15.1 : ラプラス変換の定義

The Laplace transform is a powerful mathematical tool that simplifies differential equations by converting them into algebraic expressions.

Generally, the Laplace transform of a function is represented by the symbol L[x(t)], expressed by the following equation where 's' is a complex variable with both real and imaginary components, σ and ω respectively.

There are two types of Laplace transforms - the bilateral and the unilateral.

The Bilateral transform allows time functions to be non-zero for negative time, making it useful for both causal and non-causal signals.

Conversely, the Unilateral transform, which is more common in practice, assumes a zero function for negative time, focusing solely on positive-time signals.

A unique property of the Laplace transform is its ability to convert a time-domain function into a frequency-domain function. This conversion constitutes a Laplace transform pair.

The inverse of this operation is called an inverse Laplace transform and is given by the following expression.

The Laplace transform is widely used in signal analysis, control engineering, communication, system analysis, and differential equations.