2019ApJ...879..112J

X-ray point sources in galaxies are dominated by X-ray binaries (XRBs) that are variables or transients, and whether their variability would alter the X-ray luminosity functions (XLFs) is still in debate. Here we report on NGC 7331 as an example to test this with seven Chandra observations. Their detection limit is 7 x 10³⁷ erg s^–1 in the energy range 0.3-8.0 keV by assuming a power-law (PL) spectral model with a photon index of 1.7. We detected 55 X-ray sources. Thirteen of them are variables, of which three are transients, and some of the sources possess a bimodal feature in their luminosity-hardness ratio, which is often observed among XRBs. Nine more ultraluminous X-ray sources are found in comparison with previous studies and eight are likely to be low-mass or high-mass XRBs. Twenty-one optical counterpart candidates are found based on the Hubble Space Telescope images, but we cannot rule out the possibility of positional coincidence. The spectral analysis of SN 2014C shows a trend of increasing soft photons and decreasing hydrogen column densities as its outer shell expands. We fit the seven incompleteness-corrected XLFs to both a PL and a PL with an exponential cut-off (PLC) model using the Bayesian method, which is used for the first time in XLF fitting. The hierarchical PLC model can describe the XLF of NGC 7331 best with a slope of ∼0.5 and a luminosity cut-off around 8 x 10³⁸ erg s^–1. This study proves that multi-epoch observations decrease the deviation due to the variable luminous sources in XLFs.