Analog and Hybrid Computer Programming, 2nd Edition

A The Laplace transform

One of the standard tools for solving differential equations analytically is the LAPLACE transform, a parameter integral named after PIERRE SIMON LAPLACE. Although this is not directly connected to analog computer programming, the basic ideas and techniques are quite interesting and can sometimes be used to cross check the analog computer solution of a problem.²⁰⁶

It is defined as $F(s)=\mathcal{L}(f(t))={\displaystyle {\int }_0^{\infty }f}(t){\text{e}}^{-st} \text{d}t(A.1)$

with s ∈ C and transforms differential equations into algebraic equations which can typically be solved much more easily. The result obtained is then transformed back by means of an inverse transform L⁻¹ (F(s)), which is normally done by using tabulated functions.

A. 1 Basic functions

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