Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW)
Citation:
Kapteyn, W., On the integral equation of Fredholm, in:
KNAW, Proceedings, 13 II, 1910-1911, Amsterdam, 1911, pp. 734-741
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-1-
( 734 )
see where Ihe ClU'ves Ii'l(a l) ulnd!t 2 (1~2) meet. The meeting point
indicates the pole P (!?,(I) of the l'ccJllil'ed seca.nt plane.
IJl fig. 6 the gmphicaJ sol 11 tioll of t" 0 of the ab0\'e dil:lcnssed
problems is I'epl'esenled. If Olle tnkes (cf. fig. 2) I.lle l'ilomboeloclecaheclronplanc l110) as eqllatol'plane, then tbe planes (101) anc! (011) .fol'm
angles Witll it eqnal to (~l
60" anel ([2 ~ 60 . Tlle secant plane S
gives hl
60°, Il 2
60°; y
-109°28'17". lf the diagl'tl,m for
u 1 is l'emo\·ed :1 09° 1 / 2 with regal'd to the diagram fol' U 2 , and both
are placed OH etl,ch villel', 1.l1ell· it appears, tlw,t tIle curves hl (60°)
anc! !t 2 ( - - 60°) meet onl,)' iu the 5th octant. In the figlJl'e the curves
for
0 anel (j
0 are dl'awn side by side. If the azimuth of
/l is = 0, the ClLl've AB (a 2 = 6()o) indicates lhe poles fol' lt 2
60
with (I
0; Be I.he polcs fol' Il 2
60' with (j
0, AG those
fol' h2 60° wUlt (j O. The azimuth of D
109° 1 / 2 ; FD gives
the poles fol' hl = - 60° with
0; FE ~l,nd DE give those for
ft l
60° resp. hl
60° with (j
O. The meeting point of the
CUl'ves BG anel FE giyes I.he pole of the requil'ed secant plane with
!? = 54°3/ 4 wHh regard to A, Ol' !?
-54":1/4 with l'egm'd lo f),
and û
54°:1/ 4 _
The second pl'oblem refers 10 Ihe mnphihole-cl'ystal spoken of on
page 782, whicb is cut hy tbe section plane in sllch a wa,.\' thaI the
appal'ent angles [Jetween the plalles (liO): (110) n,nd (010): (110)
a.mount l'espe('tively 10 !ti
43° and 1t 2 = - 79°.
In fig. 6 tbe zono-axis (!.) = 0) is indicated by j{; the Clll'VeS
L01);J and HJ.VA indicate the Ioem, of the pol es fol' !tl
43°
with (j
0 (1 ~t octant) and (j
0 (8 tll octant); the cnl'\'es 10 and
IN indicate the locns of tlle poies fot, !t 2 = - 79° with (J
0
(1 st octant) anel {j
0 (8 h octant). The meeting point of tlw (,lll'vesLM: 10 and HA: flV gives the points 0 (!? = :35°, fi = 31°) anel
N(!?=-35°, a=-31 C ) which vn,lllcS likcwise cOl'respond en1.i1'e1,\'
to those fOtllld n,uove in I,he a,nalytical wa,)'.
=
(J>
>
= -
=
<
= --
<
=
=
=-
(J>
= --
=
<
=
0
<
=
=-
=
>
=
<
>
<
Mathematics. -'" Un tlw fntegral equation I)f FmiDuoT,lIl." 13y
Prof. VVo KAP1'J'1YN.
1.
Let
IJ
cp(m)
J
= f(,v) + ).
K(a:s) cp ($) d$
(1)
be (,he integnll eqnation of Fm<mnou\[, in which tlle constants (/" b, À,
and the t'tmctions f(m) and J( (,1)//) [we known, and cp(m) is the fnnct.ion
to be de1.el'mined.
-2-
( n5 )
We wjlJ suppose tha,(, I(x' is continnous in tiJe inlerval Cl:;:V::; b
anel tbat j( (tCV) is finite in tbe square a ~ ,1J ~ b, a ~ V ~ b.
The method of NEUlIIANN gives ihen immediately the s01ution
IJ
J
+ J.
p(,v) =f(,v)
r(tGsJ.)f(s) ds
a
"rhere
(2)
anel
=JK(,vs)
b
](11
(my)
1(11-1
(sy) ds.
.
a
The disadva,lltage of this solution is ,thai it only converges fol'
ceriain values of J.. But a much better soIntion was discove1'ed by
FREDlJOLM in which the function f' (tC Y ).) is exh ibited as the ratio of
two ]Jo:\ve1' series which are convergent fol' all values of I..
