Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 32

32
in (32) or (32a). This quadratic equation may always be
written as including 3 determinants of third order or
a0 0 0 ax a0 0 Cl^ Öq i
^2 A ~f~ x A2 Aj A0 + X2 + +
B2 Bx B0 B2 bx b0
(59)
where v40, Av A2, B0, Bx and B2, are found by suitable reduc-
tion of the determinants in (32a) which are of higher order
usually. The ratios AJA0, A2/A0, BJB0 and B2/B0 approach
more and more definite values as r increases. If we wish
to use this method to solve a cubic equation or an equa-
tion of higher degree we may consider x”1 as the unknown
as well as x and thus select the largest roots as well as
the smallest ones, in analogy with what we have shown
in (58a).
The Bessel function has no poles excepting at the origin.
The problem of finding x that makes y infinitely large is
therefore out of question. Nevertheless it seems to me
worth while to remind of the fact that the asymtotic me-
thods set forth here enables us to determine x, for which
y becomes infinite, if such values exist. This may be accom-
plished by putting the determinant on the left hand side
of (32a) equal to zero whence x is to be found approximately
for which y becomes infinitely large. As a rule we choose
r— 1 and s oo and therefore:
lim
s —> oo
x
:«+i as
= 0
(60)
After elimination of the respective factor from the series
we repeat the same procedure as often as necessary.
Sometimes, however, it is more practical to use the formula: