\psi - \log \psi - 1 = 4 \left ( \Xi^{-1} - \log {\Xi^{-1}} - 1 \right )

When \Xi^{-1} >> 1 we have

-\log \psi = 4\Xi^{-1} or \psi = e^{-4/\Xi}

When \Xi^{-1} << 1 we have

\psi = -4 \log \Xi^{-1} = 4 \log \Xi

When \Xi \sim 1 we have

\psi - (\psi - 1) + (\psi-1)^2/2 - 1 = 4(\Xi^{-1} - (\Xi^{-1}-1)+ (\Xi^{-1}-1)^2/2 - 1)

(\psi-1)^2/2 = 2 (\Xi^{-1}-1)^2

\psi= 1 + 2 (\Xi^{-1}-1)