OUl' first, object will be io show that tbe lat ter solution may be
cleelllCed ft'om the former in a very simple way, Supposing that the
iinite fUl1ction J( (,VV) eall be expanded in a finite series of the form
K(,v'!})
= Xl(,v) Yl(y) + X (,v) Y
2
2
(y)
+ ..
XlI(,v) Yn(,v) .
.
(3)
ii may be shown that a linea1' 1'elation with constant coefficients
t successive fnllctiol1s J(i(,vy)
exists between n
+
a,J(j+l (m'!}) - a ll-1 KV+2 (,vy)
+
ZJ = 0, 1, 2, . ,.
(-1)n-1 aJ(p+1I
+ ..
+ (-l)nKp+n+l (,DY) =
Kl (.'lJy) == J( (,vy).
(my)
0
(4)
whel'ein
anel
Thus it is evident that the series (2) is a reciprocal one which
may be representeel as the ratio of two polynornia
r(my).) =
Bo-Bl)'
+ B 2).s-., •. + (_I)'I-IBlI_1 )..11-1
+ a).:- .. + (--1)1I1X1l I.n
I-al)'
(5)
whel'e
Bo = R. l
BI
= aJ(I-!(2
B2
=
((21(1 -aJ(2
(6)
+ 1(a
B1I-l = (tn-lJ(I-CXn-2K2
+ ."+ (-1)>l-2a)~!_1 +(_I)n-l!~!
anel cOllsielel'ing' ihe limit' of this quotient 1'01' n =
immccliately Uw l'esllH of ~'ltlWUOl,l\J.
CIJ
we obtain
2. To, prove t.he l'elation (4), we expand the determinant..
48
Pl'oceedings H.oyal Acad. Amslerdam. Vol. XIII.
-3-
( 736 )
J(lVy) [(lClV I) , , , J(lVlV n )
K(mly) J(,'V/C l) , " J(lV I,'l:l1 )
J(t/]l1Y) J(XnlV I) , " ](''V1I''C n)
according to the elements of the fil'st 1'0W, and integrate over tile
variables X I /1'2 ' ,XII between the limits a and b, This gives, as has
been l'emal'ked already by FREDHOJ,l\I
b
b
b
b
j~fJ((tlJ~1
"~Il)cllVI"d'UIl = J(.'CY)JJJ(('~I
...~n)d'IJI"d'UIl
• YllJl··llJ n
lV ··t'V
1 11
a
a
a
a
a
1
a
ll
a
By l'epeating this. process we obtain
b
ff
a
a
b
IJ
b
= E('r:IY) r.JJ(('~l"lVn-l)d''/Jl"d''V1l
K('rl:'CI":'/Jn-l)dlVl"d''/Jn_l
\y·'Cl"lVn-l
'J '
a a
1
1U1"lUn-1
Pursuing in this way, and putting
b
k! ak
b
ff](('~l"'~k)
d,'V I,.d''Clc
{IJk
lUl'
a
a
we get, aftel' IJ operations
In = n! C(nK(lVY) - n! CCn-l K 2 (tlJY) +n! a ll -2K3 (xy)-,,+( -l)p -ln!KA''VY) +
b
+ (-l)pn(n-l) .. (n-p+
b
l1f
I((,'/Jt'JJ(('r: 1 'r:2 ) "K('Tp--I'r:/J)(lr l "dt'jJ X
a
lt
and, if 17 = n --1
ln=-:n![a,J((:lJy)-an_1J(2(,'Vy)+",+(-1)1I-laJ~I(,'/Jy)+(-1)nI~1+1 (lUy)]
(7) •
fi'om whieh at onee the values Bp may be detel'mined, for
b
Bp
= 1J1 =
pI
b
2.Jf
pI
a
a
K (''/J'V l '''VP) dJJI"d,vp '
Y[IJ 1",'Vp
-4-
(8)
( 737 )
}{euml'king now Lhat
X1(m) X 2 (iV) .. Xu(m) 1
..
•
•
•
.
t
.
