Deck 11: Orthogonal Functions and Fourier Series

Question 1

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$ is

A) $\lambda = ( n - 1 / 2 ) \pi , y = \cos ( ( n - 1 / 2 ) \pi x ) + \sin ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$
B) $\lambda = ( n - 1 / 2 ) \pi , y = \cos ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$
C) $\lambda = ( n - 1 / 2 ) \pi , y = \sin ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$
D) $\lambda = ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } , y = \cos ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$
E) $\lambda = ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } , y = \sin ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$

Answer 1

$\lambda = ( n - 1 / 2 ) ^ { 2 } \pi ^ { 2 } , y = \sin ( ( n - 1 / 2 ) \pi x ) , n = 1,2,3 , \ldots$

Question 2

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges at $x$ , to $( f ( x + ) + f ( x - ) ) / 2$ which of the following conditions need to be satisfied? Select all that apply.

A) $f$ is continuous on $[ a , b ]$
B) $f ^ { \prime }$ is continuous on $[ a , b ]$
C) $f$ is piecewise continuous on $[ a , b ]$
D) $f ^ { \prime }$ is piecewise continuous on $[ a , b ]$
E) $f$ is integrable on $[ a , b ]$

Answer 2

$f$ is piecewise continuous on $[ a , b ]$
$f ^ { \prime }$ is piecewise continuous on $[ a , b ]$
$f$ is integrable on $[ a , b ]$

Question 3

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

A) $p$ , $q$ , $r$ piecewise continuous on $[ a , b ]$
B) $r ( x ) > 0$ and $p ( x ) > 0$ on $[ a , b ]$
C) $r ( x ) < 0$ and $p ( x ) > 0$ on $[ a , b ]$
D) $A _ { 1 } B _ { 1 } \neq 0$
E) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$

Answer 3

$r ( x ) > 0$ and $p ( x ) > 0$ on $[ a , b ]$
$A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$

Question 4

The square norm of the function $f ( x ) = \cos ( 3 x )$ on the interval $[ 0 , \pi / 2 ]$ is

A) 1
B) $\pi$
C) $\pi / 2$
D) $\pi / 4$
E) 0

Question 5

The Fourier series of an even function might Select all that apply.

A) contain sine terms
B) contain cosine terms
C) contain a constant term
D) contain sine and cosine terms
E) contain sine, cosine, and constant terms

Question 6

The square norm of the function $f ( x ) = x ^ { 2 }$ on the interval $[ 0,1 ]$ is

A) 1/2
B) 1/3
C) 1/5
D) 1
E) 0

Question 7

The differential equation $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ is

A) Legendre's equation
B) Bessel's equation
C) the Fourier-Bessel
D) the hypergeometric
E) none of the above

Question 8

The Fourier series of the function $f ( x ) = \left\{ \begin{aligned}x & \text { if } x < 0 \\2 - x & \text { if } x > 0\end{aligned} \right\}$ on $[ - 2,2 ]$ are Select all that apply.

A) contains only cosine terms
B) contains only sine terms
C) contains sine and cosine terms
D) contains a constant term
E) contains sine, cosine, and constant terms

Question 9

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( \pi ) = 0$ is

A) $\lambda = n ^ { 2 } , y = \cos ( n x ) , n = 0,1,2 , \ldots$
B) $\lambda = n ^ { 2 } , y = \sin ( n x ) , n = 1,2,3 , \ldots$
C) $\lambda = n , y = \cos ( n x ) , n = 0,1,2 , \ldots$
D) $\lambda = n , y = \sin ( n x ) , n = 1,2,3 , \ldots$
E) $\lambda = n , y = \cos ( n x ) + \sin ( n x ) , n = 1,2,3 , \ldots$

Question 10

The Fourier coeficients of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ are Select all that apply.

A) $a _ { 0 } = 2 / 3$
B) $a _ { n } = 0$
C) $a _ { n } = 4 ( - 1 ) ^ { n } / \left( n ^ { 2 } \pi ^ { 2 } \right)$
D) $b _ { n } = 0$
E) $b _ { n } = 4 ( - 1 ) ^ { n } / \left( n ^ { 2 } \pi ^ { 2 } \right)$

Question 11

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is not a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

A) $r = 1 / ( x - a )$ on $( a , b )$
B) $q ( x ) = 0$ on $[ a , b ]$
C) $p ( x ) = x - a$ on $[ a , b ]$
D) $A _ { 1 } A _ { 2 } = 0$
E) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$

Question 12

The solution of the eigenvalue problem $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$ where $y$ is bounded on $[ - 1,1 ]$ , is

A) $\lambda = n , y = P _ { n } ( x ) , n = 1,2,3 ,$
B) $\lambda = n - 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$
C) $\lambda = n + 1 , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$
D) $\lambda = n ^ { 2 } , y = P _ { n } ( x ) , n = 1,2,3 , .$
E) $\lambda = n ( n + 1 ) , y = P _ { n } ( x ) , n = 1,2,3 , \ldots$

Question 13

Which of the following differential equations are in self-adjoint form? Select all that apply.

