Find the bosonic partition function


 * 1) Write down the general expression for the partition function: \(Z = \operatorname{trace}{\,e^{-\beta\left( \hat{\mathcal{H}} - \mu\hat{N}\right)}}\)
 * 2) Expand the trace function: \(Z = \sum_n \langle n|e^{-\beta\left( \hat{\mathcal{H}} - \mu\hat{N}\right)}|n\rangle\)
 * 3) Write as an integral of the wavefunction: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} e^{-\bar{\phi}\phi} \langle \phi|e^{-\beta\left( \hat{\mathcal{H}} - \mu\hat{N}\right)}|\phi\rangle\)
 * 4) Expand the number operator: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} e^{-\bar{\phi}\phi} \langle \phi|e^{-\beta\left( \epsilon - \mu\right)\hat{a}^{\dagger}\hat{a}}|\phi\rangle\)
 * 5) Substitute in \(|\phi\rangle = \sum \limits^{\infty}_{n=0} \frac{1}{\sqrt{N!}} \phi^n |n\rangle\): \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} e^{-\bar{\phi}\phi} \sum \limits^{\infty}_{n=0} \frac{1}{\sqrt{N!}} \langle \phi|e^{-\beta\left( \epsilon - \mu\right)n}\phi^n |n\rangle\)
 * 6) Simplify: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} e^{-\bar{\phi}\phi} \langle \phi|\sum \limits^{\infty}_{n=0} \frac{\left(\phi e^{-\beta\left( \epsilon - \mu\right)}\right)^n}{\sqrt{N!}} |n\rangle\)
 * 7) Write in wavefunction form.
 * 8) Multiply out the operator: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} e^{-\bar{\phi}\phi} \langle \phi|\phi e^{-\beta\left( \epsilon - \mu\right)}\rangle\)
 * 9) Write the Dirac bracket as an exponential: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} \exp{\left[\bar{\phi}\phi + \bar{\phi}\phi e^{-\beta\left( \epsilon - \mu\right)}\right]}\)
 * 10) Simplify: \(Z = \int \frac{d\bar{\phi}\,d\phi}{2\pi i} \exp{\left[\bar{\phi}\phi \left( 1 + e^{-\beta\left( \epsilon - \mu\right)}\right)\right]}\)
 * 11) Integrate approximately: \(Z \approx \frac{1}{1 + e^{-\beta\left( \epsilon - \mu\right)}}\)