This lesson introduces the Laplace transform as a tool for converting linear differential equations into algebraic equations in the complex frequency domain.

Definition and common transforms

F(s) = ℒ{f(t)} = ∫₀∞ e−stf(t)dt, where the integral converges for the region of interest.

• ℒ{1} = 1/s
• ℒ{tⁿ} = n!/sn+1
• ℒ{eat} = 1/(s − a)

Linearity

For constants a and b, ℒ{af(t) + bg(t)} = aF(s) + bG(s).

Practice

Use the table and linearity to find ℒ{3 + 2e4t}. Check that your result is 3/s + 2/(s − 4).