The Saddle-Point Method

The function F(s) has a saddle point at s = Sq defined by z (Sq) = 0. It is shown below that Sq is a point on the imaginary axis. Since F(s) is an analytic function the integration path can be deformed in such a way as to pass through Sq and F(Sq) is a maximum for that path. Since N is large, this maximum is very sharp. The saddle-point method essentially consists of developing F(s) around the saddle point thus we write 

---

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

The integrals in the numerator and denominator can be estimated by using the saddle point method again. By expanding g (x) around the maximum contribution at x , we get, up to second order. 

Taking into account relationships (75) and (76), over the long time interval z> 1 the integration of (73) may be performed asymptotically by the saddle-point method [135]. The main term of the asymptotic expansion can be obtained as the product of the power and stretched exponential universal relaxation laws [38] ... 

The integrals (XI.4.14) and (XI.4.15) can be evaluated with good accuracy by the method appropriate to exponentially peaked functions, namely the saddle-point method (Marcus, loc. cit). 

Finally, let us try to formulate the scattering matrix within the present semiclassical treatment. Using the adiabatic wave functions obtained by the saddle point method,... 

AG (T) have been derived a particularly useful one is obtained using the saddle-point method ". This expression is ... 

Kinetic coefficients. The kinetic adsorption and desorption coefficients can be estimated [82, 219, 412] if the form of the potential (Z) of interaction between a particle (a surfactant molecule) and the solution surface is known here Z is the coordinate measured from the surface into the bulk of liquid. If the function (Z) has the form of a potential barrier with a potential well, then the saddle-point method [261] implies... 

This observation is most helpful for our asymptotical treatment of the atomic systems suggesting that the saddle point approximation (Mathews Walker, 1970), is suitable to fairly analytical perform the involved integrals. According with the saddle point method, to evaluate an integral of type (4.325) the intermediate form (4.326) is approximated by the saddle-point recipe (3.154) specialized here as (see also the Appendix of the present volume) ... 

In this case, the saddle point method is inapplicable for the calculation of the integral over a in Eq. (27). The major contribution to the integral comes rather from the large values of o-. This leads to the following expression for the transition probability... 

In the domain of long times this integral can be evaluated with the use of the saddle-point method. Finally, one has [142] ... 

See Problem C. Mathematically, Eq. (6.7) says that Qk(T) is the Laplace transform of p(E)k(E). It is natural to want to use an inverse Laplace transform to get k(E) from a measured k(T). Computationally this is not a safe route because taking an inverse Laplace transform is numerically rather unstable and is very sensitive to errors in the measured k(T). What is more satisfactory is to evaluate the Laplace transform by the saddle-point method as introduced in Chapter 3. See Problem P. 

---