Q1: Show the Fourier Sine Transform of the given functions.
f(x)
fs (W) = 7 g(f)
- cos aw
1
si <
if 0<x<a
otherwise
W
2
1/VX
1/Vw
Answers
Answer:
The expression you obtained is incorrect. It is missing a factor of a in front of the cosine term.
Assume a>0. (Otherwise the integral does not converge.)
∫∞0cos(kx)e−axdx=Re∫∞0eikxe−axdx=Re∫∞0e(ik−a)xdx=Ree(ik−a)xik−a∣∣∞0
Since a>0, limx→∞e(ik−a)x is zero, and we are left with:
∫∞0cos(kx)e−axdx=Re−1ik−a=aa2+k2
As weird as it may sound for a question which is evidently purely-mathematical, one way to catch an error such as the missing a above is by dimensional analysis. Imagine that the units of x are length. Then for consistency, a and k must have units of inverse length. That means you can't add or subtract two terms like ksin(kx) and cos(kx), since they have different units. (One has units of inverse length, the other is dimensionless.) If you arrive at such an expression, then there must have been a mistake somewhere.