Laplacetranszformációihoz

The Laplace transform is an integral transform used to convert a function of time into a function of complex frequency. For a function f(t) defined for t ≥ 0, the unilateral Laplace transform F(s) is defined by F(s) = ∫_0^∞ f(t) e^{-st} dt, where s is a complex number with real part large enough for convergence. The inverse Laplace transform recovers f(t) from F(s) via f(t) = (1/2π i) ∫_{γ - i∞}^{γ + i∞} F(s) e^{st} ds, with γ chosen in the region of convergence.

Laplace transforms are widely used to solve linear ordinary differential equations with given initial conditions, converting differentiation into multiplication by s and integration into division by s. They also support properties such as linearity, time-domain shifting, and frequency-domain shifting, as well as the convolution theorem: the Laplace transform of the convolution f * g is F(s)G(s). The transform is particularly effective for systems governed by linear time-invariant differential equations and for control theory.

Common transforms include L{1} = 1/s, L{t^n} = n!/s^{n+1}, L{e^{at}} = 1/(s - a), L{sin(bt)} = b/(s^2 + b^2), and L{cos(bt)} = s/(s^2 + b^2), valid for appropriate regions of convergence. The unilateral transform focuses on t ≥ 0, while the bilateral transform extends to all real t with its own region of convergence. The Laplace transform is connected to the Fourier transform by evaluating s on the imaginary axis and is a standard tool in engineering, physics, and applied mathematics for analyzing linear systems and signals.