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In this video, the author explains complex Fourier series and how they are used to create animations. Fourier series can describe any real-world phenomena in terms of a sum of sinusoidal waves with different frequencies. The author explains how to solve the heat equation using Fourier series.

**00:00:00**In this video, the author explains complex Fourier series and how they are used to create animations like the one shown. Fourier series can describe any real-world phenomena in terms of a sum of sinusoidal waves with different frequencies. The author explains how to solve the heat equation using Fourier series.**00:05:00**Fourier series are a way to sum sinewaves in a way that makes the sum continuous and discontinuous. Fourier thought to ask a question which seems absurd: How do you express any initial temperature distribution as a sum of sine waves. And it's more constrained than just that! You have to restrict yourself to adding waves which satisfy a certain boundary condition, which as we saw last video means working only with these cosine functions whose frequencies are all some whole number multiple of a given base frequency.**00:10:00**This video covers the basics of Fourier series. It explains that a Fourier series is a sum of rotating vectors, and that complex exponential functions are a special case of this. It goes on to show how to write down the formulas for these vectors, and how to think of them as rotating arrows pointing in one direction.**00:15:00**Fourier series describe the motion of vectors that rotate around a center point. The control we have over these vectors is by adjusting the coefficients of their complex numbers.**00:20:00**In this video, a Fourier series is explained, and it is shown that the complex-valued function is a summation of sine and cosine waves. The video also demonstrates how to compute the coefficients of a Fourier series by integrating.

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