sheng-fang huang. 11.3 even and odd functions. half-range expansions the g is even if g(–x) =...
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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 Fig. 263. Odd function
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Fourier Cosine Series THEOREM 1
The Fourier series of an even function of period 2L is a “Fourier cosine series”
(1)
with coefficients (note: integration from 0 to L only!)
(2)
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Fourier Sine Series THEOREM 1
The Fourier series of an odd function of period 2L is a “Fourier sine series”
(3)
with coefficients
(4)
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Sum and Scalar Multiple
THEOREM 2
The Fourier coefficients of a sum ƒ1 + ƒ2 are the sums of the corresponding Fourier coefficients of ƒ1 and ƒ2.
The Fourier coefficients of cƒ are c times the corresponding Fourier coefficients of ƒ.
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Example 1: Rectangular PulseThe function ƒ*(x) in Fig. 264 is the sum of
the function ƒ(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 RectifierThe function u(t) in Example 3 of Sec.
11.2 has a Fourier cosine series plus a single term v(t) = (E/2) sin ωt. We conclude from this and Theorem 2 that u(t) – v(t) must be an even function.
u(t) – v(t) with E = 1, ω = 1
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Example 3: Sawtooth WaveFind the Fourier series of the function ƒ(x)
= x + π if –π < x < π and ƒ(x + 2π) = ƒ(x).
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Solution.
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Half-Range ExpansionsHalf-range expansions are Fourier series (
Fig. 267). To represent ƒ(x) in Fig. 267a by a Fourier
series, we could extend ƒ(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 ExpansionsFor our given ƒ we can calculate Fourier
coefficients from (2) or from (4) in Theorem 1. This is the even periodic extension ƒ1 of ƒ
(Fig. 267b). If choosing (4) instead, we get (3), the odd periodic extension ƒ2 of ƒ (Fig. 267c).
Half-range expansions: ƒ is given only on half the range, half the interval of periodicity of length 2L.
493
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Fig. 267. (a) Function ƒ(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 ExpansionsFind 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 ƒ(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 ƒ(x).
(6)
For a function of period 2L our reasoning gives the complex Fourier series
(7)
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Example 1: Complex Fourier SeriesFind the complex Fourier series of ƒ(x) =
ex if –π < x < π and ƒ(x + 2π) = ƒ(x) and obtain from it the usual Fourier series.
Solution.
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Example 1: Complex Fourier SeriesSolution.
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Fig. 270. Partial sum of (9), terms from n = 0 to 50