Saddle point method

A method for computing the asymptotic expansion of integrals of the form

$$ \tag{* } F( \lambda ) = \int\limits _ \gamma f( z) e ^ {\lambda S( z) } dz, $$

where $ \lambda > 0 $, $ \lambda \rightarrow + \infty $ is a large parameter, $ \gamma $ is a contour in the complex $ z $- plane, and the functions $ f( z) $ and $ S( z) $ are holomorphic in a domain $ D $ containing $ \gamma $. The zeros of $ S ^ \prime ( z) $ are called the saddle points of $ S( z) $. The essence of the method is as follows. The contour $ \gamma $ is deformed to a contour $ \widetilde \gamma $ with the same end-points and lying in $ D $ and such that $ \max _ {z \in \widetilde \gamma } \mathop{\rm Re} S( z) $ is attained only at the saddle points or at the ends of $ \widetilde \gamma $( the contour of steepest descent). The asymptotics of the integral (*) along the path of steepest descent are calculated by means of the Laplace method and are equal to the sum of the contributions from the saddle points. The contribution $ V _ {z _ {0} } ( \lambda ) $ from the point $ z _ {0} $ is an integral of the form of (*) taken over a small arc of $ \widetilde \gamma $ containing the point $ z _ {0} $. If $ z _ {0} $ is an interior point of $ \widetilde \gamma $ and $ z _ {0} $ is a saddle point with $ S ^ {\prime\prime} ( z _ {0} ) \neq 0 $, then

$$ V _ {z _ {0} } ( \lambda ) = \sqrt {- \frac{2 \pi }{\lambda S ^ {\prime\prime} ( z _ {0} ) } } e ^ {\lambda S( z _ {0} ) } [ f( z _ {0} ) + O( \lambda ^ {-} 1 )]. $$

The contour of steepest descent has a minimax property; on it,

$$ \min _ {\gamma ^ \prime } \max _ {z \in \gamma ^ \prime } \mathop{\rm Re} S( z) $$

is attained, where the minimum is taken over all contours $ \gamma ^ \prime $ lying in $ D $ having the same end-points as $ \gamma $. The main difficulty in using the method is to select the saddle points, i.e. to choose the $ \widetilde \gamma $ corresponding to $ \gamma $.

The method is due to P. Debye [1], although the ideas in the method were suggested earlier by B. Riemann [2]. See [3]–[9] for the calculation of the contributions from the saddle points and from the end-points of the contour.

The method is in essence the only method for calculating the asymptotic expansions of integrals of the form (*). It can be used to derive the asymptotic expansions for Laplace, Fourier and Mellin transforms, as well as for transforms of exponentials of polynomials and many special functions.

Let $ z \in \mathbf C ^ {n} $, let $ \gamma $ be a bounded manifold with boundary of dimension $ n $ and of class $ C ^ \infty $, let functions $ f( z) $ and $ S( z) $ be holomorphic in a certain domain $ D $ containing $ \gamma $, and let $ dz = dz _ {1} \dots dz _ {n} $. Suppose that $ \max _ {z \in \gamma } \mathop{\rm Re} S( z) $ is attained at a single point $ z ^ {0} $ which is an interior point for $ \gamma $ and a non-singular saddle point for $ S( z) $, i.e. $ \Delta _ {S} ( z ^ {0} ) \equiv \mathop{\rm det} S ^ {\prime\prime} ( z ^ {0} ) \neq 0 $. Then the contribution from $ z ^ {0} $ is

$$ F( \lambda ) = \left ( \frac{2 \pi } \lambda \right ) ^ {n/2} (- \Delta _ {S} ( z ^ {0} )) ^ {-} 1/2 e ^ {\lambda S( z ^ {0} ) } [ f( z ^ {0} ) + O( \lambda ^ {-} 1 )]. $$

References

[1]	P. Debye, "Näherungsformeln für die Zylinderfunktionen für grosse Werte des Arguments und unbeschränkt veranderliche Werte des Index" Math. Ann. , 67 (1909) pp. 535–558
[2]	B. Riemann, "Mathematische Werke" , Dover, reprint (1953)
[3]	A. Erdélyi, "Asymptotic expansions" , Dover, reprint (1956)
[4]	N.G. de Bruijn, "Asymptotic methods in analysis" , Dover, reprint (1981)
[5]	M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach (1962) (Translated from Russian)
[6]	E.T. Copson, "Asymptotic expansions" , Cambridge Univ. Press (1965)
[7]	F.W.J. Olver, "Asymptotics and special functions" , Acad. Press (1974)
[8]	E.Ya. Riekstyn'sh, "Asymptotic expansions of integrals" , 1–2 , Riga (1974–1977) (In Russian)
[9]	M.V. Fedoryuk, "The saddle-point method" , Moscow (1977) (In Russian)

References

[a1]	R. Wong, "Asymptotic approximations of integrals" , Acad. Press (1989)