The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False? Explain why.

CoormaBak9

CoormaBak9

Answered question

2020-11-08

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True of False?

Answer & Explanation

unett

unett

Skilled2020-11-09Added 119 answers

Answer: False. The product of the Laplace transforms of the two functions that make up a function is not the function itself.
Explanation: Laplace transform of a function f(t) is given by:
F(s)=0estf(t)dt
Then, for any two functions f(t) and g(t) the Laplace transform of the product of two functions is:
L{f(t)g(t)}
Integral form is: 0est[f(t)g(t)]dt
Which is not equal to 0estf(t)dt0estg(t)dt
Therefore, the given statement if false.
However, it is equivalent to the convolutional product of the Laplace transforms of each function.

If F(s) and G(s) are the Laplace transforms of the functions f(t) and g(t), then,
Laplace transform of f(t)g(t) is:
L{f(t)g(t)}=F(s)G(s)
Where, indicates convolution of the two functions, given by
fg(t)=u=0tf(u)g(tu)du
As a result, the statement is untrue.

fudzisako

fudzisako

Skilled2023-06-19Added 105 answers

The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function. True or False?
The statement is false. In general, the Laplace transform of the product of two functions is not equal to the product of their individual Laplace transforms. Mathematically, if F(t) and G(t) are two functions, then the Laplace transform of their product F(t)·G(t) is given by:
{F(t)·G(t)}{F(t)}·{G(t)}
In most cases, the Laplace transform of a product involves convolution rather than simple multiplication. Therefore, the statement is false.
Jazz Frenia

Jazz Frenia

Skilled2023-06-19Added 106 answers

Step 1:
To determine whether the statement ''The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function'' is true or false, we need to consider the definition and properties of the Laplace transform.
The Laplace transform of a function f(t) is defined as:
F(s)={f(t)}=0estf(t)dt
where F(s) is the Laplace transform of f(t).
Step 2:
Now, let's consider two functions f(t) and g(t), and their Laplace transforms F(s) and G(s), respectively. We want to determine if the Laplace transform of their product, h(t)=f(t)·g(t), is equal to the product of their individual Laplace transforms, H(s)=F(s)·G(s).
To do this, we will compute the Laplace transform of h(t) and see if it matches the expression H(s). Using the definition of the Laplace transform, we have:
{h(t)}={f(t)·g(t)}=0est(f(t)·g(t))dt
Step 3:
Now, we can rewrite the integral as a product of two separate integrals:
0est(f(t)·g(t))dt=0estf(t)dt·0estg(t)dt
This shows that the Laplace transform of the product of two functions is equal to the product of their individual Laplace transforms.
Therefore, the statement ''The Laplace transform of the product of two functions is the product of the Laplace transforms of each given function'' is true.
xleb123

xleb123

Skilled2023-06-19Added 181 answers

The statement you provided is True. The Laplace transform of the product of two functions is indeed the product of the Laplace transforms of each given function.
Let's denote the Laplace transform of a function f(t) as F(s). Using this notation, the Laplace transform of the product of two functions, f(t) and g(t), can be expressed as:
{f(t)·g(t)}=F(s)·G(s)
Here, F(s) represents the Laplace transform of f(t), and G(s) represents the Laplace transform of g(t).
To emphasize this relationship, we can write the statement in LaTeX markup as:
{f(t)·g(t)}=F(s)·G(s)
Therefore, the Laplace transform of the product of two functions is indeed the product of the Laplace transforms of each given function.

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