GVj t for Displaced-Distorted Oscillator

With the contour-integral representation of the Hermite polynomial  

Continue the derivation starting from Eq. (4.60). Defining the new variables X] — —cojt and — co jt, the TCF Gav AVj(t) is further written as following  

consider the displaced oscillator case, i.e., oj = coj, flj — fij and — —Xj. By using the contour-integral representation of the Hermite polynomials (see Eq. (A.4)), 

By using the Mehler s formula in the sinusoidal-function form, 

The terms in the exponentials in the integrand can be simplified by observing that 

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