Rit (Vísindafélag Íslendinga) - 01.06.1946, Page 29

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tor of (50) becomes zero for x=0 if k = + m and then it
is advisable to substitute for y:
y — x"‘ u + x~m v (52)
where u and v are functions which are easily expanded
into series in x. I shall here make the assumption that
u = 0, and u assumes then the most common form of Bes-
sel’s functions:
y = u0x
m
n — o (m + n)!n!
(53)
and
íi +(7+2") jT
dx (m + n)! n\ \ 2 ' I
(54)
The series (53) and (54) are convergent and therefore
usable for calculating purposes if x is small. However, the
asymptotic formulas (32) and (32a) are as a rule more ex-
pedient and if we wish to compute the lowest x corre-
sponding to zero of y, the last mentioned formulas are
far more practical.
• We put in (32a) s = 1 and for y = 0 we have approxi-
mately:
1 0 0 0............ 0
m\ m\
1 0.......... 0
2! (m + 2)! (m + 1)!
m\ m\ m\
(r+ 1)! (m + r+ 1)!.r! (m + r)! (m+1)!