motivating the Laplace transform?

“Laplace transform is nothing else but the continuous version of power series”

The substitution x=e^(-s) makes the integral look like a formal power series, f(t) being the coefficient of x^t.

Similarly, the Fourier transform is a continuous version of the (discrete) fourier (power) series.

We know that solutions to a differential equation has to be continuous. By Weierstrass theorem, any continuous function can be uniformly approximated by a polynomial. So why not look for solutions to the DE as a polynomial/power series whose coefficients we shall determine. (series solutions to a DE) ??But the Laplace transform is not the solution yet. We need to inverse the Laplace transform.

There are several nice things about choosing g(t;s)=e^−st as the set of basis (kernel) functions to use, the prime one being that dg(t;s)ds=−se^−st, which is a neat property if you’re dealing with derivatives. The Laplace transform of a derivative simply becomes an algebraic function of the Laplace transform of the function itself.

The Fourier transform resolves a function or signal into its modes of vibration, while the Laplace transform resolves a function into its moments. (transform from time domain to moments domain)

Its power:

The Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system.

An integral transform “maps” an equation from its original “domain” into another domain. Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.

the Laplace transform converts integral and di®erential equations into algebraic equations. Laplace transform turns convolution into multiplication.

More history:

From 1744, Leonhard Euler investigated (indefinite) integrals of the form

 z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx

as solutions of differential equations. These types of integrals (from probability theory) seem first to have attracted Laplace’s attention in 1782.

Laplace went a critical step forward when, rather than just looking for a solution in the form of an integral, he started to use the integrals as transforms (convergent improper integrals) and looked for solutions of the transformed equation. (but how did he know this?) He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.

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