Causality and the response function, the Kramer?s-Kronig relations are consistent with the principle that an effect not precedes its cause. Consider a delta function force applied at time t = 0: whence F_w = 1/2?.?

(a) Show by direct integration or by use of the KK relations that the oscillator response function gives zero displacement, x(t) = 0, for t

(b) Evaluate x(t) for t > 0. Note that a(w) has poles at ? (w²_0?? ? p²)^1/2 ? ?iP, both in the lower half-plane.

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F(t) = 8(t) = 2/1/² e -Ion dw a(w) = (w²w² - iwp)-1