Chapter 7: Q6P (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 $\frac{1}{2} + \frac{2 i}{π} (\sum_{n = - \infty}^{\infty} \frac{e^{i n x}}{n}) n = 2 + 4 k, k \in Z$ .

Step by step solution

Given data

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

Concept of Fourier series

An infinite sum of sines and cosines is used to represent the expansion of a periodic function f(x) into a Fourier series.

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

Find the coefficients

The $c_{0}$ coefficient is, $c_{0} = \frac{1}{2}$ .

The other coefficients are as follows:

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

Where, $n = 2 + 4 k, k \in Z$ .

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

Check and simplify the expression

For check, change $n \to - n$ .

$f (x) = \frac{1}{2} - \frac{2 i}{π} [\sum_{n = - \infty}^{- 1} \frac{e^{i n x}}{n} + \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n}] f (x) = \frac{1}{2} - \frac{2 i}{π} [\sum_{n = 1}^{\infty} \frac{e^{i n x} - e^{- i n x}}{n}] f (x) = \frac{1}{2} + \frac{4}{π} [\sum_{n = 1}^{\infty} \frac{\sin (n x)}{n}]$

Therefore, the expansion is $\frac{1}{2} + \frac{2 i}{π} (\sum_{n = - \infty}^{\infty} \frac{e^{i n x}}{n}) n = 2 + 4 k, k \in Z$ .

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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

Check and simplify the expression

One App. One Place for Learning.

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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

Check and simplify the expression

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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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.