| dbp:proof
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- thumb|center|An illustration to the derivation of equation (8)
First, we deform the contour into a new contour passing through the saddle point and sharing the boundary with . This deformation does not change the value of the integral . We employ the Complex Morse Lemma to change the variables of integration. According to the lemma, the function maps a neighborhood onto a neighborhood containing the origin. The integral can be split into two: , where is the integral over , while is over . Since the latter region does not contain the saddle point , the value of is exponentially smaller than as ; thus, is ignored. Introducing the contour such that , we have
Recalling that as well as , we expand the pre-exponential function into a Taylor series and keep just the leading zero-order term
Here, we have substituted the integration region by because both contain the origin, which is a saddle point, hence they are equal up to an exponentially small term. The integrals in the r.h.s. of equation can be expressed as
From this representation, we conclude that condition must be satisfied in order for the r.h.s. and l.h.s. of equation to coincide. According to assumption 2, is a negatively defined quadratic form implying the existence of the integral , which is readily calculated
: (en)
- The following proof is a straightforward generalization of the proof of the real Morse Lemma, which can be found in. We begin by demonstrating
:Auxiliary statement. Let be holomorphic in a neighborhood of the origin and . Then in some neighborhood, there exist functions such that where each is holomorphic and
From the identity
:
we conclude that
:
and
:
Without loss of generality, we translate the origin to , such that and . Using the Auxiliary Statement, we have
:
Since the origin is a saddle point,
:
we can also apply the Auxiliary Statement to the functions and obtain
Recall that an arbitrary matrix can be represented as a sum of symmetric and anti-symmetric matrices,
:
The contraction of any symmetric matrix B with an arbitrary matrix is
i.e., the anti-symmetric component of does not contribute because
:
Thus, in equation can be assumed to be symmetric with respect to the interchange of the indices and . Note that
:
hence, because the origin is a non-degenerate saddle point.
Let us show by induction that there are local coordinates , such that
First, assume that there exist local coordinates , such that
where is symmetric due to equation . By a linear change of the variables , we can assure that . From the chain rule, we have
:
Therefore:
:
whence,
:
The matrix can be recast in the Jordan normal form: , were gives the desired non-singular linear transformation and the diagonal of contains non-zero eigenvalues of . If then, due to continuity of , it must be also non-vanishing in some neighborhood of the origin. Having introduced , we write
:
Motivated by the last expression, we introduce new coordinates
:
The change of the variables is locally invertible since the corresponding Jacobian is non-zero,
:
Therefore,
Comparing equations and , we conclude that equation is verified. Denoting the eigenvalues of by , equation can be rewritten as
Therefore,
From equation , it follows that . The Jordan normal form of reads , where is an upper diagonal matrix containing the eigenvalues and ; hence, . We obtain from equation
:
If , then interchanging two variables assures that . (en)
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