Dated: 30-10-2024

Laplace Transform

\[\mathscr L (e^{3t}) = \frac {1} {s - 3}\]

\[\implies \mathscr L^{-1} \left(\frac 1 {s - 3}\right) = e^{3t}\]

\[\mathscr L (t^3) = \frac{3!}{s^4}\]

\[\implies \mathscr L^{-1} \left(\frac{3!}{s^4}\right) = t^3\]

Definition

The laplace transform of a function¹ \(F(t)\) denotes as \(\mathscr L (F(t))\) is defined as

\[\mathscr L (F(t)) = \int_0^\infty e^{-st} F(t) dt\]

In all cases, the parameter \(s\) is assumed to be positive and large enough to ensure that the product \(F(t) \cdot e^{-st}\) converges to \(0\) as \(t \to \infty\)

Example

\[\mathscr L (a) = \int_0^\infty a e^{-st} dt\]

\[= a \int_0^\infty e^{-st} dt\]

\[=a\frac{e^{-st}}{-s}\bigg|_{0}^{\infty}=-\frac{a}{s}(0-1)=\frac{a}{s}\]

Complex Numbers

A number of the following form is called a complex number.

\[z = a + b\iota\]

where \(a\) is the real part and \(b\) is the imaginary part.

Conjugate of Complex Numbers

If \(z = a + b \iota\) is a complex number then its conjugate will be \(\overline z = \overline{a + b \iota} = a - b \iota\)

Several Standard Results

\[\mathscr L (a) = \frac a s\]
\[\mathscr L (e^{at}) = \frac 1 {s - a}\]
\[\mathscr L (t^n) = \frac{n!}{s^{n+1}}\]
\[\mathscr L (\sin (at)) = \frac{a}{s^2 + a^2}\]
\[\mathscr L (\sinh (at)) = \frac{a}{s^2 - a^2}\]
\[\mathscr L (\cos (at)) = \frac{s}{s^2 + a^2}\]
\[\mathscr L (\cosh (at)) = \frac{s}{s^2 - a^2}\]