Exercise 10.2 Define the Fourier transform of an integrable function f by Of .y/ D R

R f .x/e????ix dx.

Show that if f .x; t/ is a soluton of the telegraph equation, then Of D g is a solution of the ordinary differential equation gtt C2gt C2g D 0. Assume as much smoothness and decay as you need.

The general solution of (10.13) is obtained by first finding the characteristic exponents, solutions of the algebraic equation r2 C2rC2 D 0; r D ????1

q 1????2 in case jj < 1 and with a corresponding formula if jj > 1.

The general solution of (10.13) for jj < 1 is written in terms of hyperbolic functions:

OPij./ D Aij./e????t cosht q

1????2 CBij./e????t sinh t q

1????2:

For jj > 1, the hyperbolic functions can be replaced by suitable trigonometric functions.

In this way, we can obtain the representation of the distribution function in terms of its Fourier transform.