A) $r ( x ) y ^ { \prime \prime } + r ^ { \prime } ( x ) y ^ { \prime } + \lambda y = 0$
B) $y ^ { \prime \prime } + y ^ { \prime } + \lambda y = 0$
C) $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$
D) $y ^ { \prime \prime } + \lambda y = 0$
E) $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0$

Question 14

The Fourier Series of a function $f ( x ) = x$ defined on $[ - 1,1 ]$ is $f ( x ) = a _ { 0 } / 2 + \sum _ { n - 1 } ^ { \infty } \left( a _ { n } \cos ( n \pi x ) + b _ { n } \sin ( n \pi x ) \right)$ where Select all that apply.

A) $a _ { 0 } = 1$
B) $a _ { n } = 0$
C) $a _ { n } = \int _ { - 1 } ^ { 1 } x \sin ( n \pi x ) d x$
D) $b _ { n } = 0$
E) $b _ { n } = \int _ { - 1 } ^ { 1 } x \sin ( n \pi x ) d x$

Question 15

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 + 5 x & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 1$ to

A) 7
B) 1
C) 1/2
D) $- 3$
E) unknown

Question 16

Using the eigenfunctions of the previous problem, the Fourier-Legendre series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } P _ { n } ( x )$ , where

A) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$
B) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x$
C) $c _ { n } = ( 2 n + 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$
D) $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } f ( x ) P _ { n } ( x ) d x / 2$
E) $c _ { n } = ( 2 n - 1 ) \int _ { - 1 } ^ { 1 } x f ( x ) P _ { n } ( x ) d x$

Question 17

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 - x & \text { if } x > 0\end{array} \right\}$ is Select all that apply.

A) odd
B) even
C) neither even nor odd
D) continuous on $[ - \pi , \pi ]$
E) discontinuous on $[ - \pi , \pi ]$

Question 18

An example of a regular Sturm-Liouville problem is $y ^ { \prime \prime } + \lambda y = 0$ with boundary conditions Select all that apply.

A) $y ( 0 ) + y ^ { \prime } ( 1 ) = 0 , y ( 1 ) = 0$
B) $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$
C) $y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$
D) $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$
E) $y$ is bounded on $[ - 1,1 ]$

Question 19

The function $f ( x ) = \left\{ \begin{array} { c c } x & \text { if } x < 0 \\2 + 5 x & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to

A) 0
B) 1
C) 1/2
D) 2
E) unknown

Question 20

The Fourier series of the function $f ( x ) = x ^ { 2 }$ on $[ - 1,1 ]$ is

A) $\sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right) + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
B) $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
C) $\sum _ { n = 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
D) $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \sin ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$
E) $1 / 3 + \sum _ { n - 1 } ^ { \infty } 4 ( - 1 ) ^ { n } \cos ( n \pi x ) / \left( n ^ { 2 } \pi ^ { 2 } \right)$

Question 21

The Fourier Series of a function $f$ defined on $[ - p , p ]$ is $f ( x ) = a _ { 0 } / 2 + \sum _ { n - 1 } ^ { \infty } \left( a _ { n } \cos ( n \pi x / p ) + b _ { n } \sin ( n \pi x / p ) \right)$ where Select all that apply.

A) $a _ { 0 } = \int _ { -p } ^ { p } f ( x ) d x / p$
B) $a _ { n } = \int _ { -p} ^ { p } f ( x ) \cos ( n \pi x / p ) d x / p$
C) $a _ { n } = \int _ { - p } ^ { p } f ( x ) \sin ( n \pi x / p ) d x / p$
D) $b _ { n } = \int _ { - p } ^ { p } f ( x ) \cos ( n \pi x / p ) d x / p$
E) $b _ { n } = \int _ { - p } ^ { p } f ( x ) \sin ( n \pi x / p ) d x / p$

Question 22

Using the eigenfunctions of the previous problem, written as $g _ { n } ( x )$ , the Fourier-Bessel series for the function $f ( x )$ is $\sum _ { n = 1 } ^ { \infty } c _ { n } g _ { n } ( x )$ , where

