Computation of oscillatory integrals

Let us recall that in the case of computing the oscillatory integrals (3.25) in which the function F has a degenerate critical point, the stationary phase method, described in Section 3.4.2, fails. There are two basic methods of computation of such integrals which will be exemplified by the Airy function. The information provided below is brief and intended to facilitate the reader an access to suitable references (see bibliographical remarks at the end of this chapter). 

The second method consists in writing an oscillatory integral in the form 

Investigations of the heartbeat have revealed that the heart may occur in two fundamental states the state of decontraction (diastole) and the state of contraction (systole). Responding to an electrochemical stimulation, each fibre of the cardiac muscle rapidly contracts, remaining in this state momentarily, followed by a rapid return to the state of decontraction. 

In the case of transmission of nerve impulses, the dynamics is different. The state of a transmitting nerve fibre, axon, is determined by the electrochemical potential between inner and outer fibres of the axon. In the absence of a perturbation the potential remains at a constant level. In the case of impulse transmission, the potential abruptly changes, followed by a slow a return to the initial state. 

Zeeman s concept of modelling these phenomena involves their qualitative description by differential equations phase portraits of these equations must only meet some qualitative requirements consistent with the above characteristic of the systems being modelled. The processes described above have certain significant characteristics, which have to be taken into account in the model  

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