By Sellwood, Astrophysical discs.
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When only one of the two terms in the linear combination is significant, we delete the compound symbol corresponding to the insignificant term; see Fig. 1(c). The compound symbols in Fig. 1 constitute ‘vehicles’ for the movement of which in the complex z-plane one can formulate certain ‘traffic rules’. 4 in Fr¨oman and Fr¨oman (1965). 5. 6. Thus, when for instance the term with the coefficient a2 becomes insignificant, the symbol in Fig. 1(b) is replaced by the symbol in Fig. 1(c). Using these ‘traffic rules’ one can easily trace a desired solution of the Schr¨odinger equation along Stokes and anti-Stokes lines in the complex z-plane.
3a,b) one can obtain fairly simple asymptotic expressions with respect to a ‘small’ parameter. Thus one obtains the phase-integral approximation generated from an unspecified base function. , solutions with the whole asymptotic series in the exponent, one is not able to obtain asymptotic approximations with a constant Wronskian, nor to express these solutions in terms of a single function. 16) where u(z) and w(z) are real functions, when z and R(z) are real. 16) is the same as the function w(z) in the phase-integral approximation generated from an unspecified base function with Q 2 (z) = R(z), but the function u(z) is very complicated in the higher orders of the Carlini approximation; see eqs.
21b) Thus, the Stokes lines are lines along which the absolute value of exp[iw(z)] increases or decreases most rapidly, while the anti-Stokes lines are level lines for constant absolute values of exp[iw(z)]. 4. 6. 3 F-matrix method 29 in the neighbourhood of the transition points, it is in general not difficult to get their overall pattern. 15), where t is a transition point of odd order, we define a Stokes line emerging from t as a Stokes line on which w(z) is purely imaginary, and we define an anti-Stokes line emerging from t as an anti-Stokes line on which w(z) is real.
An EC summer school by Sellwood, Astrophysical discs.