Transforms

Mathematical Principles & Theorems

The unilateral Laplace Transform \(\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty f(t) e^{-st} dt\) converts differential equations into algebraic equations in complex frequency \(s = \sigma + j\omega\). Key properties: Linearity, Differentiation \(\mathcal{L}\{f'(t)\} = sF(s) - f(0)\), Integration \(\mathcal{L}\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}\), and Convolution \(\mathcal{L}\{f * g\} = F(s)G(s)\). Transfer function \(H(s) = \frac{Y(s)}{X(s)}\) poles determine system stability (stable if \(\text{Re}(p_i) < 0\)).

Operating Instructions

• Select test function \(f(t)\) (Step, Impulse, Exponential, Sinusoid, Polynomial) or enter transfer function \(H(s)\).
• Configure system parameters and initial conditions using numeric inputs.
• Click Transform to view analytic Laplace domain representation \(F(s)\).
• Inspect the impulse response \(h(t)\), step response curves, and pole-zero s-plane constellation.