Q13P Repeat the example using the sam... [FREE SOLUTION]

Chapter 7: Q13P (page 358)

Repeat the example using the same Fourier series but at $x = 蟺 / 2$ .

Short Answer

Expert verified

At, $x = \frac{蟺}{2}$ :

$f (\frac{1}{2}) = (\begin{matrix} \frac{1}{4} + \frac{1}{蟺} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{蟺} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{蟺}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

Step by step solution

Given

The given function is $f (x) = \{\begin{cases} 0, - 蟺 < x < 0 \\ \sin x, 0 < x < 蟺 \end{cases}$

The given point is $x = \frac{蟺}{2}$ .

Definition of Fourier series

The Fourier series for the function $f (x)$ :

$f (x) = \frac{a_{0}}{2} + {鈭�}_{n - 1}^{鈭�} (a_{n} \cos nx + b_{n} sinx) a_{0} = \frac{1}{蟺} {鈭�}_{- x}^{x} f (x) d x a_{n} = \frac{1}{蟺} {鈭�}_{- x}^{x} f (x) c o s n x d x b_{n} = \frac{1}{蟺} {鈭�}_{- x}^{x} f (x) s i n n x d x$

If $f (x)$ is an even function:

$b_{n} = 0 f (x) = \frac{a_{0}}{2} + {鈭�}_{n - 1}^{鈭�} a_{n} c o s n x$

If $f (x)$ is an odd function:

$a_{0} = a_{n} = 0 f (x) = {鈭�}_{n - 1}^{鈭�} b_{n} s i n n x$

Write the Fourier series of the function and input the value

The Fourier series of function is shown below.

$f (x) - (\begin{matrix} \frac{1}{4} + \frac{1}{蟺} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{蟺} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{蟺}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

Now, at $x - \frac{蟺}{2}$ .

role="math" localid="1659238212014" $f (\frac{1}{2}) - (\begin{matrix} \frac{1}{4} + \frac{1}{蟺} (\frac{1}{1} \cos 3 x + \frac{1}{5} \cos 5 x - \frac{1}{7} \cos 7 x + . . .) \\ + \frac{1}{蟺} (\frac{1}{1} \sin x - \frac{2}{2} \sin 2 x + \frac{1}{3} \sin 3 x + \frac{1}{5} \sin 5 x + \frac{2}{6} \sin 6 x + \frac{1}{7} \sin 7 x + . . .) \end{matrix}) \frac{蟺}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + . . .$

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Repeat the example using the same Fourier series but at $x = 蟺 / 2$ .

Short Answer

Step by step solution

Given

Definition of Fourier series

Write the Fourier series of the function and input the value

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91影视

Repeat the example using the same Fourier series but at x=蟺/2.

Short Answer

Step by step solution

Given

Definition of Fourier series

Write the Fourier series of the function and input the value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Repeat the example using the same Fourier series but at $x = 蟺 / 2$ .