Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 42

42
out, we get rid of the questions concerning the convergence
of the series and, what is of more importance, we ar able
to deal with the series a^xn in many cases where the
H = 0
00
divergence of a,txn makes the use of the last mentioned
n = 0
series impossible.
On the other hand if we are dealing with divergent series
by the asymptotic methods set out here, we have to re-
call that the results always are given as real. If, therefore,
the quantity sought for actually is imaginary or complex
we cannot expect that the asymtotic methods give the
quantity correctly. This fact 'is indicated by oscillating or
even divergent results when we attempt to attain greater
accuracy by including more terms in our calculations.
REFERENCES
1) Thorkell Thorkelsson, Approximate Integration, Tímarit
Verkfræðingafélags íslands. XXII,
p. 36 (1937).
2) — — Serial Relations II, Soc. Sci. Isl.,
Greinar I, p. 181, (1940).
3) — — Serial Relations, Soc. Sci. Isl., Grei-
nar I, p. 99, (1937).
4) — — DifferentialSeriesof EulerianType,
Soc. Sci. Isl., Greinar I. p. 201,
(1940).
5) — — Divergent Series, Tímarit Verk-
fræðingafélags íslands, XVIII, p. 23,
English Summary, p. 30, (1933).
6) J. G. Brainerd and C. N. Weygandt, Solutions of Mathieu’s
Equation, Phil. Mag., XXX, p. 474
(1940).
7) E. Jahnke and F. Emde, Funktionentafeln, Leipzig, 1909.
8) Thorkell Thorkelsson, Divergent Power Series, Soc. Sci.
Islandica, XVII. pp. 23, (1934).