Differential equations MOC

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 ≤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)=∫∞0𝑒−𝑠𝑡𝑓(𝑡)𝑑𝑡

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 ≤𝛾, then 𝐹(𝑠) =L{𝑓}(𝑠) exists for all 𝑠 ≥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.

See also

For a list of the Laplace transforms of some common functions, see Table of Laplace transforms.

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