Matrix continued fractions susceptibility

We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation... 

Equations (210) and (211) constitute the exact solution of our problem formulated in terms of matrix continued fractions. Having determined the Laplace transform Y(co) and noting that CY(lco) = f (i )/fj (0), one may calculate the susceptibility xT( ) from Eq. (201). 

Exact Solution for the Complex Susceptibility Using Matrix Continued Fractions Approximate Expressions for the Complex Susceptibility Numerical Results and Comparison with Experimental Data Conclusions... 

C. Exact Solution for the Complex Susceptibility Using Matrix Continued Fractions... 

The complex susceptibility x( ) yielded by Eq. (9), combined with Eq. (22) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms combined with the matrix continued fraction solution for the fixed center of oscillation cosine potential model treated in detail in Ref. 25. Thus we shall merely outline that solution as far as it is needed here and refer the reader to Ref. 25 for the various matrix manipulations, and so on. On considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential, we have... 

Figure 2. Imaginary part of the complex susceptibility X (co) versus normalized frequency rjco for various values of the reaction field parameter. SoUd lines correspond to the matrix continued fraction solution, Eqs. (37) and (51) circles correspond to the smaU oscillation solution, Eq. (26) dashed lines correspond to the approximate small oscillation solution Eq. (53) and dotted lines correspond to the solution based on the dynamic mobility, Eq. (58).

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