Question

Prove that if $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{\sum }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{i}\mathbf{n}\mathbf{x}}$,then the average value of${\left|f\left(x\right)\right|}^{\mathbf{2}}$is${\mathbf{\sum }}_{\mathbf{-}\mathbf{\infty }}^{\mathbf{\infty }}{\mathit{c}}_{\mathbf{n}}{\mathit{c}}_{\mathbf{-}\mathbf{n}}$.Show by problem7.12 that for real f(x)this becomes (11.5).

Short answer

For a given function $f\left(x\right)={\sum }_{-\infty }^{\infty }{c}_n{e}^{inx}$, the average value of${\left|f\left(x\right)\right|}^2$ is proven to be${\sum }_{-\infty }^{\infty }{c}_n{c}_{-n}$

Step-by-step solution

Definition of Fourier series and Kronecker delta

When sine and cosine swells are added together, a periodic function is represented by a Fourier series. Each surge in the sum, or peak, has a frequency that is an integer multiple of the abecedarian frequency of the periodic function. Harmonic analysis can be used to determine each harmonious phase and width.

The Kronecker delta in mathematics is a function of two variables, which are typically merely non-negative integers. If the variables are equal, the function is 1, otherwise it is 0, otherwise it can be used with Iverson classes, in which case the Kronecker deltais a piecewise function of the variables i and j.

Given Parameters

Given the function$f\left(x\right)={\sum }_{-\infty }^{\infty }{c}_n{e}^{inx}$

It is to be proven that the average value of${\left|f\left(x\right)\right|}^2$ is${\sum }_{-\infty }^{\infty }{c}_n{c}_{-n}$ using technique used in problem 7.12 that is use of Kronecker delta${\delta }_{ij}$

Use concept of average value of function squared and Kronecker delta to solve and get required answer.

It is known that the average value of the square of given function can be calculated as

Now replace the n in the second term to -m and get

…… (1)

Now, first solve the integral from equation (1)

$\underset{-\pi }{\overset{\pi }{\int }}{e}^{ix(n-m)}dx=\frac{e^{i\pi (n-m)}-e^{-i\pi (n-m)}}{i(n-m)}$ …… (2)

It is known that$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$

Compare, substitute the above value in equation (2) and get

$$\begin{gathered} \underset{-\pi }{\overset{\pi }{\int }}{e}^{ix(n-m)}dx=\frac{2}{n-m}\sin \left(\pi \right(n-m\left)\right) \\ \approx 2\pi {\delta }_{nm} \end{gathered}$$

where${\delta }_{nm}$ is the Kronecker delta which is zero until n=m

Substitute the above value in equation (1)

Write$\sum _{m=-\infty }^{\infty }{c}_{-m}{\delta }_{nm}$ as$c_{-n}$

Thus, …… (3)

Step 4:Substitute c-n as role="math" localid="1664296553204" cn¯ in equation (3) and get

Therefore, the average value of${\left|f\left(x\right)\right|}^2$ is${\sum }_{-\infty }^{\infty }{c}_n{c}_{-n}$ using technique used in problem 7.12 that is use of Kronecker delta${\delta }_{ij}$