Laplace

Laplace Transform Definition

The Laplace transform of a function f(t) is defined as the integral from 0 to infinity of e*(-st) * f(t) dt.

Laplace Transform Notation

The Laplace transform of f(t) is denoted as L{f(t)} or F(s).

Transform Variable

The Laplace transform F(s) is a function of the complex variable s, not the time variable t.

Piecewise Continuous

A function must be piecewise continuous on every finite interval for its Laplace transform to potentially exist.

Improper Integral

The definition of the Laplace transform involves an improper integral.

Linearity Property

The Laplace transform is a linear operation: L{af(t) + bg(t)} = aF(s) + bG(s).

Transform of 1

The Laplace transform of the constant function 1 is 1/s, for s > 0.

Transform of e

(at)

The Laplace transform of e*(at) is 1/(s-a), for s > a.

Transform of sin(at)

The Laplace transform of sin(at) is a/(s² + a²).

First Shifting Theorem

If L{f(t)} = F(s), then L{e(at)f(t)} = F(s - a).

Handles Nonhomogeneous ODEs

The Laplace transform method solves nonhomogeneous ODEs without first solving the homogeneous ODE, provided initial conditions are given.

Automatic Initial Conditions

The Laplace transform method automatically incorporates initial conditions into the solution process.

s-Shifting

The First Shifting Theorem is also known as the s-shifting property.

Convolution Theorem

The Laplace transform of the convolution integral of f(t) and g(t) is F(s)G(s).

Application to Systems

Laplace transforms can be used to solve systems of linear differential equations with initial conditions.

Solution Uniqueness

For IVPs where the Laplace transform method applies, the solution obtained is unique.

Existence of Transform

A sufficient condition for the existence of the Laplace transform is that the function is piecewise continuous and of exponential order.