# Math Help - Help understanding a Fourier Transform question

1. ## Help understanding a Fourier Transform question

Calculate the Fourier Transform of $g(x)$, where:

$g(x) =
\begin{cases}
x & 0 \le x \le 1 \\
2-x & 1 \le x \le 2 \\
0 & \text{otherwise}
\end{cases}
$

The formula that I've got is the Fourier transform $\hat{f}(s)$ of a function $f(x)$ is a function defined by:

$\hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx$

My question is, how would I go about calculating the Fourier Transform for the above function? Which value of $f(x)$ would I use, and which limits?

Thanks in advance.

2. Just had a thought. Would I calculate the two integrals separately and then add them together?

Also if this is the case, is this something you can always do?

Thanks again

3. Originally Posted by craig
Calculate the Fourier Transform of $g(x)$, where:

$g(x) =
\begin{cases}
x & 0 \le x \le 1 \\
2-x & 1 \le x \le 2 \\
0 & \text{otherwise}
\end{cases}
$

The formula that I've got is the Fourier transform $\hat{f}(s)$ of a function $f(x)$ is a function defined by:

$\hat{f}(s) := \left\int^{\infty}_{-\infty}\right f(x) e^{-isx} dx$

My question is, how would I go about calculating the Fourier Transform for the above function? Which value of $f(x)$ would I use, and which limits?

Thanks in advance.
I'm pretty sure that it would be safe to say that $\hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx$ due to the piecewise definition of g(x).

I hope this helps!

4. Originally Posted by Chris L T521
I'm pretty sure that it would be safe to say that $\hat{g}(s) = \displaystyle\int_0^1 xe^{-isx}\,dx+\int_1^2(2-x)e^{-isx}\,dx$ due to the piecewise definition of g(x).

I hope this helps!
Hi thanks for the reply.

So I'm guessing that this would apply for any function defined like this?

5. Originally Posted by craig
Hi thanks for the reply.

So I'm guessing that this would apply for any function defined like this?
I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.

6. Originally Posted by Chris L T521
I want to say that the function should be piecewise continuous in order to do this; I'm pretty sure this is the case, but I'm not 100% sure.
This is true for any integrable function.

7. Thankyou