Page 460 - Advanced engineering mathematics

440 CHAPTER 13 Fourier Series

Compare these coefficients in the orthogonal projection with the Fourier coefficients of f on
[−L, L]. First
L
f · C 0 −L f (x)dx
=
L
C 0 · C 0 dx
−L
1 L 1
= f (x)dx = a 0 .
2L −L 2
Next,
L
f · C n −L f (x)cos(nπx/L)dx
=
L
2
C n · C n cos (nπx/L)dx
−L
1 L
= f (x)cos(nπx/L)dx = a n ,
L −L
and similarly,
1 L
f · S n
= f (x)sin(nπx/L)dx = b n .
S n · S n L −L
Thus, the orthogonal projection f S of f onto S is exactly the Nth partial sum of the Fourier series
of f on [−L, L].
This broader perspective of Fourier series will provide a unifying theme when we consider
general eigenfunction expansions in Chapter 15.

SECTION 13.2 PROBLEMS

In each of Problems 1 through 12, write the Fourier series 10. f (x) = cos(x/2) − sin(x),−π ≤ x ≤ π
of the function on the interval and determine the sum of
11. f (x) = cos(x),−3 ≤ x ≤ 3
the Fourier series. Graph some partial sums of the series,
compared with the graph of the function.
1 − x for −1 ≤ x ≤ 0
12. f (x) =
1. f (x) = 4,−3 ≤ x ≤ 3 0 for 0 < x ≤ 1
2. f (x) =−x,−1 ≤ x ≤ 1
In each of Problems 13 through 19, use the convergence
3. f (x) = cosh(πx),−1 ≤ x ≤ 1 theorem to determine the sum of the Fourier series of the
4. f (x) = 1 −|x|,−2 ≤ x ≤ 2 function on the interval. It is not necessary to write the
series to do this.

−4for −π ≤ x ≤ 0
5. f (x) = ⎧
4 for 0 < x ≤ π ⎪ 2x for −3 ≤ x < −2
⎪
⎨
6. f (x) = sin(2x),−π ≤ x ≤ π 13. f (x) = 0 for −2 ≤ x < 1
⎪ 2
⎩x for 1 ≤ x ≤ 3
⎪
2
7. f (x) = x − x + 3,−2 ≤ x ≤ 2

−x for −5 ≤ x < 0 2x − 2for −π ≤ x ≤ 1
8. f (x) = 14. f (x) =
1 + x 2 for 0 ≤ x ≤ 5 3 for 1 < x ≤−π

1for −π ≤ x < 0 x 2 for −π ≤ x ≤ 0
9. f (x) = 15. f (x) =
2for 0 ≤ x ≤ π 2 for 0 < x ≤ π
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October 14, 2010 14:57 THM/NEIL Page-440 27410_13_ch13_p425-464