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 ๐น๐๐-๐๐ฅ of f(x) = ๐-๐๐ฅ where a>0
Question 2 Select one:
- 2๐๐๐ยฒ - ๐คยฒ
- 2๐๐๐ยฒ + ๐คยฒ
- 2๐ - ๐๐ยฒ + ๐คยฒ
- -2๐๐๐ยฒ + ๐คยฒ
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.
Similar Questions
The goal of this exercise is to determine the Fourier transform of the function๐(๐ฅ)={๐ฅifย โ1<๐ฅ<10otherwisef(x)={ x0โ ifย โ1<x<1otherwiseโ
Find the value of the real numberย ๐0ย of the Fourier series if f(x) = ๐ฅ2ย for 0 โค x โค 2๐.Question 1Select one:-8๐23-4๐234๐238๐23
Obtain the Fourier series of the function๏จ ๏ฉf x given as follows:๐(๐ฅ) = { ๐ฅ, 0 โค ๐ฅ โค ๐0, ๐ โค ๐ฅ โค 2๐ }
Find the Fourier Transform of the following functions and draw the spectrum:Part d - Max 400 words.u(t).CosaootSgn(t)
Find the Fourier transform off (t) ={ 1, |t| < 1;0, |t| > 1.Hence evaluate the integral โซ โ0 sin tt dt
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.