## Related questions with answers

The subject of Fourier series is concerned with the representation of a

$2 \pi \text { -periodic }$

function f as the following infinite linear combination of the set of functions

$\{ 1 , \sin n x , \cos n x \} _ { n = 1 } ^ { \infty } : \\ f ( x ) = \frac { 1 } { 2 } a _ { 0 } + \sum _ { n = 1 } ^ { \infty } \left( a _ { n } \cos n x + b _ { n } \sin n x \right) . \quad ( 5.2 .5 )$

In this problem, we investigate the possibility of performing such a representation. (a) Use appropriate trigonometric identities, or some form of technology, to verify that the set of functions

$\{ 1 , \sin n x , \cos n x \} _ { n = 1 } ^ { \infty }$

is orthogonal on the interval

$[ - \pi , \pi ].$

(b) By multiplying (5.2.5) by cos mx and integrating over the interval

$[ - \pi , \pi ],$

show that

$a _ { 0 } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) d x \text { and } a _ { m } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \cos m x\ d x.$

(c) Use a similar procedure to show that

$b _ { m } = \frac { 1 } { \pi } \int _ { - \pi } ^ { \pi } f ( x ) \sin m x\ d x.$

It can be shown that if f is in

$C ^ { 1 } ( - \pi , \pi ),$

then Equation (5.2.5) holds for each

$x \in ( - \pi , \pi ).$

The series appearing on the right-hand side of (5.2.5) is called the Fourier series of f, and the constants in the summation are called the Fourier coefficients for f. (d) Show that the Fourier coefficients for the function

$f ( x ) = x , - \pi < x \leq \pi , f ( x + 2 \pi ) = f ( x ),$

are

$\begin{array} { l l } { a _ { n } = 0 , } & { n = 0,1,2 , \ldots } \\ { b _ { n } = - \frac { 2 } { n } \cos n \pi , } & { n = 1,2 , \ldots } \end{array}$

and thereby determine the Fourier series of f. (e) Using some form of technology, sketch the approximations to f(x) = x on the interval

$( - \pi , \pi )$

obtained by considering the first three terms, first five terms, and first ten terms in the Fourier series for f. What do you conclude?

Solution

Verifieda) We have the general results

$\begin{equation} \int \sin (m x) \sin (n x) \, dx=\frac{n \sin (m x) \cos (n x)-m \cos (m x) \sin (n x)}{m^2-n^2} \end{equation}$

$\begin{equation} \int \cos (m x) \cos (n x) \, dx=\frac{m \sin (m x) \cos (n x)-n \cos (m x) \sin (n x)}{m^2-n^2} \end{equation}$

$\begin{equation} \int \sin (m x) \sin (m x) \, dx=\frac{x}{2}-\frac{\sin (2 m x)}{4 m} \end{equation}$

$\begin{equation} \int \cos (m x) \cos (m x) \, dx=\frac{2 m x+\sin (2 m x)}{4 m} \end{equation}$

$\begin{equation} \int \sin (m x) \, dx=-\frac{\cos (m x)}{m} \end{equation}$

$\begin{equation} \int \cos (m x) \, dx=\frac{\sin (m x)}{m} \end{equation}$

$\begin{equation} \int \cos (m x) \sin (n x) \, dx=\frac{m \sin (m x) \sin (n x)+n \cos (m x) \cos (n x)}{m^2-n^2} \end{equation}$

Now from the above results we can obtain, on the interval $(-\pi,\pi)$ the results

$\begin{equation} \int_{-\pi }^{\pi } \sin (m x) \sin (m x) \, dx=\pi \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \cos (m x) \cos (m x) \, dx=\pi \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \sin (m x) \sin (n x) \, dx=0 \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \cos (m x) \cos (n x) \, dx=0 \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \sin (m x) \, dx=0 \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \cos (m x) \, dx=0 \end{equation}$

$\begin{equation} \int_{-\pi }^{\pi } \cos (m x) \sin (n x) \, dx=0 \end{equation}$

Then, the set $\{(1,\sin(nx),\cos(nx))\}_{n=1}^{\infty}$ is orthogonal on the interval $(-\pi,\pi)$.

## Create an account to view solutions

## Create an account to view solutions

## Recommended textbook solutions

#### Differential Equations and Linear Algebra

2nd Edition•ISBN: 9780131860612 (4 more)Beverly H. West, Hall, Jean Marie McDill, Jerry Farlow#### Differential Equations and Linear Algebra

4th Edition•ISBN: 9780134497181 (2 more)C. Henry Edwards, David Calvis, David E. Penney#### Linear Algebra and Differential Equations

1st Edition•ISBN: 9780201662122Gary Peterson, James Sochacki#### Differential Equations and Linear Algebra

4th Edition•ISBN: 9780321964670Scott A. Annin, Stephen W. Goode## More related questions

- prealgebra

1/4

- prealgebra

1/7