intuition for Fourier coefficients

We can most easily understand the origin of Fourier coefficients by examining the compact exponential form of the Fourier series:

where .

We know that . Let’s see how this integral yields :

All the terms except will be zero because they complete at least one full cycle and we know that the average of a point on the circle is the origin.

Therefore:

If we want to calculate other coefficients such as , notice that which shifts all the above integrals by . And we get

References

• 3blue1brown - But what is a Fourier series? From heat flow to circle drawings