Question

Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. Find ℒ{f(t)}. (Write your answer as a function of s.)

Answer 1

`\mathcal{L}\{f(t)\} = (f(0))/(2\cdot s)`

Explanation:

The definition of the Laplace Transform lead to the following form:

`\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dt = -(1)/(s)\cdot e^(-s\cdot t)\cdot f(t)|_(0)^(\infty)-\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dx`

`\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dt = -(1)/(2\cdot s)\cdot e^(-s\cdot t)\cdot f(t)|_(0)^(\infty)`

`\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dt = -(1)/(2\cdot s)\cdot e^(-s\cdot t)\cdot f(\infty)+(1)/(2\cdot s)\cdot f(0)`

`\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dt = 0+(1)/(2\cdot s)\cdot f(0)`

`\int\limits^(\infty)_(0) {e^(-s\cdot t)\cdot f(t)} \, dt = (1)/(2\cdot s)\cdot f(0)`

`\mathcal{L}\{f(t)\} = (f(0))/(2\cdot s)`