Cole-Cole decomposition conductivities

complex

After Tarasov and Titov, 2013:

\hat{\sigma}(\omega) &= \sigma_\infty \left(1 - \sum_k\frac{m_k}{1 + (j \omega \tau_k)^c}\right)\\

real and imaginary parts

&= \sigma_\infty \left( 1 - \sum_k m_k \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2} - j (\omega \tau)^c \sin \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} \right)\\ &= \sigma_\infty \left( 1 - \sum_k m_k \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} \right) + j \sigma_\infty \sum_k m_k \frac{(\omega \tau)^c \sin \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}}\\ \Rightarrow \sigma' &= \sigma_\infty \left( 1 - \sum_k m_k \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} \right)\\ \sigma'' &= j \sigma_\infty \sum_k m_k \frac{(\omega \tau)^c \sin \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}}\\ m &= \frac{\sigma_\infty - \sigma_0}{\sigma_\infty}\\ \sigma_0 &= (1 - m) \cdot \sigma_\infty

partial derivatives

There are partial derivatives of the real and the imaginary part respect to all variables.

real parts

\frac{\partial \hat{\sigma'}(\omega)}{\partial \sigma_\infty} &= 1 - \sum_k m_k \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}}\\ \frac{\partial \hat{\sigma'}(\omega)}{\partial m_i} &= -\sigma_\infty \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2}}{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}}\\ \frac{\partial \hat{\sigma'}(\omega)}{\partial \tau_k} &= \sigma_\infty m_k \left[ \frac{c \omega (\omega \tau)^{c - 1} \cos \frac{c \pi}{2}} {1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} - \frac{1 + (\omega \tau)^c \cos \frac{c \pi}{2}} {\left[{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} \right]^2} \cdot \frac{2 c \omega (\omega \tau)^{c-1} \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}}{1} \right]\\ \frac{\partial \hat{\sigma'}(\omega)}{\partial c} &= \sigma_\infty m_k \left[ \frac{\ln(\omega \tau) (\omega \tau)^{c} \sin \frac{c \pi}{2} + (\omega \tau)^c \cos \frac{c \pi}{2} \frac{\pi}{2}} {1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} - \frac{(\omega \tau)^c \sin \frac{c}{\pi}{2}} {\left[{1 + 2 (\omega \tau)^c \cos \frac{c \pi}{2} + (\omega \tau)^{2 c}} \right]^2} \right. \cdot\\ & \left. \frac{2 \ln (\omega \tau) (\omega \tau)^c \cos \frac{c \pi}{2} - 2 (\omega \tau)^2 \frac{\pi}{2} \sin \frac{c \pi}{2} + 2 \ln(\omega \tau) (\omega \tau)^{2 c}}{1} \right]

imaginary parts

\frac{\partial \hat{\sigma}''(\omega)}{\partial \sigma_\infty} &= - \frac{m (\omega \tau)^c \sin(\frac{c \pi}{2})}{1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}}\\ \frac{\partial \hat{\sigma''}(\omega)}{\partial m} &= - \sigma_\infty m (\omega \tau)^c \frac{sin(\frac{c \pi}{2})}{1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}}\\ \frac{\partial \hat{\sigma''}(\omega)}{\partial \tau} &= \sigma_\infty \frac{-m \omega^c c \tau^{c-1} \sin(\frac{c \pi}{2} }{1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}} +\\ &\sigma_0 \frac{\left[-m (\omega \tau)^c \sin(\frac{c \pi}{2} \right] \cdot \left[ 2 \omega^c c \tau^{c-1} \cos(\frac{c \pi}{2}) + 2 c \omega^{2 c} \tau^{2 c - 1}\right]}{\left[1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}\right]^2}\\ \frac{\partial \hat{\sigma''}(\omega)}{\partial c} &= \sigma_0 \frac{-m \sin(\frac{c \pi}{2}) \ln(\omega \tau)(\omega \tau)^c - m (\omega \tau)^c \frac{\pi}{2} \cos(\frac{\pi}{2}}{1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}} +\\ &\sigma_0 \frac{\left[-m (\omega \tau)^c \cos(\frac{c \pi}{2}) \right] \cdot \left[ -2 \ln(\omega \tau) (\omega \tau)^c \cos(\frac{c \pi}{2}) + 2 (\omega \tau)^c \frac{\pi}{2} \cos(\frac{c \pi}{2}) \right] + \left[2 \ln(\omega \tau) (\omega \tau)^{2 c}\right]}{\left[1 + 2 (\omega \tau)^c \cos(\frac{c \pi}{2}) + (\omega \tau)^{2 c}\right]^2}

Partial derivatives respect to log_{10}(x):

There are also partial derivatives of the real and the imaginary part respect to the common logarithm of all variables:

\frac{\partial F(X)}{\partial log_{10}(x)} &= \frac{\partial F(X)}{\partial x} \cdot x \cdot log(x)