Application of the Laplace transform in time-domain optical spectroscopy and imaging

The Laplace transform is a powerful mathematical tool used to convert functions from the time domain into the frequency domain, greatly simplifying the analysis and solution of linear time-invariant systems. This transformation is facilitated by several universal properties: Linearity, Time-Scaling, Time-Shifting, and Frequency Shifting.
The Linearity property is foundational to the Laplace transform. It states that the transform of a linear combination of functions is equivalent to the same...

The Laplace transform is an indispensable mathematical technique for simplifying the resolution of differential equations by converting them into more manageable algebraic expressions. The Laplace transform of a function is denoted by L[x(t)], where x(t) is the time-domain function. The laplace transform is mathematically expressed as

Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...