1. The Laplace transform of a function $f(t)$, defined for $t \geq 0$, is given by the integral formula: $$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) \, dt$$ where $s$ is a complex number parameter. 2. Some common Laplace transform formulas include: - $\mathcal{L}\{1\} = \frac{1}{s}$ for $\operatorname{Re}(s) > 0$ - $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}$ for integer $n \geq 0$ - $\mathcal{L}\{e^{at}\} = \frac{1}{s - a}$ for $\operatorname{Re}(s) > a$ - $\mathcal{L}\{\sin(bt)\} = \frac{b}{s^2 + b^2}$ - $\mathcal{L}\{\cos(bt)\} = \frac{s}{s^2 + b^2}$ 3. These formulas allow transforming differential equations into algebraic equations in the $s$-domain, simplifying their solution. Final answer: The Laplace transform is defined by $$\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt$$ and common transforms include those for constants, powers of $t$, exponentials, sine, and cosine functions.