Q2Advanced Engineering Mathematics I

Question

Obtain Fourier transform of $f (x) = {x^{2} 0 for ∣ x ∣ \leq a for ∣ x ∣ > a$ . Hence evaluate $\int_{0}^{\infty} cos (\frac{a s}{2}) [\frac{( a ^{2} s ^{2} - 2 ) s i n a s + 2 a s c o s a s}{s ^{3}}] d s$ .

Answer 1

Fourier transform: $F (s) = \int_{- a}^{a} x^{2} e^{- i s x} d x$ .

Since $x^{2}$ is even, $F (s) = 2 \int_{0}^{a} x^{2} cos (s x) d x$ .

Using integration by parts twice: $F (s) = 2 [x^{2} \frac{s i n s x}{s} + 2 x \frac{c o s s x}{s ^{2}} - 2 \frac{s i n s x}{s ^{3}}]_{0}^{a}$

$= 2 [\frac{( a ^{2} s ^{2} - 2 ) s i n a s + 2 a s c o s a s}{s ^{3}}]$ .

To evaluate the given integral, we use the Inverse Fourier Transform definition at a specific value of $x$ (like $x = a /2$ ), which corresponds to the $cos (a s /2)$ term in the integral.

Q2Advanced Engineering Mathematics I

Question

Answer