Engineering Mathematics Class

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Engineering Mathematics Class 14 Fourier
Series, Integrals, and Transforms (Part 2)

• Sheng-Fang Huang

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11.3 Even and Odd Functions. Half-Range
Expansions

• The g is even if g(x) g(x), so that its graph
  is symmetric with respect to the vertical axis.
• A function h is odd if h(x) h(x).
• The function is even, and its Fourier series has
  only cosine terms. The function is odd, and its
  Fourier series has only sine terms.

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Fig. 262. Even function
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Fourier Cosine Series
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Fourier Sine Series
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Example 1 Rectangular Pulse

• The function (x) in Fig. 264 is the sum of the
  function (x) in Example 1 of Sec 11.1 and the
  constant k. Hence, from that example and Theorem
  2 we conclude that

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Example 2 Half-Wave Rectifier

• The function u(t) in Example 3 of Sec. 11.2 has a
  Fourier cosine series plus a single term v(t)
  (E/2) sin ?t. We conclude from this and Theorem 2
  that u(t) v(t) must be an even function.

u(t) v(t) with E 1, ? 1
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Example 3 Sawtooth Wave

• Find the Fourier series of the function (x) x
  p if p lt x lt p and (x 2p) (x).

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• Solution.

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Half-Range Expansions

• Half-range expansions are Fourier series ( Fig.
  267).
• To represent (x) in Fig. 267a by a Fourier
  series, we could extend (x) as a function of
  period L and develop it into a Fourier series
  which in general contain both cosine and sine
  terms.

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Half-Range Expansions

• For our given we can calculate Fourier
  coefficients from (2) or from (4) in Theorem 1.
• This is the even periodic extension 1 of (Fig.
  267b). If choosing (4) instead, we get (3), the
  odd periodic extension 2 of (Fig. 267c).
• Half-range expansions is given only on half
  the range, half the interval of periodicity of
  length 2L.

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Fig. 267. (a) Function (x) given on an interval
0 x L
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Fig. 267. (b) Even extension to the full range
(interval) L x L
(heavy curve) and the periodic
extension of period 2L to the x-axis
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Fig. 267. (c) Odd extension to L x L (heavy
curve) and the periodic extension of period 2L to
the x-axis
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Example 4 Triangle and Its Half-Range
Expansions

• Find the two half-range expansions of the
  function (Fig. 268)

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• Solution. (a) Even periodic extension.

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• Solution. (b) Odd periodic extension.

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Fig. 269. Periodic extensions of (x) in Example
4
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11.4 Complex Fourier Series.

• Given the Fourier series
• can be written in complex form, which
  sometimes simplifies calculations. This complex
  form can be obtained by the basic Euler formula

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Complex Fourier Coefficients

• The cn are called the complex Fourier
  coefficients of (x).
• (6)
• For a function of period 2L our reasoning gives
  the complex Fourier series
• (7)

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Example 1 Complex Fourier Series

• Find the complex Fourier series of (x) ex if
  p lt x lt p and (x 2p) (x) and obtain from
  it the usual Fourier series.
• Solution.

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Example 1 Complex Fourier Series

• Solution.

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Fig. 270. Partial sum of (9), terms from n 0
to 50