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
ANSWERED
asked 2020-12-16
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) \(f(t)=e^{t^2}\sin(\omega t)\)

Answers (1)

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.

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