Laplace Transforms¶
Job: Convert linear constant-coefficient IVPs and piecewise/impulse forcing into algebraic equations on the half-line; invert via residues or tables.
Core¶
(Unilateral) Laplace transform of a locally integrable \(f:[0,\infty)\to\mathbb{C}\) of exponential type: $$ F(s)=\mathcal{L}{f}(s)=\int_0^\infty e^{-st}f(t)\,dt,\qquad \operatorname{Re}s>\sigma_0. $$ Analytic in a right half-plane; growth of \(F\) encodes regularity of \(f\) (Paley–Wiener-type statements for the bilateral transform).
| Property | Formula |
|---|---|
| Linearity | \(\mathcal{L}\{af+bg\}=aF+bG\) |
| Derivative | \(\mathcal{L}\{f'\}=sF-f(0^+)\) |
| \(n\)th derivative | \(s^n F-s^{n-1}f(0)-\cdots-f^{(n-1)}(0)\) |
| Integral | \(\mathcal{L}\{\int_0^t f\}=F(s)/s\) |
| Shift (t) | \(\mathcal{L}\{e^{at}f\}=F(s-a)\) |
| Shift (Heaviside) | \(\mathcal{L}\{u_c(t)f(t-c)\}=e^{-cs}F(s)\) |
| Convolution | \(\mathcal{L}\{f*g\}=FG\), \((f*g)(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau\) |
| Dirac | \(\mathcal{L}\{\delta(t-c)\}=e^{-cs}\) (\(c\ge0\)) |
IVP method: \(ay''+by'+cy=g\) \(\mapsto\) rational (or meromorphic) equation for \(Y(s)\); invert.
Inversion: Bromwich integral $$ f(t)=\frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{st}F(s)\,ds $$ (\(\gamma>\sigma_0\)). For rational \(F\), residues at poles. Partial fractions + table for elementary cases.
Final / initial value theorems (under suitable hypotheses: \(f'\) Laplace-transformable, limits exist): $$ \lim_{t\to0^+}f(t)=\lim_{s\to\infty}sF(s),\qquad \lim_{t\to\infty}f(t)=\lim_{s\to0^+}sF(s). $$
Transfer functions: LTI system \(\mapsto H(s)=Y(s)/U(s)\) (zero IC); poles determine stability (all in left half-plane for asymptotic stability of the free response).
See it¶
\(y''+y=\delta(t-\pi)\), \(y(0)=y'(0)=0\). Then \((s^2+1)Y=e^{-\pi s}\), so $$ Y=\frac{e^{-\pi s}}{s^2+1}\Rightarrow y(t)=u_\pi(t)\sin(t-\pi)=-u_\pi(t)\sin t. $$ Impulse at \(t=\pi\) kicks velocity; free oscillation thereafter.
Check¶
Why is \(\mathcal{L}\{f*g\}=FG\) restricted to causal supports on \([0,\infty)\)?
The unilateral convolution theorem uses \(\int_0^t\); bilateral Laplace / Fourier convolution needs full-line supports and different abscissae of convergence. Mixing unilateral formulas with noncausal signals is a common error.
If \(F(s)=1/(s-1)\), what is \(f\), and does the final-value theorem apply?
\(f(t)=e^{t}\). Final-value hypothesis fails (no finite limit as \(t\to\infty\)); formally \(sF\to0\) is meaningless as a limit of \(f\).
Pitfalls¶
- \(f(0^+)\) vs \(f(0^-)\) in impulsive systems: state jumps; use consistent IC conventions.
- Region of convergence matters for inverse uniqueness (identical rational expressions, different ROCs \(\Rightarrow\) different time functions in bilateral theory).
- Distributional derivatives: \(\mathcal{L}\) extends to Laplace-transformable distributions (Schwartz).
Refs¶
Doetsch, Introduction to the Theory and Application of the Laplace Transformation; Schwartz, Théorie des distributions; control texts (transfer functions).