A) $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$
B) $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$
C) $c _ { n } = \int _ { 0 } ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$
D) $c _ { n } = \int _ { 0 } ^ { 2 } x f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x g _ { n } ^ { 2 } ( x ) d x$
E) $c _ { n } = \int _ { 0 } ^ { 2 } x ^ { 2 } f ( x ) g _ { n } ( x ) d x / \int _ { 0 } ^ { 2 } x ^ { 2 } g _ { n } ^ { 2 } ( x ) d x$

Question 23

The solution of the eigenvalue problem $r R ^ { \prime \prime } + R ^ { \prime } + r \lambda R = 0 , R ( 0 )$ is bounded, $R ^ { \prime } ( 3 ) = 0$ is $\left( J _ { 0 } ^ { t } \left( z _ { n } \right) = 0 \right)$

A) $\lambda = z _ { n } ^ { 2 } , R = J _ { 0 } \left( z _ { n } r \right) , n = 1,2,3 , \ldots$
B) $\lambda = z _ { n } ^ { 2 } / 9 , R = J _ { 0 } \left( z _ { n } r / 3 \right) , n = 1,2,3 , \ldots$
C) $\lambda = z _ { n } , R = J _ { 0 } \left( z _ { n } r \right) , n = 1,2,3 , \ldots$
D) $\lambda = z _ { n } / 3 , R = J _ { 0 } \left( z _ { n } r / 3 \right) , n = 1,2,3 , \ldots$
E) $\lambda = z _ { n } , R = J _ { 0 } ( r ) , n = 1,2,3 , \ldots$

Question 24

The square norm of the function $f ( x ) = 1 - x$ on the interval $[ 0,2 ]$ is

A) 2/3
B) $\sqrt { 2 / 3 }$
C) 1/3
D) $\sqrt { 1 / 3 }$
E) 0

Question 25

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ( 0 ) = 0 , y ( 1 ) = 0$ is

A) $\lambda = n \pi , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$
B) $\lambda = n \pi , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \cos ( n \pi x ) , n = 0,1,2 , \ldots$
D) $\lambda = n ^ { 2 } \pi ^ { 2 } , y = \sin ( n \pi x ) , n = 1,2,3 , \ldots$
E) $\lambda = n \pi , y = \cos ( n \pi x ) + \sin ( n \pi x ) , n = 1,2,3 , \ldots$

Question 26

Consider the differential equation $y ^ { \prime \prime } + \lambda y = 0$ . Examples of boundary conditions for this equation that make a regular Sturm-Liouville problem are Select all that apply.

A) $y ( 0 ) = 0 , y ( 1 ) = 0$
B) $y ( 0 ) = 0 , y ^ { \prime } ( 1 ) = 0$
C) $y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 0$
D) $y ^ { \prime } ( 0 ) = 0 , y ( 1 ) + y ^ { \prime } ( 1 ) = 0$
E) $y ( 0 ) + y ( 1 ) = 0 , y ^ { \prime } ( 1 ) = 0$

Question 27

The Fourier series of the function $f ( x ) = x$ on $[ - 1,1 ]$ is

A) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 2 \cos ( n \pi x ) / ( n \pi )$
B) $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \cos ( n \pi x ) / ( n \pi )$
C) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n } 2 \sin ( n \pi x ) / ( n \pi )$
D) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } 2 \sin ( n \pi x ) / ( n \pi )$
E) $\sum _ { n - 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \sin ( n \pi x ) / ( n \pi )$

Question 28

The square norm of the function $f ( x ) = \sin x$ on the interval $[ 0 , \pi ]$ is

A) 1
B) $\pi$
C) $\pi / 2$
D) $\pi / 4$
E) 0

Question 29

In order to be assured by a theorem that the Fourier Series of $f$ on $[ a , b ]$ converges to $f$ , which of the following conditions need to be satisfied? Select all that apply.

A) $f$ is continuous on $$[ a , b ]$$
B) $f ^ { \prime }$ is continuous on $[ a , b ]$
C) $f$ is piecewise continuous on $[ a , b ]$
D) $f ^ { \prime }$ is piecewise continuous on $[ a , b ]$
E) $f$ is integrable on $[ a , b ]$

Question 30

The function $f ( x ) = | x |$ is Select all that apply.

A) odd
B) even
C) neither even nor odd
D) continuous on $[ - \pi , \pi ]$
E) discontinuous on $[ - \pi , \pi ]$

Question 31

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under certain conditions, including Select all that apply.