Xl (iV Il )X2 (iV lI ) .. X,,(mn)l
Y l (iV,,) Y2(iVlI)"
}~1(''IJ,,)0
t follows tbat the firsi numbeL' of (7) is zero. This pl'oves the
(4) when ZJ = O. Writing th is l'esuH
(.(,J(l(m.s) - (,(n-1J(2(''lJö)
(-1)n- 1aJ(1I(mö)
(-l)llI~I+l(mö)
0,
~qnation
+ ... +
=
+
.nultipl)'ing br J( (sy) ds anel integrating between Lhe limits a anel
J, ,ye get
J(2(''lJY) - ((n-lJ(3(my)
'X 1
+ ... + (-1)12-1aJ~I_1(mv) + (-1)nJ~I+2(,'lJH) = O.
Repeating ihis process it is evident that equation (4) holds for
:tIl values of p.
If now n is infinite, the equation (5) lIla)' be wi-ÏUen
. ) _ D(,vV J.)
r (mv).
---)D(J.
8.
:mcl
.
a
.
(10)
a
Fol' the proof tlmt the first of these 'series converges absol u tely
fl,ucl nniform 1)' in the squl1l'e, allel thai the second con vel'ges absolutely
fol' all values of ), we refer to tlle original memoil' of FUEDHOLl\I.
4. The prececlillg method enables us a1so io obiain the coefficients
of both series in the form which has been discovel'eel by PLEl\mLJ.
Expanding in tbe same way as before we have
b
b
K
ff
a a
b
b
('~1,~~",~p)d'V2
.. d.'IJp=J((ml ,VI) (fJ( ('~2,,~p)d,'l)2"cl.'IJp
,V ",'lJ
~.
,V ,,,Vp
1.'IJ
2
p
2
a
a
48*'
-5-
( 738 )
etc.
Thus, putting
b
J
Kk (,va:) da:
== al"
a
anel integrating over Xl between a and b
p!«p = (p-l)! a1{l:p-l - (P-l)! a2 u p-2 + (v-I)! a3 «p-3 - ..
+ (-I)/l (p-l)! ap-l«l + (--l)p+! (p-l)! ap
Ol'
ap=ap-l«1-ap-2rt2
•
(11)
+ ap-3ct3- .. · + (-1)Pa lXp-l + (-l)p+lpup.
l
Rence
al = al
a2 = al u l -2a 2
I
a 3 = a 2«1-al «2
+ 3«3
ap-l = ap-2~1-ap-3u2+ap-4a3 - ' .
(-l)ppu p
+ Ctp
= ap-lul-ap-2ct2+ap-3a3 -
anel eliminating from these u l a 2
p! up==
..
..
+ (-l)p(p-l)«p_l
+ (-I)p alap-l
Up-l
al
1
0
... 0
al
a~
a)
2
... 0
a2
(tl
p-2
... 0
a3
a2
al
... 0
a3
(t 2
al
... 0
Ctp-l Ctp-2 ap-3 ... p-1
ap-l
(t
ap
ap
ap_l Ctp -2 ... al
ap_.l Ctp_2 ... al
Eliminating in the same way
al =
UI
U2
..
... 0
p-l 0
(12)
p-2 ap_3 ... 1
a p from the
lJ
+1
equations
((1
a 2 = al ((1-2rt 2
a 3 =Ct2Ql-al«2+3aa
Ctp-l = ap-2u1-ap-3C< 2+ap-4ua-.. -H-l)/l(p-l)((p-l
all
= ap-!(~I-(tp-2(t2+a1l-3 «3-.. +(-1)/la l (tZJ-l +(-1)Z+1znl'p
(_l)p-lBp+lCz+l = Kp «1-J(p-lU2+J(p-2a8-..+(-1)IJK~ap-l +(-l)I+IKlap
=ww"'"m
-6-
( 739 )
the last being one of the equations (6) we find
1)
! Bp ==
0
... 0
[(1
p
J(2
Ct l
p-l ... 0
](3
Ct J
al'"
0
J(p
ap-1
Ctp-2
...
1
11/)+1
ap
ap_1
... al
(13)
Tbe formulae (12) anel (13) agTee with those of
5.
PLEl\1ELJ.
If the kemel /( (xy) be defined thl'oughout the square so that
I (,'Cy)
J((,vy)=-(,'C--y)""
where I (IJ V) is firlÎte in the square and a
n-l
< --,
n
thel1 it ma)' be
rcadily proved thai the ileraled kemels /(2 IC .. /(n-1 are all infinite
fol' a; = 'Ij, and that the l{el'nels I~" I~I+l, ... are all finite in the
wbole square. Likewise all the integl'als
b
fI~/l (,'Cs) f(,) ds
= 1,2,3 ... (0)
(m
a
are finite throughout the sqnare.
