Question

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Explain Lapace transform of integration property..................
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Answer 1

Answer:

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by

where s is a complex number frequency parameter

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0, ∞). For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood to be a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergent improper integral at ∞. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral.