Q12E Find all Fourier coefficients of... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q12E (page 261)

Find all Fourier coefficients of the absolute value function
$f (t) = | t |$

Short Answer

Expert verified

The Fourier coefficients are

$a_{0} = \frac{蟿蟿}{\sqrt{2}} b_{k} = 0 c_{k} = \frac{- 4}{蟿蟿 k^{2}}, 0$

k is even.

Step by step solution

The Fourier coefficient of the function f(t)=|t|

Here it needs to find out the function $f (t) = |t|$ . Recall that the coefficients are given by

$a_{0} = f (t), \frac{1}{\sqrt{2}} b_{k} = <f (t), \sin (k t)> c_{k} = <f (t), \cos (k t)>$

The given function is $f (t) = |t|$ . So,

$a_{0} = |t|, \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2 蟿蟿}} {鈭�}_{- 蟿蟿}^{蟿蟿} |t| d t = \frac{2}{\sqrt{2 蟿蟿}} {鈭�}_{0}^{蟿蟿} t d t = \frac{蟿蟿}{2}$

Similarly, the value of $b_{k} = <f (t), \sin (k t)>$ is,

$b_{k} = <|t|, \sin (k t)> d t = \frac{1}{蟿蟿} |t|, \sin (k t) d t = 0$

And now for the last coefficient $c_{k}$ consider the solution,

$c_{k} = <|t|, \cos (k t)> = \frac{1}{蟿蟿} {鈭�}_{- 蟿蟿}^{蟿蟿} |t| \cos (k t) d t = \frac{2}{蟿蟿} {鈭�}_{0}^{蟿蟿} |t| \cos (k t) d t = \frac{2}{蟿蟿} {鈭�}_{0}^{蟿蟿} t \cos (k t) d t$

Further solve the above expression,

$c_{k} = \frac{2}{蟿蟿} {[\frac{1}{k} t \sin (k t)]}_{0}^{蟿蟿} - {鈭�}_{0}^{蟺} - \sin (k t) d t = \frac{2}{蟿蟿} . \frac{1}{k^{2}} {[\cos (k t)]}_{o}^{蟿蟿} = \frac{2}{蟿蟿 k^{2}} (\cos (k 蟿蟿) - 1) = \{\begin{cases} \frac{- 4}{蟿蟿 k^{2}} & k i s e v e n \\ 0 & k i s e v e n 0 \end{cases}$

Hence, the Fourier coefficients are $a_{0} = \frac{蟿蟿}{\sqrt{2}}, b_{k} = 0$ and $c_{k} = \frac{- 4}{蟿蟿 k^{2}}, 0$ .

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91影视

Find all Fourier coefficients of the absolute value function
$f (t) = | t |$

Short Answer

Step by step solution

The Fourier coefficient of the function f(t)=|t|

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91影视

Find all Fourier coefficients of the absolute value functionf(t)=|t|

Short Answer

Step by step solution

The Fourier coefficient of the function f(t)=|t|

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Mechanics Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Find all Fourier coefficients of the absolute value function
$f (t) = | t |$