漸進行為與拉普拉斯轉換的關係

Abstract

在這篇論文中，我們主要討論拉普拉斯轉換如何影響一個給定函數的漸進行為。研究發現，一個函數f(t)在t很大的行為會影響L(f)(s)在s很小的漸進行為。更進一步在某些狀況下我們可以藉由L(f)(s)的漸進行為去推得f(t)在t很大的漸進行為。這篇論文主要使用實分析技巧。我們還將此結果與Abelian and Tauberian Theorem of Laplace transform 進行比較。

In the present thesis, we concern the relation between the function and its Laplace transform. More precisely, the asymptotic behavior when t is large (small) implies asymptotic behavior of its Laplace transform when s is small (large) and vice versa, especially, for critical exponent we provide a simple sufficient and necessary condition. Our approach only involves real analysis. We will also compare the result with Abelian and Tauberian Theorem of Laplace transform.