Figure 4.1: Complex integration contour used to establish the Kronig-Kramers relations ty 2: We h To establish this property we need the Riemann-Lebesgue lemma[142, which states that if f(r) is absolutely integrable on the interval (a, b) where a and b are finite or infinite constants, then lim f(t)e- o dt=0 From this we see that o(r, te ja dt=0 ime(r,o)-∈0=0 To establish the Kronig-Kramers relations we examine the integral where r is the contour shown in Figure 4.L. Since the points $2=0, o are excluded, the integrand is analytic everywhere within and on T, hence the integral vanishes by the Cauchy-Goursat theorem. By Property 2 we have e(r,92)一∈0 2001 by CRC Press LLCFigure 4.1: Complex integration contour used to establish the Kronig–Kramers relations. Property 2: We have lim ω→±∞ ˜ c (r,ω) − 0 = 0. To establish this property we need the Riemann–Lebesgue lemma [142], which states that if f (t) is absolutely integrable on the interval (a, b) where a and b are finite or infinite constants, then lim ω→±∞ b a f (t)e− jωt dt = 0. From this we see that lim ω→±∞ σ(˜ r,ω) jω = lim ω→±∞ 1 jω ∞ 0 σ(r, t  )e− jωt dt = 0, lim ω→±∞ 0χe(r,ω) = lim ω→±∞ 0 ∞ 0 χe(r, t  )e− jωt dt = 0, and thus lim ω→±∞ ˜ c (r,ω) − 0 = 0. To establish the Kronig–Kramers relations we examine the integral   ˜ c(r, ) − 0  − ω d where  is the contour shown in Figure 4.l. Since the points  = 0, ω are excluded, the integrand is analytic everywhere within and on , hence the integral vanishes by the Cauchy–Goursat theorem. By Property 2 we have lim R→∞ C∞ ˜ c(r, ) − 0  − ω d = 0,