What is modified complex action function and how does it apply to an integral?

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What is modified complex action function and how does it apply to an integral? I am very confused on where to start? Could Recommend some Literature to see how to learn what it is,the refenences are too hard to understand for me(I am an undergrad) https://royalsocietypublishing.org/doi/10.1098/rspa.2014.0943 equation (1.17)