Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 42
[3]
42
z — 'y' (l+A’?A,)n —><p(r))— —^tp(^l) (32b)
u = n u
n = o
=yg“"g»?-»(p(T|) =
]—e<oÐr, ->cp(ri) (32c)
n = o
From (32b) we derive
A77/0) -»z(r))=—<p(ii)
(33)
If, therefore, 2(11) and co are known we readily find tp(r|)
from (33). The problem of finding 2 if ro and q>(r)) is known
is more difficult and we shall, therefore, look a little closer
into the matter.
Let z=f(r\) satisfy (33), then f(x\)-\-c where c is a con-
stant also will satisfy (33). The determínation of this con-
siant cannot be obtained without closer definition of (32).
With reference to our proviso p. 6 periodic functions with
the period co are here left out of consideration. However,
in special cases we have to keep in mind this possibility.
When we are unable to see, immediately, which function,
substituted for z, satisfies (33) we can in (32b) and (32c)
make the following substitution:—
q?(Ti) =£>^0^7-^qp(ri) = £>77J cp(r))dr\ (34)
Now (32c) leads to the following well-known solutions:—