MATH1012 - Lecture 25

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MATH1012 - Lecture 24
Skeleton Slides

Euler’s Formulas for Functions of Any Period

• In general, for a function where one cycle is defined over $[-L,L)$, and thus is $2L$-periodic, its Fourier series is: ${\text{FS}}_{f(t)}=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }{a}_n\cos (\frac{n\pi t}{L})+{\sum }_{n=1}^{\infty }{b}_n\sin (\frac{n\pi t}{L})$
  • i.e. $\pi$ is replaced b y $L$ (for most terms)
• Coefficients:
  $a_0=\frac{1}{L}{\int }_{-L}^{L}f(t)dt$
  $a_n=\frac{1}{L}{\int }_{-L}^{L}f(t)\cos (\frac{n\pi t}{L})dt$
  $b_n=\frac{1}{L}{\int }_{-L}^{L}f(t)\sin (\frac{n\pi t}{L})dt$

As $n$ increases, the Fourier series becomes a better approximation of the periodic function, i.e. the curve becomes smoother

• Important to note that $\frac{a_0}{2}$ is the average value of the function
  • You can use this to check if your calculations are correct
• If you identify the function as odd or even, see lecture 26

Convergence of the Fourier Series

• For a fixed value $t$, the value of ${\lim }_{n\to \infty }{\text{FS}}_{f(t)}$ should converge to $f(t)$
  • However, this is not always the case at points of discontinuity
• These points likely exist on bounds of a function defined over $(-L,L)$
  • i.e. where the original function does not exist
• At a point of discontinuity, the Fourier series converges to: $\frac{{\lim }_{t\to c^{-}}f(t)+{\lim }_{t\to c^{+}}f(t)}{2}$
  • i.e. the average between the limits from both sides

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MATH1012 - Lecture 26
Annotated Slides