EXERCISES 10.3
10.3.1. 
Show that the Fourier transform is a linear operator; that is, show that

(a) $FT[c_1 f (x) + c_2 g(x)] = c_1 FT[f] + c_2 FT[g]$

(b) $FT[f(x) g(x)] \ne FT[f(x)] FT[g(x)]$

10.3.2. 
Show that the inverse Fourier transform is a linear operator; that is, show
that

(a) $FT^{1}[c_1FT[f] + c_2FT[g]] = c_1f(x) + c_2g(x)$

(b) $FT^{-1}[F(w)G(w)] \ne f(x)g(x)$

10.3.3. 
Let $F(w)$ be the Fourier transform of $f (x)$. Show that if $f (x)$ is
real, then $F^*(w) = F(-w)$, where $*$ denotes the complex conjugate.

10.3.4. 
Show that $FT\left[\int f(x;\alpha)d\apha\right] = \int F(w,\alpha)d\alpha$.

10.3.5. 
If $F(w)$ is the Fourier transform of $f (x)$, show that the inverse Fourier
transform of $e^{iw\beta} F(w)$ is $f (x - \beta)$. This result is known
as the {\bf shift theorem} for Fourier transforms.

*10.3.6. 
If $f(x)=\left\{\begin{array}{ll} 0 & |x|>a, \\ 1 & |x|<a,\end{array}\right.$
then determine the Fourier transform of 
$f (x). 
[The answer is given in the table of Fourier transforms in H10, Section 4.4.]

*10.3.7. 
If $F(w) = e^{-|w|\alpha}$, $\alpha>0$, then determine the inverse
Fourier transform of $F(w)$.
[The answer is given in the table of Fourier transforms in H10, Section 4.4.]

10.3.8. 
If $F(w)$ is the Fourier transform of $f (x)$, show that $-i\,dF/dw$ is
the Fourier transform of $xf (x)$.

10.3.9. 
(a) Multiply (10.3.6) (assuming that $\gamma = 1$) by $e^{-iwx}$ and
integrate from $-L$ to $L$ to show that
$$\int_{-L}^{L} F(w)e^{-iwx}dw = \frac{1}{2\pi} \int_{-\infty}^\infty 
f(u)\frac{2\sin(L(u-x))}{u-x}du.\mbox{~~(10.3.13)}
$$
(b) Derive (10.3.7). For simplicity, assume that $f (x)$ is continuous.
[Hints:
Let $f (u) = f (x) + f (u) - f (x)$. Use the sine integral, $\int_0^\infty \frac{\sin s}{s}ds=\frac{\pi}{2}$
Integrate (10.3.13) by parts and then take the limit as $L\to\infty$.

*10.3.10
 Removed due to Bessel functions.

10.3.11. 
(a) If $f (x)$ is a function with unit area, $\int_{-\infty}^\infty
f(x)dx=1$, show that the scaled and stretched
function $(1/\alpha) f (x/\alpha)4 also has unit area.

(b) If $F(w)$ is the Fourier transform of $f (x)$, show that $F(\alpha w)$
is the Fourier transform of $(\alpha)f(x/\alpha)$.

(c) Show that part (b) implies that broadly spread functions have sharply
peaked Fourier transforms near $w = 0$, and vice versa.

10.3.12. 
Removed due to complex integration.

10.3.13. 
Evaluate $\int_0^\infty e^{-kw^2t}\cos(wx)dw$ in the following way.
Determine $\partial I/\partial x$, and then integrate by parts.

10.3.14.
The gamma function r(x) is defined as follows:
$$\Gamma(x) = \int_0^\infty t^{x-1}e^{-t}dt.$$
Show that
(a)$\Gamma(1)=1$ & (b) $\Gamma(x+1)=\Gamma(x)$ \\
(c) $\Gamma(n + 1) = n!$ &  (d) $\Gamma(1/2) = 2\int_0^\infty e^{-t^2}dt = \sqrt{\pi}$ \\
(e) What is $\Gamma(3/2)$?

10.3.15. 
(a) Using the definition of the gamma function in the previous Exercise,
show that
$$\Gamma(x) = 2\int_0^\infty u^{2x-1}e^{-u^2}du.$$
(b) Using double integrals in polar coordinates, show that
$$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z).$$
[Hint: It is known from complex variables that
$2\int_0^{\pi/2} (\tan\theta)^{2x-1}d\theta = \frac{\pi}{\sin(\pi z)}$.]

*10.3.16. 
Evaluate
$\int_0^\infty y^pe^{-ky^n}dy$
in terms of the gamma function (see Exercise 10.3.14).

10.3.17. 
Removed due to complex variable theory.

10.3.18.
Removed due to complexity and Dirac unit impulse function.