Chapter 11 Fourier Series Dennis G Zill Differential

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Chapter 11 Fourier Series Dennis G. Zill, Differential Equations with Boundary Value Problems, 9

Chapter 11 Fourier Series Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 2

Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 2

FIGURE 11. 2. 1 Piecewise-continuous function f in Example 1 Dennis G. Zill, Differential

FIGURE 11. 2. 1 Piecewise-continuous function f in Example 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 3

FIGURE 11. 2. 2 Piecewise-continuous derivative f’ in Example 2 Dennis G. Zill, Differential

FIGURE 11. 2. 2 Piecewise-continuous derivative f’ in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 4

FIGURE 11. 2. 3 Periodic extension of function f shown in Figure 11. 2.

FIGURE 11. 2. 3 Periodic extension of function f shown in Figure 11. 2. 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 5

FIGURE 11. 2. 4 Partial sums of Fourier series (13) in Example 1 Dennis

FIGURE 11. 2. 4 Partial sums of Fourier series (13) in Example 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 6

FIGURE 11. 3. 1 Even function; graph symmetric with respect to y-axis Dennis G.

FIGURE 11. 3. 1 Even function; graph symmetric with respect to y-axis Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 7

FIGURE 11. 3. 2 Odd function; graph symmetric with respect to origin Dennis G.

FIGURE 11. 3. 2 Odd function; graph symmetric with respect to origin Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 8

FIGURE 11. 3. 3 Odd function in Example 1 Dennis G. Zill, Differential Equations

FIGURE 11. 3. 3 Odd function in Example 1 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 9

FIGURE 11. 3. 4 Periodic extension of function shown in Figure 11. 3. 3

FIGURE 11. 3. 4 Periodic extension of function shown in Figure 11. 3. 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 10

FIGURE 11. 3. 5 Odd function in Example 2 Dennis G. Zill, Differential Equations

FIGURE 11. 3. 5 Odd function in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 11

FIGURE 11. 3. 6 Partial sums of sine series (7) Dennis G. Zill, Differential

FIGURE 11. 3. 6 Partial sums of sine series (7) Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 12

FIGURE 11. 3. 7 Even reflection Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 11. 3. 7 Even reflection Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 13

FIGURE 11. 3. 8 Odd reflection Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 11. 3. 8 Odd reflection Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 14

FIGURE 11. 3. 9 Identity reflection Dennis G. Zill, Differential Equations with Boundary Value

FIGURE 11. 3. 9 Identity reflection Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 15

FIGURE 11. 3. 10 Function f in Example 3 is neither odd nor even.

FIGURE 11. 3. 10 Function f in Example 3 is neither odd nor even. Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 16

FIGURE 11. 3. 11 Same function on (0, L) but different periodic extensions Dennis

FIGURE 11. 3. 11 Same function on (0, L) but different periodic extensions Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 17

FIGURE 11. 3. 12 Periodic forcing function for spring/mass system in Example 4 Dennis

FIGURE 11. 3. 12 Periodic forcing function for spring/mass system in Example 4 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 18

FIGURE 11. 3. 13 Graph for Problem 39 Dennis G. Zill, Differential Equations with

FIGURE 11. 3. 13 Graph for Problem 39 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 19

FIGURE 11. 3. 14 Graph for Problem 40 Dennis G. Zill, Differential Equations with

FIGURE 11. 3. 14 Graph for Problem 40 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 20

FIGURE 11. 3. 15 Graph for Problem 41 Dennis G. Zill, Differential Equations with

FIGURE 11. 3. 15 Graph for Problem 41 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 21

FIGURE 11. 3. 16 Graph for Problem 42 Dennis G. Zill, Differential Equations with

FIGURE 11. 3. 16 Graph for Problem 42 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 22

FIGURE 11. 3. 17 Graph for Problem 50 Dennis G. Zill, Differential Equations with

FIGURE 11. 3. 17 Graph for Problem 50 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 23

FIGURE 11. 4. 1 Positive roots x 1, x 2, x 3, . .

FIGURE 11. 4. 1 Positive roots x 1, x 2, x 3, . . . of tan x = −x in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 24

FIGURE 11. 5. 1 Graphs of two partial sums of the Fourier-Bessel series in

FIGURE 11. 5. 1 Graphs of two partial sums of the Fourier-Bessel series in Example 2 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 25

FIGURE 11. 5. 2 Partial sum S 5(x) of the Fourier-Legendre series in Example

FIGURE 11. 5. 2 Partial sum S 5(x) of the Fourier-Legendre series in Example 3 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 26

FIGURE 11. R. 1 Graph for Problem 18 Dennis G. Zill, Differential Equations with

FIGURE 11. R. 1 Graph for Problem 18 Dennis G. Zill, Differential Equations with Boundary Value Problems, 9 e, © 2018 27