The Laplace–Carson transform is an integral transform and a variation of the standard Laplace transform. It is primarily utilized in the fields of operational calculus, engineering, and mathematical physics. The transform is named after the French mathematician Pierre-Simon Laplace and the American transmission engineer John Renshaw Carson.
Definition
For a function $f(t)$ defined for $t \ge 0$, the Laplace–Carson transform $f^(p)$ is defined by the integral: $$f^(p) = p \int_{0}^{\infty} e^{-pt} f(t) , dt$$ where $p$ is a complex parameter. This transform is directly related to the standard unilateral Laplace transform, $F(p)$, by the relation $f^*(p) = pF(p)$.
Historical Context
John R. Carson introduced this specific form in the early 20th century to provide a more rigorous mathematical framework for the operational calculus developed by Oliver Heaviside. While the standard Laplace transform became the dominant tool in Western mathematical circles for solving differential equations, the Laplace–Carson transform remained a standard convention in Russian and Soviet mathematical literature, often referred to simply as the "operational" form of the Laplace transform.
Characteristics and Comparison
The principal distinction between the Laplace–Carson transform and the standard Laplace transform lies in the treatment of constants. In the standard Laplace transform, the transform of a constant $c$ is $c/p$. In the Laplace–Carson transform, the transform of a constant $c$ remains $c$: $$f^*(p) = p \int_{0}^{\infty} e^{-pt} c , dt = c$$ Because of this property, the Laplace–Carson transform preserves the units or dimensions of the original function $f(t)$, which can be advantageous in certain engineering and physical applications.
Applications
The Laplace–Carson transform is frequently applied in:
- Operational Calculus: It serves as a tool for converting linear differential equations into algebraic equations.
- Viscoelasticity and Mechanics: It is used to describe the time-dependent behavior of materials, where the correspondence principle often utilizes this transform to link elastic and viscoelastic solutions.
- Probability Theory: It is occasionally used in the study of stochastic processes and the distribution of random variables, where it relates to the concept of the Laplace-Stieltjes transform.
- Electrical Engineering: It was historically used in circuit analysis to justify Heaviside's symbolic methods for transient phenomena.