Fourier Sine Theorem

Spectral representations like Eq.(4.77) yield pairs of functions which are transforms of each other. Let $f(x)$ be an integrable function $f(x)$ defined on the real interval $0\le x <\infty$ . Multiply Eq.(4.77) by $f(\xi)$ and integrate over the half line $0\le \xi<\infty$ . The result is

$$f(x)=\sqrt{\frac{2}{\pi}} \int_0^\infty F(k)\sin kx~dk ~~,$$

where

$$F(k)=\sqrt{\frac{2}{\pi}} \int_0^\infty f(\xi)\sin k\xi~d\xi$$

These two function are the Fourier sine transforms of each other.

Exercise 411.1 (VERY LONG STRING: STEADY STATE RESPONSE)
Over the interval $-\infty<x<\infty$ consider

$${d^2G\over dx^2}+\lambda G = -\delta (x-\xi)$$

$${d^2 u\over dx^2}+\lambda u = -f(x)~~{\rm and}~~{d^2\varphi\over dx^2}+\lambda\phi=0.$$

We are looking for solutions in ${\cal{L}}^2(-\infty, \infty)$ and assume that $f$ is in ${\cal{L}}^2(-\infty, \infty)$ .

(a)

Show that there are two candidates for $G$ , namely

$$G=G^{out}(x\vert \xi;\lambda) = {i\over 2\sqrt{\lambda}} \exp(-i\sqrt{\lambda} \xi)\exp(i\sqrt{\lambda}x)~~~\xi<x$$

$$= {i\over 2\sqrt{\lambda}} \exp(-i\sqrt{\lambda} x)\exp(i\sqrt{\lambda}\xi)~~~x<\xi$$

$$= {i\over 2\sqrt{\lambda}}\exp(i\sqrt{\lambda}\vert x-\xi\vert).$$

and

$$G=G^{in}(x\vert \xi;\lambda) = {-i\over 2\sqrt{\lambda}}\exp(-i\sqrt{\lambda}\vert x-\xi\vert).$$

(b)

Given the fact that $\sqrt{\lambda}=\alpha+i\beta$ with $\beta>0$ , point out why only one of them is square-integrable.

(c)

Consider the contour integral $\oint G(x\vert \xi;\lambda)\, d\lambda$ over a large circle of radius $R$ . Demonstrate that

$$\lim_{R\rightarrow\infty} \frac{1}{2\pi i} \oint G(x\vert \xi;\lambda)\, d\lambda=-\delta(x-\xi)~.$$

(d)

Next deform the contour until it fits snugly around the branch cut of $\sqrt{\lambda}$ , and show that

$$\delta(x-\xi)=\int^{\infty}_0\cdots~~ d\lambda \qquad~~\qquad~~(\ast)$$

and then show that $(\ast)$ can be rewritten as

$$\delta(x-\xi)={1\over 2\pi}\int^{\infty}_{-\infty}e^{i\omega(x-\xi)}d\omega \qquad \textrm{for}~x<\xi ~~\textrm{and}~~\xi<x~.$$

(d)

Express $u(x)$ as a Fourier integral in terms of $f$ .

(e)

Express $G(x\vert \xi;\lambda)$ in the same way, i.e. obtain the bilinear expansion for $G$ .

Exercise 411.2 (STEADY STATE RESPONSE VIA FOURIER)
Again consider

$${d^2G\over dx^2}+\lambda G = -\delta (x-\xi)$$

$${d^2 u\over dx^2}+\lambda u = -f(x)~~{\rm and}~~{d^2\varphi\over dx^2}+\lambda\phi=0.$$

over the interval $-\infty<x<\infty$ , but leave the boundary conditions as-yet-unspecfied.

(a)

Express $u(x)$ as a Fourier integral in terms of $f$ .

(b)

Express $G(x\vert \xi;\lambda)$ in the same way, i.e. obtain the bilinear expansion for $G$ .

(c)

How, do you think, should one incorporate boundary conditions into these expressions?

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