Find the Fourier Sine transform of the function;𝑓𝑥=𝑘, 0<𝑥<𝑎0, 𝑥>𝑎
Question
Find the Fourier Sine transform of the function;𝑓𝑥=𝑘, 0<𝑥<𝑎0, 𝑥>𝑎
Solution
The Fourier Sine Transform of a function f(x) is given by the integral from 0 to infinity of f(x)sin(wx)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: one for 0 < x < a and one for x > a.
The Fourier Sine Transform F(w) is then given by:
F(w) = ∫ from 0 to a of ksin(wx)dx + ∫ from a to ∞ of 0sin(wx)dx
The second integral is zero because the integrand is zero. So we only need to compute the first integral.
∫ from 0 to a of k*sin(wx)dx = [-k/w * cos(wx)] from 0 to a
Evaluating this from 0 to a gives:
F(w) = -k/w * cos(wa) - (-k/w * cos(0)) = -k/w * cos(wa) + k/w
This is the Fourier Sine Transform of the given function.
Similar Questions
Find the Fourier Cosine transform of the function;𝑓𝑥=𝑘, 0<𝑥<𝑎0, 𝑥>𝑎
Find the Fourier Cosine transform 𝐹𝑐𝑒-𝑎𝑥 of f(x) = 𝑒-𝑎𝑥 where a>0
Find the Fourier transform off (t) ={ 1, |t| < 1;0, |t| > 1.Hence evaluate the integral ∫ ∞0 sin tt dt.Solution: F(ω) = 2 sin ωω ,∫ ∞0sin tt dt = π2 .2. Find the Fourier sine and cosine transform of f (t) = e−at, a > 0.Solution: Fs(ω) = ωω2 + a2 , Fc(ω) = aω2 + a2 .3. Find the Fourier transform off (t) =−(1 + t), −1 ≤ t ≤ 0;t − 1, 0 < t ≤ 1;0, |t| > 1.Solution: F(ω) = 2(cos ω − 1)ω2 .4. Find the inverse Fourier transform ofF(ω) = e−iω2(1 + iω) .Solution: f (t) = 12 e−(t−1) H(t − 1), where H is the Heaviside step function.5. Find the Fourier transform off (t) ={ cos t, −l ≤ t ≤ l;0, |t| > l.Solution: F(ω) = 2ω cos l sin ωl − sin l cos ωlω2 − 1 .
Find the Fourier transform off (t) ={ 1, |t| < 1;0, |t| > 1.Hence evaluate the integral ∫ ∞0 sin tt dt
Find the Fourier sine and cosine transform of f (t) = e−at, a > 0
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.