The Laplace transform is a linear integral operator that maps functions defined on time to complex-valued functions in the s-plane by decomposing them into superpositions of exponential modes $e^{st}$. Theoretically, this transformation relies on the property that derivatives become multiplications by $s$, effectively converting differential equations into algebraic equations within a specific domain of convergence where $\text{Re}(s) > \sigma_c$. Within complex analysis and system theory, the transformed function is characterized analytically via analytic continuation beyond its region of absolute convergence, revealing hidden structural information as isolated singularities known as poles.
The Laplace transform is a linear integral operator that maps functions defined on time to complex-valued functions in the s-plane by decomposing them into superpositions of exponential modes $e^{st}…