Chapter 7: Q26P (page 385)

Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem.

Short Answer

Expert verified

By using the problem 15 and Fourier series cosine transform given function can be proved.

Step by step solution

Definition of Fourier series

The for Fourier series formula gives an expansion of a periodic function f (x) in terms of an horizonless sum of sines and cosines. It's used to putrefy any periodic function or periodic signal into the sum of a group of straightforward oscillating functions, videlicet sines and cosines.

Step 2:Given parameters

The given functions are

$\int_{0}^{\infty} \frac{1 - \cos α}{α^{2}} d α = \frac{π}{2}$

There need to prove the given functions using problem 15.

Use problem 15

The problem 15 says that

$\begin{matrix} f (x) = \frac{4}{π} \int_{0}^{\infty} \frac{1 - \cos (α x)}{α^{2}} \cos (α x) d α \\ = \{\begin{cases} \begin{array}{l} 2 (a - x) & , x \in [0, a] \end{array} \\ \begin{array}{l} 2 (x + a) & , x \in [- a, 0] \end{array} \end{cases} \end{matrix}$

Set the values of and used in problem

Set the values that is $a = 1$ and $x = 0$ This gives:

$\begin{matrix} f (0) = 2 \\ = \frac{4}{π} \int_{0}^{\infty} \frac{1 - \cos α}{α^{2}} d α \\ \frac{π}{2} = \int_{0}^{\infty} \frac{1 - \cos α}{α^{2}} d α \end{matrix}$

Thus, using $x = 0$ and $a = 1$ in the problem 15 the given functions are proved.

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Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem.

Short Answer

Step by step solution

Definition of Fourier series

Step 2:Given parameters

Use problem 15

Set the values of and used in problem

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