The Laplace transform represents one of the most powerful and versatile mathematical tools in
engineering and applied sciences, fundamentally transforming how we approach and solve complex
differential equations. Named after the renowned French mathematician Pierre-Simon Laplace (1749
1827), this integral transformation converts complicated differential equations in the time domain into algebraic equations in the complex frequency domain, thereby significantly simplifying their solution. This chapter explores the theoretical foundations, properties, and applications of the Laplace transform, beginning with its definition for elementary functions and proceeding through its essential properties, shifting theorems, and applications to derivatives and integrals. We will also examine how the transform behaves under operations such as multiplication and division by powers of the independent variable. Throughout this discussion, we emphasize the conceptual understanding of these transformations, their historical development, and their profound significance in fields ranging from electrical engineering and control systems to physics and applied mathematics. By...