Question

state and prove existence theorem of laplace transformation​

Answer 1

Explanation:

The Laplace transform is defined as:

F(s)=∫+∞0e−stf(t)dt

Your first question: As one can see the limit of the integral is from 0 to ∞. So, it is inherently assumed that f(t) is zero for t<0. As a result, when we talk about f(t)=t, it is actually f(t)=t,t≥0, which is a piecewise continuous function.

Second question: A function f(t) is said of exponential order if there exists a constant a and positive constants t0 and M such that |f(t)|<Meat, for all t>t0 at which f(t) is defined. This condition should hold because otherwise the Laplace transform will not exist. To prove it, lets first split the integral into two parts:

F(s)=∫+∞0e−stf(t)dt=∫t00e−stf(t)dt+∫∞t0e−stf(t)dt

It is easy to show that the first part exists. Now, we need to show that the existence of the laplace transform depends on the convergence of the second part:

|∫∞t0e−stf(t)dt|≤∫∞t0|e−stf(t)|dt≤∫∞t0Me−steatdt=M∫∞t0e−(s−a)tdt

This integral converges if s>a. So, we conclude that if f(x) is exponentially ordered, there exists a constant a in which F(s) exists for s>a.