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A non-degenerate saddle point, , of a holomorphic function is a critical point of the function (i.e., ) where the function's Hessian matrix has a non-vanishing determinant (i.e., ).

The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:

Complex Morse lemma

The Morse lemma for real-valued functions generalizes as follows for holomorphic functions:

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