  Passive microrheology typically deduces shear elastic loss and storage moduli
from displacement time series or mean-squared displacement (MSD) of thermally
fluctuating probe particles in equilibrium materials. Common data analysis
methods use either Kramers-Kronig (KK) transformations or functional fitting to
calculate frequency-dependent loss and storage moduli. We propose a new
analysis method for passive microrheology that avoids the limitations of both
of these approaches. In this method, we determine both real and imaginary
components of the complex, frequency-dependent response function $\chi(\omega)
= \chi^{\prime}(\omega)+i\chi^{\prime\prime}(\omega)$ as direct integral
transforms of the MSD of thermal particle motion. This procedure significantly
improves the high-frequency fidelity of $\chi(\omega)$ relative to the use of
KK transformation, which has been shown to lead to artifacts in
$\chi^{\prime}(\omega)$. We test our method on both model data and experimental
data. Experiments were performed on solutions of worm-like micelles and dilute
collagen solutions. While the present method agrees well with established
KK-based methods at low frequencies, we demonstrate significant improvement at
high frequencies using our symmetric analysis method, up to almost the
fundamental Nyquist limit.
