About equality of the cosine transform to the sine transform of Fourier and the transform of Laplace

TL;DR

This paper investigates the relationship between the Laplace and Fourier transforms, proving regularity properties and exploring the transform class for functions with specific regularity conditions near zero.

Contribution

It establishes a connection between Laplace and Fourier transforms, demonstrating regularity results and characterizing the transform class for functions with limited regularity at zero.

Findings

Proves regularity of Laplace transform in an open area near zero.

Shows the class of Laplace transforms derived from Fourier transforms for functions with limited regularity.

Analyzes the relationship between Laplace and Fourier transforms for functions without regularity at zero.

Abstract

Regularity of the transform of Laplace in the opened area of 0 is proved with help of some methods of the transform of Fourier. The class of the transform of Laplace from the transform of Fourier is considered from some functions without a regularity in null. The functions are regular in the opened area of 0.