Find the Fourier sine and cosine transform of f (t) = e−at, a > 0

The Fourier sine and cosine transforms of a function f(t) are defined as follows:

Fourier Cosine Transform (FCT): F_c(ω) = ∫_0^∞ f(t) cos(ωt) dt Fourier Sine Transform (FST): F_s(ω) = ∫_0^∞ f(t) sin(ωt) dt

Given f(t) = e^(-at), a > 0, we can calculate the Fourier sine and cosine transforms as follows:

1. Fourier Cosine Transform (FCT):

F_c(ω) = ∫_0^∞ e^(-at) cos(ωt) dt

To solve this integral, we can use the method of integration by parts, where u = e^(-at) and dv = cos(ωt) dt. Then du = -a e^(-at) dt and v = (1/ω) sin(ωt).

Applying the formula for integration by parts (uv - ∫v du), we get:

F_c(ω) = [e^(-at) * (1/ω) sin(ωt)]_0^∞ - ∫_0^∞ (1/ω) sin(ωt) * -a e^(-at) dt

Solving this gives:

F_c(ω) = a / (a^2 + ω^2)

2. Fourier Sine Transform (FST):

F_s(ω) = ∫_0^∞ e^(-at) sin(ωt) dt

Again, we can use integration by parts, where u = e^(-at) and dv = sin(ωt) dt. Then du = -a e^(-at) dt and v = -(1/ω) cos(ωt).

Applying the formula for integration by parts, we get:

F_s(ω) = -[e^(-at) * (1/ω) cos(ωt)]_0^∞ + ∫_0^∞ (1/ω) cos(ωt) * a e^(-at) dt

Solving this gives:

F_s(ω) = ω / (a^2 + ω^2)

So, the Fourier cosine transform of f(t) = e^(-at) is F_c(ω) = a / (a^2 + ω^2) and the Fourier sine transform is F_s(ω) = ω / (a^2 + ω^2).