Chapter 7: Q7 7P (page 360)

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- π, π)$ and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Expert verified

The resultant expansion is $\frac{π}{4} + \frac{1}{2} (\sum_{n = - \infty}^{\infty} \frac{{(- 1)}^{n} - 1}{π n^{2}} + \frac{i}{n} {(- i)}^{n}) e^{i n x} n \neq 0$ .

Step by step solution

Given data

The given function is complex exponentials $e^{i n x}$ .

Concept of Fourier series

In terms of an infinite sum of sines and cosines, a Fourier series is an expansion of a periodic function.

The orthogonality relationships of the sine and cosine functions are used in Fourier series.

Find the coefficients

The $c_{0}$ coefficient is shown below.

$c_{0} = \frac{1}{2 π} \int_{0}^{π} x d x c_{0} = \frac{π}{4}$

The other coefficients are as follows:

$c_{n} = \frac{1}{2 π} \int_{0}^{π} x e^{- i n x} d x a c_{n} = \frac{1}{2 π} {[- \frac{x}{i n} e^{- i n x} + \frac{1}{n^{2}} e^{- i n x}]}_{- π}^{π} c_{n} = \frac{1}{2 π} (- \frac{π}{i n} e^{i n π} + \frac{1}{n^{2}} (e^{- i n π} -)) c_{n} = \frac{1}{2 π n^{2}} ({(- 1)}^{n} + 1) + \frac{i}{2 n} (- i)^{n}$

Then the function is $\frac{π}{4} + \frac{1}{2} (\sum_{n = - \infty}^{\infty} \frac{{(- 1)}^{n} - 1}{π n^{2}} + \frac{i}{n} {(- i)}^{n}) e^{i n x} n \neq 0$ .

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Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- π, π)$ and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Find the coefficients

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Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials einx on the interval(-π,π) and verify in each case that the answer is equivalent to the one found in Section 5.

Short Answer

Step by step solution

Given data

Concept of Fourier series

Find the coefficients

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Most popular questions from this chapter

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Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials $e^{i n x}$ on the interval $(- π, π)$ and verify in each case that the answer is equivalent to the one found in Section 5.