Sheng-Fang Huang

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.

Fig. 262. Even function Fig. 263. Odd function

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)

Fourier Sine Series THEOREM 1

The Fourier series of an odd function of period 2L is a “Fourier sine series”

(3)

with coefficients

(4)

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.

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

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

Example 3: Sawtooth WaveFind the Fourier series of the function ƒ(x)

= x + π if –π < x < π and ƒ(x + 2π) = ƒ(x).

Solution.

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.

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.

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Fig. 267. (a) Function ƒ(x) given on an interval 0 ≤ x ≤ L

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

Fig. 267. (c) Odd extension to –L ≤ x ≤ L (heavy curve) and the periodic extension of period 2L to the x-axis

Example 4: “Triangle” and Its Half-Range ExpansionsFind the two half-range expansions of the

function (Fig. 268)

Solution. (a) Even periodic extension.

Solution. (b) Odd periodic extension.

Fig. 269. Periodic extensions of ƒ(x) in Example 4

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

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)

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.

Example 1: Complex Fourier SeriesSolution.

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