Fresnel integrals occur in the diffraction theory...

On the Non-commutative Neutrix Product Involving Slowly Varying Functions

Let L(x) be a slowly varying function at both zero and infinity. The existence of the non-commutative neutrix convolution product of the distributions x+L(x) and x μ − is proved, where λ, μ are real numbers such that λ, μ / ∈ −N and λ+μ / ∈ −Z . Some other products of distributions are obtained. AMS Mathematics Subject Classification (2000): 46F10

Integral Transforms of Fourier Cosine and Sine Generalized Convolution Type