Chapter 15, Problem 1RQ

To determine

Choose the correct option whether the statement “each time domain function $f\left(t\right)$ has a Laplace transform” is true or false.

Answer to Problem 1RQ

The correct option from the given choices is (b) False.

Explanation of Solution

Generally, the Laplace transform of function $f\left(t\right)$ is denoted by $F\left(s\right)$ or $\mathcal{L}\left[f\left(t\right)\right]$. It is defined by,

$F\left(s\right)=\mathcal{L}\left[f\left(t\right)\right]={\displaystyle \underset{0^{-}}{\overset{\infty }{\int }}f\left(t\right)e^{-st}dt}$

Here,

s is the complex variable.

A time domain function $f\left(t\right)$ may not have a Laplace transform. For a function to have a Laplace transform, it is necessary for the integral must converge to a finite value.

Some of the time domain functions does not have a Laplace transform are as follows:

• $f\left(t\right)=\frac{1}{t}$ (Because of the singularity at $t=0$).
• $f\left(t\right)=e^{t^2}$ (Because the function grows too quickly as t becomes large).

Conclusion:

Thus, the correct option from the given choices is (b) False.