Laplace Transform

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 $t=0$.

Conditions and key results

$\mathcal L\{f'\}=sF-f(0)$ and $\mathcal L\{f''\}=s^2F-sf(0)-f'(0)$ require appropriate regularity and exponential growth control. Algebra in transform space still needs denominator domains and correct inverse pairs. The transform is not guaranteed to exist for functions growing faster than every exponential.

A reliable strategy

1. Confirm a transform is appropriate and record all initial values before transforming derivatives.
2. Transform each term using linearity, derivative formulas, and shifting or step rules.
3. Solve the resulting algebraic equation for $Y(s)$ and decompose it into invertible terms.
4. Invert, state the causal solution for $t\ge0$, and verify the original equation and initial conditions.

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.

Common mistakes

Verification and reasonableness

• Take the forward transform of the recovered time function.
• Evaluate $y(0^+)$ and, where valid, compare with $\lim_{s\to\infty}sY(s)$.
• Differentiate the result piecewise and verify the ODE away from any stated impulse or jump.

Practice

1. Compute $\mathcal L\{t^2\}$.
2. Transform $y''$.
3. For what real $s$ does $\mathcal L\{e^{3t}\}$ converge?

Further deduction

The first shifting theorem states $\mathcal L\{e^{at}f(t)\}=F(s-a)$. It shifts the transform variable, whereas the second shifting theorem $\mathcal L\{u(t-a)f(t-a)\}=e^{-as}F(s)$ delays the time signal. These operations are often confused because both are called shifts; checking whether the multiplier appears in time or in $s$ identifies the correct theorem.

Direct calculation verifies the exponential pair: for real $s>a$, $\int_0^\infty e^{-st}e^{at}dt=\int_0^\infty e^{-(s-a)t}dt=1/(s-a)$. The convergence condition is part of the result. It also shows why shifting $s$ by $a$ corresponds to multiplying the time function by $e^{at}$.

Related topics

• Inverse Laplace Transform
• Step Functions
• Initial Value Problems