# Who named the fugacity, who coined the variable name and did it already relate to complex analysis?

In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane

$$\Pi(x) = \frac{1}{2\pi i} \int \log \zeta(s)\ x^s \frac{\mathrm{d}s}{s}$$

and the zeros of $\zeta$ are consequently of interest (Riemann hypothesis).

Associated quantities relate closely to concepts in statistical physics. Planck's law of for the spectral radiance ($B_\nu(\nu, T) = 2 h c^{-2}\frac{\nu^3}{e^{h\nu/k_\mathrm{B}T} - 1}$) Mellin transforms to the Riemann zeta function ($\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x ^ {s-1}}{e ^ x - 1} \mathrm{d}x$) and physicists have their own polylog (mind the $z$) in the Fermi–Dirac integral.

Now the grand partition function $\mathcal{Z}$ relates microscopic statistical physics to thermodynamics via

$$-k_B T \ln \mathcal{Z} = \langle E \rangle - TS - \mu \langle N\rangle$$

and is often expressed as a power series in the fugacity $$z=\exp\left(\frac{\mu}{T}\right)$$ as $\mathcal{Z}(z) = \sum_{N_i} z^{N_i} Z(N_i).$

The zeros of $\mathcal{Z}$ in $z$ make the log $\ln \mathcal{Z}$ explode and this phenomenon is associated with phase transitions and people study and cook up result like the Lee-Yang-theorem. The fugacity as a parameter in $\mathbb C$, which controls the zeros, is actually on the cover of one of my favorite books. But I'd think the fugacity was defined without that context in mind.

What I wanted to know is when did "$z$" become a standard name for a //complex// variable and, more importantly, is it coincidence that the names fit?? Who names it and where there other names for the fugaciy? 