Laplace Transform

A rigorous introduction to the Laplace transform, convergence, derivative rules, and initial value problems.

Classify → choose method → apply condition. A differential equation is classified by order, linearity, and separability before a method is selected. The final condition determines constants and a residual check tests the solution.

Precise definition

For a function $f(t)$ defined on $t\ge0$, its Laplace transform is $F(s)=\mathcal L\{f\}(s)=\int_0^\infty e^{-st}f(t)dt$ where the improper integral converges. Piecewise continuity on finite intervals and exponential order are common sufficient conditions for convergence for all sufficiently large real $s$.

Notation and mathematical language

Linearity gives $\mathcal L\{af+bg\}=aF+bG$. Key pairs are $1\mapsto1/s$, $t^n\mapsto n!/s^{n+1}$, $e^{at}\mapsto1/(s-a)$, $\cos bt\mapsto s/(s^2+b^2)$, and $\sin bt\mapsto b/(s^2+b^2)$. The variable $s$ may be complex, but elementary ODE work often treats a right half-plane.

Conceptual picture

The kernel $e^{-st}$ weights later times exponentially, converting differentiation and convolution in time into algebraic operations in $s$. Initial conditions appear automatically because integration by parts produces boundary terms at $t=0$.

Fully worked example

Interpretation and application

Laplace methods are especially valuable for discontinuous forcing, impulses, and coupled initial conditions. An ideal impulse is a generalized function, so its use is mathematically precise within transform theory but physically represents a limiting approximation to a short, intense input.