A) $p$ , $q$ , $r$ are continuous on $[ a , b ]$
B) $r ( x ) > 0$ and $p ( x ) < 0$ on $[ a , b ]$
C) $r ( x ) < 0$ and $p ( x ) > 0$ on $[ a , b ]$
D) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } \neq 0$
E) $A _ { 1 } A _ { 2 } \neq 0$

Question 32

The solution of the eigenvalue problem $y ^ { \prime \prime } + \lambda y = 0 , y ^ { \prime } ( 0 ) = 0 , y ^ { \prime } ( \pi ) = 0$ is

A) $\lambda = n , y = \cos ( n x ) , n = 0,1,2 , \ldots$
B) $\lambda = n , y = \sin ( n x ) , n = 1,2,3 , \ldots$
C) $\lambda = n ^ { 2 } , y = \cos ( n x ) , n = 0,1,2 , \ldots$
D) $\lambda = n ^ { 2 } , y = \sin ( n x ) , n = 1,2,3 , \ldots$
E) $\lambda = n , y = \cos ( n x ) + \sin ( n x ) , n = 1,2,3 , \ldots$

Question 33

Which of the following differential equations are in self-adjoint form? Select all that apply.

A) $r ^ { \prime } ( x ) y ^ { \prime \prime } + r ( x ) y ^ { \prime } + \lambda y = 0$
B) $y ^ { \prime \prime } + y ^ { \prime } + \lambda y = 0$
C) $( 1 - x ) y ^ { \prime \prime } - 2 x y ^ { \prime } + \lambda y = 0$
D) $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0$
E) $y ^ { \prime \prime } + \lambda y = 0$

Question 34

The problem $\frac { d } { d x } \left[ r ( x ) y ^ { \prime } \right] + ( q ( x ) + \lambda p ( x ) ) y = 0 , A _ { 1 } y ( a ) + B _ { 1 } y ^ { \prime } ( a ) = 0 , A _ { 2 } y ( b ) + B _ { 2 } y ^ { \prime } ( b ) = 0$ is a regular Sturm-Liouville problem under which of the following conditions. Select all that apply.

A) $r = 1 / ( x - a )$ are continuous on $[ a , b ]$
B) $p ( x ) = x - a$ on $[ a , b ]$
C) $q ( x ) = 0$ on $[ a , b ]$
D) $A _ { 1 } A _ { 2 } = 0$
E) $A _ { 1 } ^ { 2 } + B _ { 1 } ^ { 2 } = 0$

Question 35

The function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 1$ to

A) 0
B) 1
C) 1/2
D) $- 1$
E) unknown

Question 36

The Fourier series of an odd function might Select all that apply.

A) contain sine terms
B) contain cosine terms
C) contain a constant term
D) contain sine and cosine terms
E) contain sine, cosine, and constant terms

Question 37

The Fourier series of the function $f ( x ) = | x |$ on $[ - 2,2 ]$ is Select all that apply.

A) contains cosine terms
B) contains sine terms
C) contains sine and cosine terms
D) contains a constant term
E) contains sine, cosine, and constant terms

Question 38

The function $f ( x ) = \left\{ \begin{array} { l l } 0 & \text { if } x < 0 \\1 & \text { if } x > 0\end{array} \right\}$ has a Fourier series on $[ - 2,2 ]$ that converges at $x = 0$ to

A) 0
B) 1
C) 1/2
D) $- 1$
E) unknown

Question 39

Consider the parameterized Bessel's differential equation $x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( \alpha ^ { 2 } x ^ { 2 } - n ^ { 2 } \right) y = 0$ along with the conditions $y ( 0 )$ is bounded, $y ( 2 ) = 0$ . The solution of this eigenvalue problem is $\left( J _ { n } \left( z _ { n } \right) = 0 \right)$

A) $\alpha = z _ { n } / 2 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$
B) $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( z _ { n } x / 2 \right) , n = 1,2,3 , \ldots$
C) $\alpha = z _ { n } , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$
D) $\alpha = z _ { n } / 2 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$
E) $\alpha = z _ { n } ^ { 2 } / 4 , y = J _ { n } \left( \sqrt { z _ { n } / 2 } x \right) , n = 1,2,3 , \ldots$

Question 40

The Fourier coeficients of the function $f ( x ) = x$ on $[ - 1,1 ]$ are

A) $a _ { 0 } = - 2 / \pi , b _ { n } = 0 , a _ { n } = 0$
B) $a _ { 0 } = 0 , b _ { n } = ( - 1 ) ^ { n } 2 / ( n \pi ) , a _ { n } = 0$
C) $a _ { 0 } = 0 , b _ { n } = ( - 1 ) ^ { n + 1 } 2 / ( n \pi ) , a _ { n } = 0$
D) $a _ { 0 } = 0 , b _ { n } = 0 , a _ { n } = ( - 1 ) ^ { n } 2 / ( n \pi )$
E) $a _ { 0 } = 0 , b _ { n } = 0 , a _ { n } = ( - 1 ) ^ { n + 1 } 2 / ( n \pi )$