Question

How do I calculate the Fourier transform of $\dfrac{a}{{2\pi }}\dfrac{1}{{{a^2} + {x^2}}}$?

Answer 1

The given equation is $\dfrac{a}{{2\pi }}\dfrac{1}{{{a^2} + {x^2}}}$
We use the Fourier transform. Compute this definite integral, consider the following complex variable function over a domain.
We use the integral function and contour. And then we use the residue theorem.
Finally we get the result.

Complete step by step answer:
The given equation is $\dfrac{a}{{2\pi }}\dfrac{1}{{{a^2} + {x^2}}}$
The$\dfrac{a}{{2\pi }}$ weight of the function or the$\dfrac{1}{{\sqrt {2\pi } }}$weight of the transform and just calculate
$\Rightarrow {I_x} = \int\limits_{ - \infty }^\infty {\dfrac{{{e^{i\omega x}}}}{{{a^2} + {x^2}}}dx}$, where$x \in \mathbb{R}$
In order to compute this define integral, consider the following complex variable function over a domain$\mathbb{C}$ :
$\Rightarrow f(z) = \dfrac{{{e^{i\omega z}}}}{{{a^2} + {z^2}}}$
And it’s associated contour integral:
$\Rightarrow \oint\limits_C {f(z)dz}$
Where $C$ is the following semi-circular contour in the complex plane with radius $R > 0$.
We will restrict $R$ to enclose the poles in the upper quadrants once we have analyzed the poles of $f(z)$. The denominator of the integrand is${a^2} + {z^2}$, and so we have simples poles when${a^2} + {z^2} = 0 \Rightarrow z = \pm ai$
We need only be concerned with the poles in ${Q_1}$ and ${Q_2}$, that (providing we make $R$ large enough) lie within our contour $C$, that is the pole $z = ai$ is positive.
So the:
$\Rightarrow \oint\limits_C {f(z)dz} = \int_{ - R}^R {f(x)dx} + \int\limits_{\gamma R} {f(z)dz}$
As $C$ encloses a single pole:
$\alpha = ai$ (Assuming $a > 0$)
Then by the residue theorem:
\Rightarrow \oint\limits_C {f(z)dz = 2\pi i} \times \left( {{\text{sum of the residues of the poles of }}f(z){\text{ within }}C} \right) = 2\pi i\left\\{ {re{s_{z = \alpha }}f(z)} \right.
And we can calculate the residues as follows:
$\Rightarrow re{s_{z = \alpha }} = \mathop {\lim }\limits_{z \to \alpha } (z - \alpha )f(Z)$
Now we substitute$\alpha$ and $f(z)$ in the residue equation, hence we get
$\Rightarrow \mathop {\lim }\limits_{z \to ai} (z - ai)\left( {\dfrac{{{e^{i\omega z}}}}{{{a^2} + {z^2}}}} \right)$
We expand the$({a^2} + {z^2})$, hence we get
$\Rightarrow \mathop {\lim }\limits_{z \to ai} (z - ai)\left( {\dfrac{{{e^{i\omega z}}}}{{(z - ai)(z + ai)}}} \right)$
Now we cancel the common term, hence we get
$\Rightarrow \mathop {\lim }\limits_{z \to ai} \left( {\dfrac{{{e^{i\omega z}}}}{{(z + ai)}}} \right)$
As $z \to ai$Substitute the expression, hence we get
$\Rightarrow \left( {\dfrac{{{e^{i\omega ai}}}}{{(ai + ai)}}} \right)$
Add the denominator, hence we get
$\Rightarrow \left( {\dfrac{{{e^{i\omega ai}}}}{{2ai}}} \right)$
$i \times i = {i^2} = - 1$
We apply the $i \times i = {i^2} = - 1$ expression in the exponent, hence we get
$\Rightarrow \left( {\dfrac{{{e^{ - \omega a}}}}{{2ai}}} \right)$
Thus:
$\Rightarrow \oint\limits_C {f(z)dz} = 2\pi i\dfrac{{{e^{ - \omega a}}}}{{2ai}}$
Now we cancel the common term, hence we get
$\Rightarrow \oint\limits_C {f(z)dz} = \dfrac{{\pi {e^{ - \omega a}}}}{a}$
Now, as we let $R \to \infty$ we get,
$\Rightarrow \oint\limits_C {f(z)dz} = \int_{ - \infty }^\infty {f(x)dx} + \int\limits_{\gamma R} {f(z)dz}$
As is often the case with contour integrals, we find that
$\Rightarrow \mathop {\lim }\limits_{R \to \infty } \int\limits_{\gamma R} {f(z)dz = 0}$
So we will end up with the result:
$\Rightarrow \int_{ - \infty }^\infty {\dfrac{{{e^{i\omega x}}}}{{{a^2} + {x^2}}}} = \dfrac{{\pi {e^{ - \omega a}}}}{a}$

Note:
The Fourier transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent. In the Fourier domain image, each point represents a particular frequency contained in the spatial domain image.
The Fourier transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.