Fot' this case it is ShOWll by POINCARÉ 1) that
still holds if the detcl'minants
1\ (''Cl
.'1]2 • •
FlmDIIOJ,l\I'S
solutioll
aip )
dl l lIJ 2 •• lVII
and
J(
({IJ Xl •• ,'Cp)
Y ''Cl' • Xp
are modified in the following way.
If by a cycle of /.; letters a, (3, r, Ó •• (.t is me::Lut 1bc pl'oclnc1
J( (u{j) J( (8r) ]( (yó) .. IC üw)
those pl'oducts, in expanding the deteeminants, mnst be omittcd which
contain cycles of less than 12 letters.
Now we wish to show that these modified coefficienis mtlY at
once be obiained fi'om those of PL}tJl\IELJ by substituting theroin
a 2 = .. an-1
O.
al
Fo!' Lhis purpose we note thaL tho equation (4) still holels if IC; bo
=
=
1) Acl. Math. Bd. 33.
-7-
( 7-10 )
l'eplnced by Cl! as ma}' be seen by. putting' y
between Cl l111c1 b.
'l'Jms
U n C&p+l -
.J
Ct.
1-
-
J.
l1nd illtegrl1ting
+ ... + (- 1)11 C&p++1 = 0 (p = 0 .1.2 .•)
pl'oves tha,t
+ a I. + a I,~ + ... is l1 l'ecipl'ocl11
((n-1 Cl p+:2
l1ncl this rell1Lion
serieEl, which ma;}' be wl'itten
al
= aJ
+
a,),
:.1
al
2
)
3
b;2
2'·
b
b
+ .. - --.:....---1I, + a
0- 1
_
11.
T1
UI
2
J... (
..
J
-
)." -
••
l)n-l bu-I ),n-l
.
+ (- l)n U n I,n
",here
bIJ
= up
al -
{(p-l
C&2 -\-
(tJl-2
aJ
-
••
+ (_1)p-1 a,+l <1) = 0.12 .. 11-1)
Ol', l1ccording 10 (:11)
D' (J.)
D (Jo)
--Ol'
-
a
"2
I. - 2
al I. -
•
D (J.) = e
W l'iting ihis equation
, + a-= ,~ + a"":),3 +
CLll'.
D (J.) e
2
J'.
2
. , •
a~
2I.
3
-
..
a-I
_11_ ),n-1
•
n-1
= 1 + (-l)n
=
I
e
/::'no
J.: _ /::'~+l J.n+~ + ... 1
n.
(n+1)!
it is evident lhat the first momber is independent of al Cl 2 •• Clll-l
If thel'efore
we have
l1nd
/::'1 0
= /::'2 0 = .. = /::'~-1 = o.
In thc Sl1me Wl1y it follows fl'om '
D (mvJ.)
= D (J,) {Kl
+ À1(2 -\- I.~ Ka + .·1
-8-
\
.
( 741 )
Ol'
ll
, .t a 2 .1."2 + a 3 ,'3.+
'11-1
. . + a _1 ,.
!ll"
2
D (my).) e
3
11-1
th at the fil'st member of this eqllation is independent of a/l J
Th ns assllming
D (,'1],1/').) = Eo -El').
+E
).2
2
).3
2/ -Ea 3.'
••
all-I.
+ .'
and
we find
that is
(15)
Having thns establishpd the l'elalions (J 4) and (15) we may conc1ude
that if
•
K (my)
= (I (,'I]Y»),
m-y"
(a < n-1)
n
fOl'mulae for D (.'1.' Y),) and D p.) where the l'oefficients are written
in PLEIIlEL.l'S form, still hold if al = 1I 2 = .. - fll/-l = O.
The same l'esult may at onee he deduced from lhe ruie given hy
POl NOAR:f:. Fol' if in::itead of Lhe eoefficient:; of FltEDnOJ,l\I we take
those of PLEMET,J and l'emal'k that the k-fold jntegral of a cJ'cle of
lc leLters alw.l1Ys gives
t he
IJ
J..ji((ufJ)
IJ
a
IJ
K({jy) .. K(tta) da ... d,lt
=JK'" (ua) dit =
a,"
a
a
thell il is evident thLl,t those tel'lllS fl'om PI,EJ\fELJ'S coefiicients mllst
be omitted which contain al a 2 • • a'1-1 Ol' wh at is equivalent, that in
these coefficients must be substituted
al = a 2 = .. = an--1 = O.
-9-