Editing Laplace's method (section)

==Median-point approximation generalization==

In the generalization, evaluation of the integral is considered equivalent to finding the norm of the distribution with density

:<math>e^{M f(x)}.</math>

Denoting the cumulative distribution <math>F(x)</math>, if there is a diffeomorphic [[Gaussian distribution]] with density

:<math>e^{-g -\frac{\gamma}{2}y^2}</math>

the norm is given by

:<math>\sqrt{2\pi\gamma^{-1}}e^{-g}</math>

and the corresponding [[diffeomorphism]] is

:<math>y(x)=\frac{1}{\sqrt{\gamma}}\Phi^{-1}{\left(\frac{F(x)}{F(\infty)}\right)},</math>

where <math>\Phi</math> denotes cumulative standard [[normal distribution]] function.

In general, any distribution diffeomorphic to the Gaussian distribution has density

:<math>e^{-g -\frac{\gamma}{2}y^2(x)}y'(x)</math>

and the [[median]]-point is mapped to the median of the Gaussian distribution. Matching the logarithm of the density functions and their derivatives at the median point up to a given order yields a system of equations that determine the approximate values of <math>\gamma</math> and <math>g</math>.

The approximation was introduced in 2019 by D. Makogon and C. Morais Smith primarily in the context of [[Partition function (quantum field theory)|partition function]] evaluation for a system of interacting fermions.<ref>{{Cite journal |last1=Makogon |first1=D. |last2=Morais Smith |first2=C. |date=2022-05-03 |title=Median-point approximation and its application for the study of fermionic systems |url=https://link.aps.org/doi/10.1103/PhysRevB.105.174505 |journal=Physical Review B |volume=105 |issue=17 |pages=174505 |doi=10.1103/PhysRevB.105.174505|bibcode=2022PhRvB.105q4505M |hdl=1874/423769 |s2cid=203591796 |hdl-access=free }}</ref>