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$$G(\phi_{f}^{*},t_{f};\phi_{0},t_{0})=\int \dfrac{d\phi_{1}^{*}d\phi_{1}}{2i\pi}\int\dfrac{d\phi_{2}^{*}d\phi_{2}}{2i\pi}\int\dfrac{d\phi_{3}^{*}d\phi_{3}}{2i\pi}......\int\dfrac{d\phi_{N-1}^{*}d\phi_{N-1}}{2i\pi}\exp\Bigg(-\phi_{f}^{*}\phi_{f}\Bigg)\exp\Bigg[\dfrac{i}{\hbar}\epsilon\sum_{i=1}^{N-1}\Bigg(\phi_{i}^{*}\dfrac{\phi_{i}-\phi_{i-1}}{\epsilon}-\omega\phi_{i}^{*}\phi_{i-1}\Bigg)\Bigg]$$. Where,$$\phi_{f}^{*},\phi_{f},\phi_{0}^{*},\phi_{0}$$ are constant complex numbers.and $\epsilon$ is infinitesimal difference between $\phi_{i}$ and $\phi_{i-1}$ and $\hbar$ is plancks real postive constant. $\omega$ is also a real positive number . Of course $i=\sqrt{-1}$.The last term $$\phi_{N}=\phi_{f}$$ which is a constant as mentioned above. To integrate this I have started as follows,$$\int\dfrac{d\phi_{1}^{*}d\phi_{1}}{2i\pi}\exp\Bigg[\epsilon\dfrac{i}{\hbar}\Bigg\{\Bigg(\phi^{*}_{1}\dfrac{\phi_{1}-\phi_{0}}{\epsilon}-\omega\phi_{1}^{*}\phi_{0}\Bigg)+\Bigg(\phi^{*}_{2}\dfrac{\phi_{2}-\phi_{1}}{\epsilon}-\omega\phi_{2}^{*}\phi_{1}\Bigg)\Bigg\}\Bigg]$$.However, I don't know how to proceed further. Can some one explains about complex gaussian integral with complex coefficients in the exponentials. How to solve this step by step by approach.

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