Laplace transform is an integral transform commonly used in engineering mathematics, also known as Laplace transform. Laplace transform is a linear transformation, which can transform a function with real number t(t≥ 0) into a function with complex number s.

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Transformation properties of function transformation pair and operation By using the definition integral, it is easy to establish the transformation pair between the original function f(t) and the image function F(s), and the corresponding relationship between the operation of f(t) in the real number field and the operation of F(s) in the complex number field. Table 1 and Table 2 respectively list some commonly used function transformation pairs and operational transformation properties.

Existence of Laplace Variations: In order to make F(s) exist, the integral formula must converge. There are the following theorems:

If the causal function f(t) satisfies that (1) is integrable in a finite interval, and (2) σ0 exists so that the limit of |f(t)|e-σt is 0 at t→∞, the Laplace integral is absolutely uniformly convergent for all σ greater than σ0.