Question

# Existence of Laplace Transform Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not) a) f(t)=t^2sin(omega t) b) f(t)=e^{t^2}sin(omega t)

Laplace transform
Existence of Laplace Transform
Do the Laplace transforms for the following functions exist? Explain your answers. (You do not need to find the transforms , just show if they exist or not)
a) $$f(t)=t^2\sin(\omega t)$$
b) $$f(t)=e^{t^2}\sin(\omega t)$$

2020-12-17

Step 1
To determine:
The existence of Laplace transforms of the functions:
a) $$f(t)=t^2\sin(\omega t)$$
b) $$f(t)=e^{t^2}\sin(\omega t)$$
Step 2
Definition of existence of Laplace transforms :
Let f be a piece – wise continuous function in $$[0,\infty)$$ and is of exponential order. The Laplace transformation F (s) of f exists for some $$s > c$$, where c is a real number depends on f.
Step 3
a)Since, the function $$f(t)=t^2 \sin(\omega t)$$ can be expressed as an exponential order. Hence the Laplace transform exists for the function $$f(t)=t^2 \sin(\omega t)$$
Step 4
b)The $$f(t)=e^{t^2}\sin(\omega t)$$ cannot be expressed as an exponential order, since it grows too fast than the exponential function and bounded by $$e^{-t}$$ and hence, the integral doesn’t converge
it would at some point exceed the $$e^{-st}$$ term.
This is the reason for the Laplace transformation of this function is undefined.