关于Laplace变换象函数在第一充分条件下解析性的一种证明

One proof of the analyticity of image function of Laplace transform under the first sufficient condition for its existence

先证明了当 f(t)在t0 处 (t0 >0 )是跳跃间断点时 ,f(t)e-st及tf(t)e-st在 (t0 ,s)处是不连续的 ,从而说明 f(t)的象函数F(s)的求导运算不能直接使用参变量广义积分求导与积分号交换次序的有关定理[1 ] ,然后在第一充分条件下 ,给出一种直接证明 F′(s) =∫+∞0dds[f(s)e-st]dt的方法 ,从而解决了这一理论上的不严密性 ,也澄清了文中所提出的事实 .

In this paper, the functions f(t) e -st and tf(t) e -st are shown to be interrupted at (t 0,s), if f(t) leaps abruptly at t 0. Thus , the related theorems dealing with the interchange of the order of differential operation and integration [1] can no louger be used . And then, under the first sufficient condition , a direct proof is given for F′(s)=∫ +∞ 0 dd s[f(s) e -st ] d t. Thus the defect on the theoretical plane is solved and the fact mentioned in those paper can also be clarified.