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Oseen C. W., Zur Hydrodynamik der Kugel.pdf


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C. W. OSEEN, ZUR HYDRODYNAMIK DER KUGEL.

Pm, o cos :J am,, o + Pm, 1 sin :J r~~, 1

3

+ {Pm, 1 cos -9 am, 1 +

+Pm, 2 sin:Jr~L 2 +Pm, o sin:Jr~, o+ Pm, osinßr~, o}cosrp +

+ {Pm, 1 cos :J ßm, 1 +

Pm, 2 sin :J o~, 2 - Pm,o sin :J ö~~l,o

+

n=m-1

+Pm, osin-90~, o}sinrp + ~ [{Pm, nCOS 9am, n +Pm, n+I sin..9r~, n+I +

+

Pm,n-1 sin :Jr~,n-1} cos nrp

+ {Pm,n cos :Jßm,n +

+ Pm,n+1 sin :Jo~,n+l + Pm,n-I sin :Jö~,n-1} sin nrp +
, 1pm, m COS n am, m + p m, m-1 Sill
. vo. l'm,
(2)
"
+
T \
m-1j
COS m 'fJ
1J

+ { p,m, cos ,() flm, m + p m, m-1 sin :J ö~, m-1} sin m rp +
+ Pm,m sin !J r~,m cos (m + 1) rp + P„n,m sin :J o~~,m sin (m + 1) (p.
'111.

Aus den Definitionsgleichungen für Pm, n folgt für

l<n<m-1:
dPm,n(x)
nx
d
= - -- - -2 Pm,n(x)
1 -x
x
dPm,n(X)
dx

l

1

m-n
+ y--Pm,n+i(x),
I-x2

nx
m+n
Pm,n-1 (x).
2 Pm,n (x)-x
V I - x2

. .t\.lso:
<JPm,n
iJ-9

aPm,n
i}:J

n cot:;. Pm,n-(m-n)Pm,n+I,

=

-

n cot:J. Pm,n

2 n cot -9 Pm,n = (m

+

+

n) Pm,n-1

(m

+

n) Pm,n-1,

+ (m- n) Pm,n+t.

Ausserdem ist:
f)Pm,o
{) .[)

=

8Pmm

ö.d

-

m p m, 1 '
p

=mcot:J m,m,

8Pm,m
B:J = -m cot-9 Pm,m
1

HEINE

I,

s.

258.

+ 2rnPm,m-t,