Learn how to interpret Laplace transform results, convergence regions, and mathematical properties
Results are displayed in standard mathematical notation using the complex variable s = σ + jω.
Input: f(t) = sin(3t)
Output: F(s) = 3/(s² + 9)
Convergence: Re(s) > 0
The region of convergence (ROC) tells you where the transform is valid in the s-plane.
Results may include information about which transform properties were used:
L{af(t) + bg(t)} = aF(s) + bG(s)
L{f(t-a)u(t-a)} = e^(-as)F(s)
L{e^(at)f(t)} = F(s-a)
L{f'(t)} = sF(s) - f(0)
Most results are ratios of polynomials in s
F(s) = (2s + 3)/(s² + 4s + 5)Time delays appear as exponential factors
F(s) = e^(-2s) · 1/sPoles (denominator = 0) determine system behavior, zeros (numerator = 0) affect magnitude.
Stability Check: All poles must have negative real parts for stability
Use value theorems to find behavior at t=0 and t=∞.
Initial Value: f(0) = lim(s→∞) sF(s)
Final Value: f(∞) = lim(s→0) sF(s)
Result: F(s) = 1/(s + 2)
Interpretation: Single pole at s = -2. This represents an exponentially decaying function f(t) = e^(-2t)u(t) with time constant τ = 1/2 = 0.5 seconds.
Result: F(s) = 5/(s² + 4s + 13)
Interpretation: Complex poles at s = -2 ± 3j. This represents a damped oscillation with damping ratio ζ = 0.67 and natural frequency ωn = √13 ≈ 3.6 rad/s.
Use our calculator to see these interpretation principles in action