Question

Find the exponential Fourier transform of the given $\mathit{f}\left(x\right)$and write $\mathit{f}\left(x\right)$as a Fourier integral.

Short answer

Answer

The exponential Fourier transform of the given function is $g\left(\alpha \right)\frac{2}{i{\mathrm{\pi \alpha }}^2}\left(\alpha a-\sin \left(\alpha a\right)\right)$and $f\left(x\right)$as a Fourier integral is$f\left(x\right)=\underset{-\infty }{\overset{\infty }{\int }}\frac{\left(\alpha a-\sin \left(\alpha a\right)\right)}{i{\mathrm{\pi \alpha }}^2}{e}^{i\alpha x}d\alpha$.

Step-by-step solution

Given information

The graph of the given function is as shown below.

In mathematical form, the function can be written as shown below.

$f\left(x\right)=\left\{\begin{array}{l}-2\left(x+a\right),x\in \left[-a,0\right]\\ -2\left(x-a\right),x\in \left[0,a\right]\end{array}\right.$.

The exponential Fourier transform of the function is to be found and the function is to be written as a Fourier integral.

The significance of Fourier transforms

The following are the formulas for the Fourier series transforms.

Here,is called the Fourier transform of.

Find the Fourier transform

Use the formula $g\left(\alpha \right)=\frac{2}{2\mathrm{\pi }}\underset{-\infty }{\overset{\infty }{\int }}f\left(x\right)e^{-i\alpha x}dx$to find the Fourier transform.

Solve further to obtain.

Now, writeas a Fourier integral using the formula.