By Adelina Georgescu (auth.)
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Extra info for Asymptotic Treatment of Differential Equations
It is seen that when the argument increases continuously, as the lines arg z = 2pn or arg z = 2(p + l)n 1 are crossed, c and c2 are subjected to jumps, and therefore the leading term of the asymptotic expansion of J v has discontinuities on these Stokes lines. However, it must be mentioned that the Bessel functions are continuous across these lines; only their asymptotic expansions, which approximate them, have discontinuities (c 1 will have a jump when the function of which it is a factor may be neglected with respect to the other; a similar situation takes place for c2 ).
1) it follows that p 1 "'1- e/2 + · · ·, p2 "' - 1- e/2 + · ·· . In this case the (exact) solutions are Pu = [ - e ± (e 2 + 4) 112]/2, such that it is easy to see that their asymptotic expansions with respect to the asymptotic sequence 1, e, e2 ... fore--+ 0 coincide with those found by means of the method presented above, and this is called the method of regular perturbations. Unfortunately, this method is not suitable for every equation. For instance, in the case of the equation ep 2 + p1 = 0, this method leads to Pi!
Both methods lead to a certain asymptotic expansion of the solution. However this fails to hold around x*. Thus the question occurs as to how to find the asymptotic expansion of the solution y(x, A) as A--+ oo in the interval (x*- B, x* +B). This problem is difficult in certain cases (corresponding to certain q0 (x)) when the asymptotic expansion from the left of x* is quickly varying while that from the right is slowly varying, or conversely. This is due to the fact that there is no elementary function describing the transition from quick variations to slow ones.
Asymptotic Treatment of Differential Equations by Adelina Georgescu (auth.)