Rit (Vísindafélag Íslendinga) - 01.06.1946, Blaðsíða 33

33
S-j-l
S — 1
°s+2 °s+l as
(60a)
Also this procedure leads to a method of solving numer-
cial equations, approximately. More details regarding this
method are to be found in my booklet Divergent Power
Series,8). However, I desire to add that when I wrote my
book I was not aware of the fact that the solution of
equations by virtue of that procedure was originally given
by D. Bernoulli (cf. N. E. Nörlund: Vorlesungen uber Diffe-
renzenrechnung, 1924, p. 299).
I shall now again return to the Bessel differential equa-
tion (49). The asymptotic methods expounded here, seem
often to be most suitable for disclosing the character of the
curve representing the differential equation, by changing the
origin i. e. by replacing the unknown x by x-f-a where a
represents the displacement. Bessel differential equation
may now be written:
, \ 2d2y . dy , 2Í , w, / m \2\ n
/ dx2 dx ^ \ /l lx + al r
(49a)
which is convenient for asymptotic differentiation. Now
we have:
—»
/t
— n) [(/r 4- n) x + (k + n — 1) a]
/1 • Lt A
y = ~-------------------------------------------------------------x" (61)
m ' '
fc I /c — 11 ot -f- A:2 x -f- x2 j x -f- a I (1 —
\ x a
and
x
dlJ 7Z
k — n x + a
kn — x 1 —
m
x-f-a
n íri
X d IJ o
n\ dxn
dx
k(k-^a + k2x + x2 (x +.a ] (1 - (^) )
(61a)