# Definition

# Why?

This makes it really easy to do many differential equations, so laplace transforms become really useful. Ignore the actual formula for now, it seems arbitrarily complex, but it is very helpful in doing many things. For example, the last post, The Eigenvalue Method, discussed solving systems of differential equations, and Laplace Transforms can do this. One of the reasons they are so good is they can turn an n'th degree equation into a first degree equation. What do I mean?

# Table

Many functions have easy laplace functions: e^at, t, 1, sin(t), etc. We're not expected to compute the integral each time, so we have table, called the **L'Chart.**

# Definition-Like Theorems

**exponential order**. That is to say, $$\boxed{\lim_{t \to \infty}\dfrac{f(t)}{e^{ct}} = 0}$$ This is because once $f(t)$ is too big, the integral is infinite, so there's no Laplace Transform.

$$\text{Corrollary: The Laplace Transform is of } s^{-1} \text{ order.}$$ $$\boxed{\lim_{s \to \infty}\mathscr{L}(f(t)) = 0}$$ Finally, Laplace transforms are

**unique**. That is to say, $$\boxed{\text{If } \mathscr{L}(f(t)) = \mathscr{L}(g(t)) \to f(t) = g(t)}$$