What is a Laplace Transform?

The Laplace transform is an integral transform that converts a function of time (t) into a function of complex frequency (s):

F(s) = L{f(t)} = ∫₀^∞ f(t)e^(-st)dt

where:

f(t) is the original function in time domain
F(s) is the transformed function in s-domain
s is the complex frequency variable
e is the base of natural logarithm

Properties of Laplace Transform

Linearity

L{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}

Time Shift

L{f(t-a)u(t-a)} = e^(-as)F(s)

Frequency Shift

L{e^(at)f(t)} = F(s-a)

Time Scaling

L{f(at)} = (1/a)F(s/a)

Differentiation

L{f'(t)} = sF(s) - f(0)

Integration

L{∫₀ᵗf(τ)dτ} = F(s)/s

Common Transform Pairs

Time Domain (f(t))

s-Domain (F(s))

1

1/s

t

1/s²

e^(at)

1/(s-a)

sin(ωt)

ω/(s²+ω²)

cos(ωt)

s/(s²+ω²)

t^n

n!/(s^(n+1))

Advanced Properties and Theorems

Convolution Theorem

L{f(t)*g(t)} = F(s)G(s)

where f(t)*g(t) = ∫₀ᵗf(τ)g(t-τ)dτ

Initial Value Theorem

lim[t→0] f(t) = lim[s→∞] sF(s)

Final Value Theorem

lim[t→∞] f(t) = lim[s→0] sF(s)

Periodic Function

For f(t) with period T:

L{f(t)} = (1/(1-e^(-sT))) ∫₀ᵀf(t)e^(-st)dt

Complex Analysis in Laplace Transforms

Region of Convergence (ROC)

Right Half-Plane: Re(s) > α for causal signals
Left Half-Plane: Re(s) < β for anti-causal signals
Strip: α < Re(s) < β for two-sided signals

Poles and Zeros

Poles determine system stability
Left-half plane poles: stable system
Right-half plane poles: unstable system
Imaginary axis poles: marginally stable

Engineering Applications

Control Systems

Transfer Functions: H(s) = Y(s)/X(s)
System Stability Analysis
PID Controller Design
Root Locus Analysis

Circuit Analysis

Impedance: Z(s) = sL (inductors)
Admittance: Y(s) = sC (capacitors)
Network Analysis
Frequency Response

Signal Processing

Filter Design
Modulation Analysis
System Response
Spectral Analysis

Numerical Methods

Numerical Integration Techniques

Trapezoidal Rule
Simpson's Rule
Gaussian Quadrature

Inverse Transform Methods

Partial Fraction Decomposition
Bromwich Integral
Heaviside's Expansion
Numerical Inversion

Laplace Transform Calculator

Understanding Laplace Transforms