The Fourier Transform the Inverted Polynomial 1/(1+t^2)

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On this page we seek the Fourier Transform for the inverted polynomial g(t):

inverted polynomial
[Equation 1]

The Fourier Transform is easily found, since we already know the Fourier Transform for the two sided decaying exponential. By using some simple properties, mainly the scaling property of the Fourier Transform, and the duality relationship among Fourier Transforms.

Hence, we can obtain (proof not shown, that's for you!) the result [G(f)]:

fourier trasnform of inverted polynomial
[Equation 2]

What I think is cool about this Fourier Transform, is that if we recall the following indefinite integral:

integral of arctangent function
[Equation 3]

We can compute the intergral from -infinity to +infinity using our Fourier Transform, and show that it matches up perfectly with the indefinite integral result:

connection of Fourier Transform result and arc-tangent integral
[Equation 4]

I like when everything works out, so I wanted to share that result in Equation [4] with you. Note that G(f) is the result from Equation [2].


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