${\displaystyle G_{WW}(\tau )/\tau =\rho (\tau )=-{{\tau _{S}} \over {\pi \tau ^{2}}}\sum \limits _{k=0}^{\infty }{{{(-1)^{k}} \over {k!}}\sin(\pi \beta k)\Gamma (\beta k+1)\left({\tau \over {\tau _{S}}}\right)}^{\beta k+1}}$

Calculated with Mathematica using:

${\displaystyle l[t\_]:=-{{\rm {tww}} \over {\rm {Pi*t}}}{\rm {*NSum[}}{{{\rm {(-1)}}^{\rm {k}}} \over {\rm {k!}}}{\rm {Sin[Pi*beta*k]*Gamma[beta*k+1]*}}\left({{\rm {t}} \over {\rm {tww}}}\right)^{\rm {beta*k+1}},\,}$

${\displaystyle {\rm {\{k,0,5000\},WorkingPrecision->50,NSumTerms->5000]}}\,}$

Based on:

• Lindsey, C. P. and Patterson, G. D. (1980). "Detailed comparison of the Williams-Watts and Cole-Davidson functions". Journal of Chemical Physics 73: 3338-3357.

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