Fourier Series

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Lecture 9

• Fourier Series
• Revision

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Lecture 9 Objectives

• Find the Fourier Series Expansion for a function
  that is periodic with period 2? (or for a
  function defined on the interval (??,?)).
• Revise the material on infinite series in a
  Question-Answer competition.

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Periodic Functions

• A function f(x) is called periodic (see picture)
  iff
• f(x) is defined for all real x, and
• for some p gt 0, f(x p) f(x) for all x.
  Graphically, the graph of f matches its left (or
  right) shift by p.
• p above is called a period of f(x).
• p is called the fundamental period of f(x) iff p
  is the smallest period of f(x).

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Examples

• sin x and cos x are periodic with fundamental
  period 2?, since e.g. sin( x 2?) sin x
• sin nx and cos nx are periodic with fundamental
  period 2?/n.
• f(x) 1 is periodic with no fundamental period.
• a0 a1cos x b1sin x a2cos 2x b2sin 2x is
  periodic with period 2?.

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Graphs of sin x and cos x
Note The graphs of sin(nx) and cos(nx) are just
horizontally compressed by a factor of n.
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Trigonometric Series

• These are infinite series of the form

Note If this series converges for all x between
?? and ?, then it converges for all real x, and
must be periodic of period 2?.
Question Can any periodic function of period 2?
be represented by a trigonometric series?
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Fourier Expansion Theorem for Periodic Functions

• Any continuous periodic function f(x) of period
  2? can be represented by the trigonometric series

This is also true for any piecewise continuous
periodic function f(x) of period 2?, such that at
any discontinuity c, f(c) f(c) f(c?)/2,
where f(c) and f(c?) are the right- and
left-hand limits of f at c.
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Finding the Fourier Expansion

• If a function f(x) can be represented by the
  trigonometric series

We can find all an and bn by multiplying both
sides by cos(mx) and sin(mx) and integrating
between 0 and 2?. We then get
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Finding the Fourier Coefficients

• For the Fourier series

the coefficients can be calculated from
Note The limits 0 to 2? can be replaced by ?? to
?.
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Example Find the Fourier series for
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Solution
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Solution
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Graphs of Partial Sums
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Example Find the Fourier series for the function
f(x) x , for ?? ? x ? ?.
Answer
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A nice Application

• The previous Fourier series represents the
  function x for ?? ? x ? ?, i.e.

Substituting x 0, we get
Thus, we get the formula
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Fourier Series and Music
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Lecture 9 Objectives (revisited)

• Find the Fourier Series Expansion for a function
  that is periodic with period 2? (or for a
  function defined on the interval (??,?)).
• Revise the material on infinite series in a
  Question-Answer competition.

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• Thank you for listening.
• Wafik