What is the Laplace Transform?

The Laplace transform is a powerful integral transform that converts a function of time ( t ) (usually ( f(t) )) into a function of a complex frequency variable ( s ), denoted ( F(s) ) or ( \mathcal{L}{f(t)} ).

Definition: [ F(s) = \mathcal{L}{f(t)} = \int_{0}^{{\infty} f(t) , e}{-st} , dt ] where ( s = \sigma + j\omega ) is complex, and the integral converges for ( \Re(s) > \sigma_0 ) (region of convergence).

It acts like a "microscope" that turns differential equations (hard in time domain) into algebraic equations (easy in ( s )-domain).

Key Properties That Make It Useful

Property	Time Domain	s-Domain	Why It’s Powerful
Differentiation	( \frac{df}{dt} )	( s F(s) - f(0^-) )	Turns derivatives into multiplication
Integration	( \int_0^t f(\tau) d\tau )	( \frac{F(s)}{s} )	Turns integrals into division
Convolution	( f(t) * g(t) )	( F(s) G(s) )	System response = input × transfer function
Time Shift	( f(t - a) u(t - a) )	( e^{-as} F(s) )	Handles delays easily
Initial/Final Value	—	( \lim_{s \to \infty} sF(s) ), ( \lim_{s \to 0} sF(s) )	Quick steady-state checks

Major Applications

1. Solving Linear Differential Equations (Control Systems & Circuits)

• Most common use.
• Example: RLC circuit or mass-spring-damper.
• Steps:

1. Take Laplace of entire ODE → algebraic equation.
2. Solve for ( X(s) ).
3. Inverse Laplace → time solution ( x(t) ).

Simple Example: Second-order system

[ \ddot{y} + 2\zeta\omega_n \dot{y} + \omega_n^2 y = \omega_n^2 u(t) ] Laplace →

[ Y(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} U(s) ] The fraction is the transfer function ( G(s) ).

Control Systems Engineering

• Analyze stability (poles of ( G(s) ) in left half-plane → stable).
• Design controllers (PID, lead-lag) in s-domain.
• Tools: Bode plots, Nyquist, root locus — all from ( G(j\omega) ).

Signal Processing & Communications

• System response to arbitrary input: ( Y(s) = H(s) X(s) ).
• Filters (low-pass, high-pass) designed in s-domain, then converted to digital (bilinear transform).

Heat Transfer, Fluid Dynamics, and PDEs

• Transform time → solve spatial ODEs.
• Example: Heat equation in semi-infinite rod → algebraic in s, inverse gives error functions.

Probability & Statistics

• Moment-generating functions are essentially Laplace transforms.
• Used in queueing theory, reliability engineering.

Mechanical & Aerospace Engineering

• Vibration analysis, flutter, servo mechanisms.
• Transient response without numerical integration.

Power Systems & Electronics

• Transient analysis of switching circuits.
• Easier than time-domain simulation for initial conditions.

Why Laplace Over Fourier?

Aspect	Fourier Transform	Laplace Transform
Frequency domain	Pure imaginary ( s = j\omega )	Complex ( s = \sigma + j\omega )
Convergence	Requires function to decay sufficiently	Handles growing exponentials (via ( \sigma ))
Transients	Poor (assumes periodic/steady)	Excellent (includes initial conditions)
Causal systems	Symmetric	Unilateral (t ≥ 0) → perfect for real systems

Fourier is a special case of Laplace on the imaginary axis.

Common Laplace Pairs (You’ll Memorize These)

( f(t) )	( F(s) )
1 (unit step)	( \frac{1}{s} )
( t )	( \frac{1}{s^2} )
( e^{-at} )	( \frac{1}{s + a} )
( \sin(\omega t) )	( \frac{\omega}{s^2 + \omega^2} )
( \cos(\omega t) )	( \frac{s}{s^2 + \omega^2} )
( e^{-at} \sin(\omega t) )	( \frac{\omega}{(s + a)^2 + \omega^2} )