## Fourier Transforms Part 2

Following on from the post that I made a couple of weeks ago introducing Fourier Transforms – this post will go into a little bit more detail and I will be looking at a simple example of a particular transform to illustrate what is happening. I will try to keep things as simple as possible – there are certain regularity conditions that would need to be applied in the general case, but here I will be choosing functions that satisfy these conditions anyway.

For a function $f(x)$ its Fourier Transform is defined to be $$\hat{f}(k)=\int_{-\infty}^{\infty}\!{f(x)e^{-2 \pi ikx}} \;\mathrm{d}x$$

The function $f(x)$ is in the time-domain with $x$ representing time. The transform $\hat{f}(k)$ is in the frequency-domain with $k$ representing the frequency.

I decided to make a video to visually present Fourier Transforms. They are something that I have always personally had difficulty visualising and making this video has helped me to understand them even better and I hope that it will help others likewise. Fourier Transforms are generally complex-valued functions and in some cases can be very difficult to find as an explicit formula. However, in the video below, I have chosen the function $\mathbf{cos}(2\pi(2x))\mathbf{e}^{-x^2}$ to work with which has a real-valued Fourier Transform that can be explicitly stated; I will typing-up the derivation of the Fourier Transform over the coming weeks.

I would love to hear any comments about how the video could be improved and any feedback on the video.