有理真分式的拉普拉斯积分变换的反演

Laplace Integral Transform Inversion of Rational Proper Fractions

有理真分式的拉普拉斯积分变换反演是利用拉普拉斯变换求解微积分方程的关键。按照有理真分式分母等于零所得的根的分类,把一般的有理真分式分解成了分项分式的形式。基于分项分式,利用拉普拉斯积分变换的线性定理对有理真分式进行了反演,给出了一般有理真分式原函数的具体形式。具体讨论了单实根、共轭复根、s重根等性质的根分别对应的原函数。

Laplace integral transform inversion of rational proper fractions is the key to solve the differential and integral equation by Laplace transform.According to the classification of roots where the denominator of a rational proper fraction equals zero,we decompose a general rational proper fraction into the form of a partial fraction.Based on the partial fraction,we use the linear theorem of Laplace integral transformation to invert the rational proper fraction,and give the concrete form of the original function of the general rational proper fraction.The original functions corresponding to the roots with properties such as single real roots,conjugated complex roots,and s-multiple roots are discussed in detail.