## Laplace Transforms

For now, I am only writing a summary of the properties of Laplace transforms. I will come back eventually and fill in all the pertinent details.

**DEFINITION | Laplace Transform**

Let \(f(t)\) be a piecewise continuous function defined on \([0, \infty)\). The

**Laplace transform**of \(f\) is defined to be\[\mathcal{L}\{f(t)\} = \int_0^{\infty}{e^{-s t} f(t) dt}.\]

Since this improper integral leads to a function of \(s\), many authors write \(\mathcal{L}\{f(t)\} = F(s)\), where \(F\) is chosen as the function name because the function we are taking the Laplace transform of is \(f(t)\). If we were taking the Laplace transform of \(h(t)\), then we would let \(\mathcal{L}\{h(t)\} = H(S)\).

**THEOREM | Linearity of the Laplace Operator**

The Laplace transform is a linear operator. That is,\[\mathcal{L}\{f(t) + c g(t)\} = \mathcal{L}\{f(t)\} + c \mathcal{L}\{g(t)\}.\]

**Basic Laplace Transforms**

- \(\mathcal{L}\{1\} = \frac{1}{s}\)
- \(\mathcal{L}\{e^{at}\} = \frac{1}{s - a}\) (shift in \(s\))
- \(\mathcal{L}\{e^{at} f(t)\} = F(s - a)\) (general shift in \(s\))
- \(\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}, n \in \mathbb{N}\)
- \(\mathcal{L}\{\sin{(\omega t)}\} = \frac{\omega}{s^2 + \omega^2}\)
- \(\mathcal{L}\{\cos{(\omega t)}\} = \frac{s}{s^2 + \omega^2}\)
- \(\mathcal{L}\{t^n f(t)\} = (-1)^n F^{(n)}(s)\)

**THEOREM | The Laplace of a Derivative**

If \(f, f', \ldots, f^{(n-1)}\) are continuous on \([0,\infty)\) (and are of exponential order), and if \(f^{(n)}(t)\) is piecewise continuous on \([0,\infty)\), then\[\mathcal{L}\{f^{(n)}(t)\} = s^n \mathcal{L}\{f(t)\} - s^{n-1}f(0) - s^{n-2}f'(0) - \cdots - s f^{(n-2)}(0) - f^{(n-1)}(0).\]

**DEFINITION | Inverse Laplace Transform**

Let \(F(s) = \mathcal{L}\{f(t)\}\) be the Laplace transform of \(f\). Then we say that \(f(t)\) is the

**inverse Laplace transform**and we denote this as\[\mathcal{L}^{-1}\{F(s)\} = f(t).\]

**THEOREM | Linearity of the Inverse Laplace Operator**

The inverse Laplace transform is a linear operator.

**DEFINITION | Heavyside Function**

The

**Heavyside function**is defined to be \[\mathcal{U}\{t - a\} = \left\{ \begin{array}{lr} 0 & t < a \\ 1 & t \geq a \end{array} \right..\]

**THEOREM | Heavyside Shift**

If \(a > 0\), then\[\mathcal{L}\{f(t - a) \mathcal{U}(t - a)\} = e^{-as}F(s).\]

**DEFINITION | Convolution**

The

**convolution**of \(f\) with \(g\) is defined to be\[f(t) \star g(t) = \int_0^t{f(\tau) g(t - \tau) d\tau}\]

**THEOREM | Commutivity of Convolutions**

\(f(t) \star g(t) = g(t) \star f(t)\)

**THEOREM |**

\(\mathcal{L}\{f(t) \star g(t)\} = \mathcal{L}\{f(t)\} \mathcal{L}\{g(t)\}\)

**THEOREM |**

\(f(t) \star g(t) = \mathcal{L}^{-1}\left\{\mathcal{L}\{f(t)\} \mathcal{L}\{g(t)\}\right\}\)