Fourier transform of sin(3pix/L)

• spacetimedude
In summary, the conversation discusses finding the sine function in terms of exponentials when taking the Fourier transform, and the use of the Dirac delta function in the process. There is a discrepancy between the use of the square root in the final equation, which may be due to different conventions or a sign error. Further clarification is needed.

The Attempt at a Solution

So we want sine in terms of the exponentials when we take the Fourier transform $$F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx$$ where $f(x)=\sin(3\pi x/L)$. Let a=3pi/L. Then $$\sin(ax)=\frac{e^{iax}-e^{-iax}}{2i}$$.
(Is this correct?)
Then we can take the Fourier transform:
$$F(k)=\int_{-\infty}^{\infty}\frac{e^{iax}-e^{-iax}}{2i}e^{-ikx}dx$$. Rearranging gives $$\frac{1}{2i}[\delta(K+a)-\delta(K-a)]$$. But my notes says there is $\sqrt{2\pi}$ in front and I'm not sure where it came from?
Any help will be appreciated.

spacetimedude said:

The Attempt at a Solution

So we want sine in terms of the exponentials when we take the Fourier transform $$F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx$$ where $f(x)=\sin(3\pi x/L)$. Let a=3pi/L. Then $$\sin(ax)=\frac{e^{iax}-e^{-iax}}{2i}$$.
(Is this correct?)
Then we can take the Fourier transform:
$$F(k)=\int_{-\infty}^{\infty}\frac{e^{iax}-e^{-iax}}{2i}e^{-ikx}dx$$. Rearranging gives $$\frac{1}{2i}[\delta(K+a)-\delta(K-a)]$$. But my notes says there is $\sqrt{2\pi}$ in front and I'm not sure where it came from?
Any help will be appreciated.

BvU
Ray Vickson said:
I don't understand how to apply the dirac delta function here? I just used the integral representation of delta to get to the last line.

Oh I am missing 2pi when I integrate. But where is the square root coming from?

You're probably using a different convention for the Fourier transform compared to what was done in your notes (or your notes are wrong). I think you also made a sign error.

1. What is a Fourier transform?

A Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies. It is used to analyze signals and data in various scientific and engineering fields.

2. How does the Fourier transform of sin(3pix/L) look like?

The Fourier transform of sin(3pix/L) is a combination of two delta functions, one at the frequency 3pi/L and one at the frequency -3pi/L. It also has a continuous component in between these two delta functions.

3. What does L represent in the Fourier transform of sin(3pix/L)?

L represents the length of the signal or the period of the function sin(3pix/L). It is used to scale the frequency components in the Fourier transform.

4. Can the Fourier transform of sin(3pix/L) be used to reconstruct the original signal?

Yes, the inverse Fourier transform can be used to reconstruct the original signal from its Fourier transform. However, the signal may not be exactly the same due to the nature of the Fourier transform.

5. What are some applications of the Fourier transform of sin(3pix/L)?

The Fourier transform of sin(3pix/L) is used in signal processing, image processing, and data analysis. It is also used in various fields such as physics, engineering, and mathematics for analyzing and understanding periodic signals and functions.