Calculus Small Changes, Big Ideas

Section 10.7 CAPSTONE: Fourier series

• Motivation: Instead of an infinite sum of power functions, represent a periodic function as an infinite sum of sinusoidal functions (music/sound connection)

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• Formula for a general Fourier series:

  \begin{equation*} a_{0}+\sum_{k=1}^{\infty} \big(a_{k}\cos kx+b_{k}\sin kx\big) \end{equation*}

  Euler-Fourier formulas for coefficients:

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  \begin{align*} a_0 \amp = \frac{1}{2\pii}\int_{-\pii}^{\pii}f(x)\dd{x} \\ a_k \amp = \frac{1}{\pii} \int_{-\pii}^{\pii}f(x)\cos(kx)\dd{x} \\ b_k \amp = \frac{1}{\pii} \int_{-\pii}^{\pii}f(x)\sin(kx)\dd{x} \end{align*}
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• This all works because:

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  \begin{align*} \int_{-\pii}^{\pii}\sin(kx)\cos(nx)\dd{x} \amp = 0 \quad \text{for all }n,k \\ \int_{-\pii}^{\pii}\cos(kx)\cos(nx)\dd{x} \amp = \pii\delta_{nk} \end{align*}
  Can use trig integrals to show this!

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• Fourier Convergence Theorem (wording from Smith/Minton):

  Theorem 8. Fourier Convergence Theorem.

  Suppose that \(f\) is periodic of period \(2l\) and that \(f\) and \(f'\) are continuous on the interval \([-l,l]\text{,}\) expect for at most a finite number of jump discontinuities. Then, \(f\) has a convergent Fourier series expansion. Further, the series converges to \(f(x)\) when \(f\) is continuous at \(x\) and to

  \begin{equation*} \frac{1}{2}\left(\limit{t\to x^+}f(t)+\limit{t\to x^-}f(t)\right) \end{equation*}

  at any points \(x\) where \(f\) is discontinuous.

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  If we introduce my notation from the improper integrals chapter, we can rewrite this last expression as

  \begin{equation*} \frac{f(x^{+})+f(x^{-})}{2}\text{.} \end{equation*}

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• Square wave:

  \begin{equation*} \frac{4}{\pii}\qty(\sin x+\frac{1}{3}\sin 3x+\frac{1}{5}\sin 5x+\frac{1}{7}\sin 7x+\cdots) \end{equation*}

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• Sawtooth wave

  \begin{equation*} \frac{2}{\pii}\qty(\sin x-\frac{1}{2}\sin 2x+\frac{1}{3}\sin 3x-\frac{1}{4}\sin 4x+\cdots) \end{equation*}

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• Triangle wave

  \begin{equation*} \frac{4}{\pii}\qty(\sin x-\frac{1}{9}\sin 3x+\frac{1}{25}\sin 5x-\frac{1}{49}\sin 7x+\cdots) \end{equation*}

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• Up-down:

  \begin{equation*} \frac{4}{\pii}\qty(\cos x+\cos 3x+\cos 5x+\cos 7x+\cdots) \end{equation*}

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• Ramp:

  \begin{equation*} \frac{\pii}{2}-\frac{\pii}{4}\qty(\cos x+\frac{1}{9}\cos 3x+\frac{1}{25}\cos 5x+\frac{1}{49}\cos 7x+\cdots) \end{equation*}

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• Square pulse:

  \begin{equation*} \frac{h}{2\pii}+\sum_{k=1}^{\infty}\frac{\sin kh}{\pii k}+\sum_{k=1}^{\infty}\frac{1-\cos kh}{\pii k} \end{equation*}

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