\tau_0 \rangle}\left(1-\frac{y}{2Re\sqrt{f}}\right.\right.{\nonumber}\\ 
&-& \left. 2\sqrt{1-\frac{y}{2Re\sqrt{f}}}\tanh^{-1}(\sqrt{1-\frac{y}{2Re\sqrt{f}}})\right){\nonumber}\\
&+& \left.  \left[\tanh^{-1}(\sqrt{1-\frac{y}{2Re\sqrt{f}}})\right]^2\right]+C^2.\end{aligned}$$ From $\tanh^{-1}(x) = \frac{1}{2} \log(\frac{1+x}{1-x})$, we see that the second term in the last formula is of leading order and we get that $$\label{eq:w2der}
\langle w^2 \rangle \sim 2C \frac{(\langle \sqrt{\tau_0} \rangle-\sqrt{\langle \tau_0 \rangle})}{\kappa \sqrt{\langle \tau_0 \rangle}}\log\left(\frac{y}{Re\sqrt{f}}\right) + h. o. t.$$ This agrees with the formula (\[eq:gloglaw\]) above. For higher order moments $\langle w^{2p} \rangle^{1/p}$ the similar term, linear in $\tanh^{-1}$ and multiplied by $2C$, is of leading order, $$\label{eq:wpder}
\langle w^{2p} \rangle^{1/p} \sim 2C \frac{\langle (\sqrt{\tau_0}-\sqrt{\langle \tau_0 \rangle})^p \rangle^{1/p}}{\kappa \sqrt{\langle \tau_0 \rangle}}\log\left(\frac{y}{Re\sqrt{f}}\right) + h. o. t.$$ These formulas establish the log dependance of the second moment of the fluctuations, with the Townsend-Perry constants, and the log dependence of the higher moments of the fluctuations, with the Generalized Townsend-Perry constants, and justify formulas Eq. (\[eq:l-lfluct\]) and Eq. (\[eq:l-pfluct\]). Together, Eq. (\[eq:l-lfluct\]) and Eq. (\[eq:l-pfluct\]) can be called the generalised log-law of the wall.

Derivation of the Generalized Townsend-Perry Constants {#sec:SCT}
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We consider the dependence of the fluctuation $w$ on the distance $x$ along the wall, to understand the Townsend-Perry constants. So far we have only considered $w(y)$ as a function of the distance $y$ from the wall, but $w(x,y)$ obviously depends 