Find the Fourier Cosine transform of the function;๐๐ฅ=๐,ย ย 0<๐ฅ<๐0,ย ย ๐ฅ>๐
Question
Find the Fourier Cosine transform of the function;๐๐ฅ=๐,ย ย 0<๐ฅ<๐0,ย ย ๐ฅ>๐
Solution
The Fourier Cosine transform of a function f(x) is given by the formula:
F(ฯ) = โ(2/ฯ) โซ from 0 to โ [f(x) cos(ฯx) dx]
Given the function f(x) = k for 0 < x < a and f(x) = 0 for x > a, we can split the integral into two parts:
F(ฯ) = โ(2/ฯ) [ โซ from 0 to a [k cos(ฯx) dx] + โซ from a to โ [0 cos(ฯx) dx] ]
The second integral is zero because the integrand is zero. So we only need to compute the first integral:
F(ฯ) = โ(2/ฯ) โซ from 0 to a [k cos(ฯx) dx]
This is a standard integral, and its solution is:
F(ฯ) = โ(2/ฯ) [ (k/ฯ) sin(ฯx) ] from 0 to a
Evaluating this at the limits gives:
F(ฯ) = โ(2/ฯ) [ (k/ฯ) sin(ฯa) - (k/ฯ) sin(0) ]
Since sin(0) = 0, this simplifies to:
F(ฯ) = โ(2/ฯ) (k/ฯ) sin(ฯa)
So the Fourier Cosine transform of the given function is F(ฯ) = โ(2/ฯ) (k/ฯ) sin(ฯa).
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