F(ω) 1√ 2π

f (x) e−iωx dx. (17.1)

Theorem 9.7 (Fourier inversion theorem) in Section 9.4 states that if f (x) is absolutely integrable on R and is piecewise smooth on every finite interval, then

f (x) .= 1√ 2π

F(ω) eiωxdω, (17.2)

in the sense that at every x0,

f (x+0 ) + f (x−0 ) 2

= 1√ 2π

F(ω) eiωx0dω. (17.3)

The Fourier transform and its inversion theorem are very useful for solving some physical problems, as we will see in Section 17.2. A reference for a rigorous proof of the Fourier inversion theorem was given in “Learn More About It” found at the end of Section 9.4. Just as the symbol L stands for the Laplace transform operator and L−1 for the inverse

Laplace transform operator, we denote the Fourier transform operator and its inverse operator by

F[ f (t)] = F(ω) 1√ 2π

f (t) e−iωtdt (17.4)

F−1[F(ω)] 1√ 2π

F(ω) eiωtdt. (17.5)

As we mentioned in Section 9.4, we caution that there are many slightly different definitions of the Fourier transform and corresponding inverse transform in current use. In this section, we concentrate on Fourier transforms and inverse transforms for which

complex variable integration methods are particularly needed or useful.