Stationary phase approximation

In calculus, the Sattelpunktsnäherung is used to integrals of the form

Approximately calculate. The method comes from Pierre Simon de Laplace (1774 ) and is sometimes named after him. It is part of the asymptotic analysis.

If the function is analytic and has a global minimum at, we obtain:

With.

The second derivative is positive, since there is a minimum. The result holds asymptotically, ie for N to infinity.

This can also be finite integration limits ( A, B).

The generalization of the Sattelpunktnäherung in the complex number plane is also called the saddle point method. From it the name of a saddle-point method or approximation explained.

Alternative formulation

It can also be considered a different sign in the exponent:

In other sign applies for

If at a global maximum exists asymptotically:

With.

Since there is a maximum of the second derivative is negative.

Grounds

Is considered to be the first case ( at minimum ) in the second case, the argument is analogous.

For large N the exponential outside around is arbitrarily small. Therefore, it is developed into a Taylor series: .

Provides inserting into the integral

.

The integral of the Gaussian distribution can be solved easily.

Applications

The Sattelpunktsnäherung and saddle point methods found various applications in theoretical physics, including statistical physics in the limit of large systems, in quantum field theory in the evaluation of path integrals or in optics.

One application is the Stirling formula:

For large N.

From the definition of the gamma function follows

With the change of variables (so that ::) is obtained:

Now you can Sattelpunktnäherung in the second form ( for maxima ) apply with

With the derivatives

F is the maximum of the second derivative value -1. Obtained with the Sattelpunktnäherung:

Generalization

The Sattelpunktnäherung is when viewed in complexes in the method of steepest descent (English: Method of steepest descent ) or the method of stationary phase (English: Method of stationary phase ) generalized (general saddle point method). The goal is the asymptotic analysis of closed Wegintegralen in the complex plane ()

For large real. This one deformed in the complex path of integration so that stationary point ( zero of the first derivative of g ) of on the path of integration lies and then goes like above ( with the additional application of Cauchy's integral theorem ). In the version of the method of steepest descent you put the path of integration in such a way that the real part u of g there is a maximum. Since the real part u of g is a harmonic function, and can not have the same sign: it is a saddle point and before you put the path of integration along the path of the " steepest descent ". Hence the name of the method.

Wherein the stationary phase method in particular Integral be considered in which the exponent of the exponential function is imaginary along the path:

U With a real function and large.

The method was first published in 1909 by Peter Debye for the estimation of Bessel functions, but also already used by Bernhard Riemann.