Laplace transform

The Laplace transform is a linear endomorphism (linear operator) on the vector space of functions, which converts a function of time domain $𝑓 (𝑡)$ defined for $𝑡 \leq 0$ to a function of imaginary frequency domain $𝐹 (𝑠) = L {𝑓} (𝑠)$ . Thus it may be thought of as a complex version of the Fourier transform.

𝐹 (𝑠) = L {𝑓} (𝑠) = \int \infty 0 𝑒 - 𝑠 𝑡 𝑓 (𝑡) 𝑑 𝑡

Existence and domain

The domain of the Laplace transform $𝐹 (𝑠)$ is the subset of $ℂ$ over which the improper integral is convergent. Given a Piecewise continuous function $𝑓 (𝑡)$ is of Exponential order $\leq 𝛾$ , then $𝐹 (𝑠) = L {𝑓} (𝑠)$ exists for all $𝑠 \geq 𝛾$ .

Properties

It follows from the linearity of the integral that the Laplace transform is a Linear endomorphism. Moreover, the Laplace transform is a linear epic endomorphism, allowing us to define the inverse Laplace transform for many functions such that

L - 1 {L {𝑓}} (𝑡) = 𝑓 (𝑡)

The properties of the Laplace transform as a linear operator, and the existence (for many functions) of the inverse, gives the transform tremendous application in solving differential equations, especially non-homogeneous linear differential equations. See Solving differential equations using the Laplace transform.

Laplace transform

Existence and domain

Properties

See also