Find the Fourier Cosine transform ๐น๐๐-๐๐ฅย of f(x) = ๐-๐๐ฅ where a>0Question 2Select one:-2๐๐๐2-๐ค22๐๐๐2+๐ค22๐-๐๐2+๐ค2-2๐๐๐2+๐ค2
Question
Find the Fourier Cosine transform ๐น๐๐-๐๐ฅย of f(x) = ๐-๐๐ฅ where a>0Question 2Select one:-2๐๐๐2-๐ค22๐๐๐2+๐ค22๐-๐๐2+๐ค2-2๐๐๐2+๐ค2
Solution
The Fourier Cosine transform of a function f(x) is given by the formula:
F_c(w) = โ(2/ฯ) โซ from 0 to โ [f(x) cos(wx) dx]
We want to find the Fourier Cosine transform of f(x) = e^-ax. Substituting this into the formula gives:
F_c(w) = โ(2/ฯ) โซ from 0 to โ [e^-ax cos(wx) dx]
This is a standard integral that can be solved using integration by parts or by looking up in a table of integrals. The result is:
F_c(w) = โ(2/ฯ) * [a / (a^2 + w^2)]
So, the Fourier Cosine transform of f(x) = e^-ax is F_c(w) = โ(2/ฯ) * [a / (a^2 + w^2)].
Therefore, the correct answer is 2๐๐/๐^2+๐ค^2.
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Find the Fourier sine and cosine transform of f (t) = eโat, a > 0
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