## 拉普拉斯变换及其应用+文献综述

Title Research on Laplace transforms and its applications
Abstract
Laplace transforms was proposed for the first time by the great French
mathematician and scientist Pierre de Laplace. Then the English engineer O. Heaviside
developed operational calculus of transform methods. It has a hugely wide range of
applications on theories and engineering. We mainly sort out and discuss the definition,
properties and some applications of Laplace transforms based on some related
knowledge of complex variables functions, mathematical analysis and some related
references.
Firstly, we discuss the definition of Laplace transforms, especially involving with
Fourier transform. Then we list some properties and theorems of Laplace transforms,
and give the proofs of some theorems. Finally, we sort out some present typical

applications, and discuss the method of solving differential equation of constant
coefficient of the initial value problem by using Laplace transforms, complex frequency
field analysis based on Laplace transforms and the way of solving some generalized
integrals through Laplace transforms. Moreovern, we make comparisons with previous ways.
Keywords: Laplace transforms; Fourier transform

1 绪论1
1.1 选题背景1
1.2 本文的工作及结构安排2
2 预备知识2
2.1 复数的基本概念2
2.2 复数的几何表示3
2.3 复数的运算法则3
2.4 复变函数的相关概念及重要定理4
3 拉普拉斯变换的基本理论6
3.1 拉普拉斯变换的定义6
3.2 拉普拉斯变换的存在定理7
3.3 拉普拉斯变换的性质、定理及相关证明7
3.4 几个常用的拉普拉斯变换.11
3.5 拉普拉斯变换的常用求法.11
3.6 拉普拉斯逆变换.12
4 三大变换的联系与区别.16
4.1 傅里叶变换与拉普拉斯变换（理论角度）.16
4.2 三大变换典型应用比较.18
5 拉普拉斯变换的典型应用.19
5.1 求解常系数微分方程.19
5.2 求解积分方程.24
5.3 求解几种重要广义积分.25
5.4 电路复频域分析法.28
6 结论.30
7 致谢.31
8 参考文献.32
1 绪论
1.1 选题背景

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