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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)