A Note on Multiple Asymptotic Series
R. D. Gregory · SIAM Journal on Mathematical Analysis · 1980
There has appeared in the literature [K. D. Shere, Introduction to multiple asymptotic series with an application to elastic scattering an attempt to extend the concept of asymptotic series to “multiple asymptotic series” of the form \[\sum_{m = 0}^\infty {\sum_{n = 0}^\infty {a_{mn} \frac{{e^{ - \lambda mx} }}{{x^n }}} .} \] Applications referring to this work have also appeared in W. Bühring [J. Mathematical Phys.,18 (1977), pp. 1121–1136], W. Buhring [Angew. Math. Mech., 57 (1977), T226–T227], K. D. Shere [SIAM J. Math. Anal., 3 (1972), pp. 263–271], and K. D. Shere [SIAM J. Math. Anal., 3 (1972), pp. 272–284]. It is the purpose of this note to show that the definition of a “multiple asymptotic series” given in K. D. Shere [J. Mathematical Phys., 12 (1971), pp. 78–82] is unsound in the sense that it fails to satisfy certain basic criteria (for instance when the series involved are convergent then they are not necessarily asymptotic, according to this definition); also even though uncountably many alternative definitions exist which overcome these difficulties, each is quite arbitrary and has little practical value.