Polynomials: t, t^2, 3t^4
Exponentials: e^(2t), e^(-3t)
Trigonometric: sin(2t), cos(5t)
Hyperbolic: sinh(t), cosh(2t)
Unit Step: u(t), u(t-3)
Delta: delta(t), delta(t-2)
L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
The Laplace transform converts a function of time f(t) into a function of complex frequency F(s), making it easier to analyze and solve differential equations.
The Laplace transform is an integral transform that converts a function of time f(t) into a function of complex frequency F(s). Named after Pierre-Simon Laplace, this mathematical tool is fundamental in engineering and physics, particularly for solving linear ordinary differential equations with constant coefficients. It transforms differentiation and integration into multiplication and division operations, significantly simplifying the analysis of linear time-invariant systems.
The transform is particularly useful in control theory, signal processing, and circuit analysis. It allows engineers to analyze the behavior of systems in the frequency domain, where complex differential equations become simple algebraic equations. The region of convergence (ROC) specifies the values of s for which the integral converges, ensuring the transform exists.
Linearity
L{af + bg} = aF(s) + bG(s)
Time Shifting
L{f(t-a)u(t-a)} = e^(-as)F(s)
Frequency Shifting
L{e^(at)f(t)} = F(s-a)
Differentiation
L{f'(t)} = sF(s) - f(0)
Note: Laplace transform calculations follow standard mathematical definitions. Results depend on correct function input and convergence conditions. This calculator handles common function types; complex expressions may require manual computation or advanced software.