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This video discusses the mathematics behind a complex Fourier series, which involves a collection of rotating arrows that can draw anything. Fourier discovered that the heat equation can be solved using an infinite family of solutions, leading to the creation of solutions for tailor-made initial conditions despite having different distributions than cosine waves. The focus shifts to breaking down functions into rotating vectors for a more generalized approach. The speaker explains how Fourier series can be used for drawing animations using rotating vectors, and relates this back to the simpler one-dimensional example of a step function. The video concludes that this foray into Fourier series is a glimpse at a deeper idea, as exponential functions play an important role in differential equations.
In this section, the narrator discusses the math behind a complex Fourier series which involves a collection of rotating arrows that draw out a specific shape by adding up together. The animation has 300 arrows rotating at a constant integer frequency, and by modifying the initial size and angle of each vector, it can draw anything. This phenomenon is described as rotating vector phenomena which is more general than the special case of functions being broken down into a sum of sine waves. The video explains that Fourier series originated from the heat equation which describes the evolution of temperature on a rod over time, and Fourier discovered that the equation can be solved using an infinite family of solutions that decay exponentially with higher frequency component decay rates. This breakthrough allowed the construction of solutions for tailor-made initial conditions despite them having different distributions than cosine waves.
In this section, the video discusses the idea of expressing any temperature distribution as a sum of sine waves with whole number multiples of a given base frequency, subject to a certain boundary condition. Fourier's discovery of this concept became far-reaching and essential in mathematics and science, and while it may seem surprising that an infinitely wavy concept could be broken down into simple oscillations, this idea of a Fourier series has many applications beyond what Fourier could have ever imagined. The video also explores the idea of an infinite sum, and in the case of the step function, is equal to an infinite sum where the coefficients are odd frequencies rescaled by 4/pi, where the sequence of partial sums approaches the value of the infinite sum. The challenge is finding the coefficients, but this formula helps to find the exact solution for how the step function evolves over time in the heat equation.
In this section, the focus shifts from breaking down functions solely into sine wave components to breaking them down into rotating vectors for a more generalized approach. Functions are viewed as drawings with the pencil tip tracing different points in the complex plane, allowing for outputs in the two-dimensional complex plane. While functions with real number outputs are essentially boring sketches, a sum of little vectors rotating at some constant integer frequency can create a sine wave that oscillates like a sine wave in the real number line. The heart and soul of Fourier series is the complex exponential, e^{i * t}, where the value of t ticks forward with time, and this value walks around the unit circle at a rate of 1 unit per second. The formulas for each rotating vector are provided, and the constant vector is described as staying at the number 1, never moving.
In this section, the concept of controlling the initial size and direction of rotating vectors using complex numbers is explained in the context of expressing an arbitrary function as a sum of these terms. The easiest term to determine is the constant term, as it represents the center of mass of the drawing. The other terms can be calculated by multiplying the function by a complex number and shifting down the frequency term in the exponent. This allows for the determination of the desired term, such as c_2, by taking the average of the modified function, which cancels out all other terms except for the desired one.
In this section, the speaker explains how Fourier series can be used for drawing animations using rotating vectors. By computing the integral for a range of values for n, the coefficients (c_n) can be found for the path to approximate the original path as the number of vectors used approaches infinity. The speaker then goes on to relate this back to a simpler one-dimensional example of a step function and pairs off the vectors rotating in opposite directions to relate it to Fourier series. The speaker concludes that this foray into Fourier series is a glimpse at a deeper idea as exponential functions play an important role in differential equations.
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