# Asymptotic expansions by A. Erdelyi

By A. Erdelyi

A variety of equipment for asymptotic assessment of integrals containing a wide parameter, and options of standard linear differential equations via asymptotic growth. initially ready for the place of work of Naval study, this significant monograph introduces quite a few tools for the asymptotic review of integrals containing a wide parameter, and ideas of standard linear differential equations through asymptotic expansions. 1956 unabridged republication of Technical record three, workplace of Naval study. Author's preface. Bibliography.

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The asymptotic study of a standard equation begins by contemplating its standard solutions at i-large values of the variable. The use of the complex plane permits one to join the i-large positive numbers and the i-large negative numbers without passing through the limited numbers; in others words, it permits one to link the asymptotic behaviour at +00 and -00 inside the asymptotic domain. 2) is the simplest equation the solutions of which have both oscillatory and exponential behaviour. 3) are Y = Asin( yCO,X + ip) and have all the same oscillatory behaviour.

LogIe:'} = Log f(;',} 0 D Log R R Fig. 9. A connected Julia set and its conformal mapping to a strip. The points Xl and X2, which occur in the text, are assumed to be distinct in the figure. Since I show that, in fact, they coincide, the figure gives a slightly false impression. Proof. Let fj be the left horizontal ray starting from one of the points of Jw the furthest to the left of Jw , and let Xo be in K~\8. Let us define 38 A. Fruchard 2. Complex analysis which maps K~\8 on a strip with height a multiple of 211" since the function fW is single valued.

What are the values of A and