# Asymptotic Treatment of Differential Equations (Applied by Adelina Georgescu (auth.) By Adelina Georgescu (auth.)

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0, ... , whence Pi! = ± 1, P; 2 = - 1/2, .... 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!

The transform z ~ z maps the non-positive singular point z 0 to infinity and consequently the corresponding new series will converge better on the new positive half z-axis. For example instead of expanding the functionf(z) = e-•1 2 (2 + 3z) 1 12 , which has the singular point z = - j we use the Euler transform z= z(2 + 3z)- 1 and expand the function[(Z) = (=f(z)). In the method ofcompletion ofseries a series known only to degree N is completed by supposing that the subsequent terms are proportional to those in the expansion of some known function whose coefficients appear to have similar behaviour.

From the integral representation of the special functions of mathematical physics we can obtain their asymptotic behaviour. 5) ooexpifl 0 f3 < argz < (3n/2) + /3, e-uUv-0. 5 ( iu)'- 0. 5) > 0; whence, with the aid of the method of steepest descent, we deduce the following asymptotic expansions as lzl-+ oo: 2 )1/2 H! :)]} I (- 1 )~(: 4 m=o (21z) m) for -n < argz < 2n and H~ 2 l(z)"' ( -2 )112 exp { -i [ z- (vn) - - (n)]} - nz 2 4 (v m) Ioo ~ m=o (2Iz) for - 2n < arg z < n. , ( - 2 ) / exp { -i [ z- (vn) - - (n)]}( oo (v m) v nz' 2 4 m=O (2iz')m THE BASICS OF ASYMPTOTICS 30 valid for 0 < argz' < 2n.