hypergeometric functions of two variables
DESCRIPTION
Hypergeometric Functions of Two VariablesTRANSCRIPT
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for the conti.g1.1Ql1S functions (9.3) . The author does not '
whether the recurrence formulas for those contiguous functions were : i:. , : , .
already disc.overed. . ..
.. . . .. :PROBLEM 1 : Find the formulas for the contiguous
functions ' : ' ' : l . . ' ' .
;- '" 'This 'problem mlght not be very difficult. In fact, the author . /
the F2 : . . , ; . l . : .. .. . / .. ,. ; I '
.(at-t)(o(- r')(C(;..1- i'+l)F2
(oc.-l, 'l/, t'',x,y)
.. (cl -'t- {'f+ .. (Cit -> p'y}F2 : . - . . ... :. t ; ! -'.. . .: l .. ' . . ' ; . - .. . . .. ....
{ c ' - 2'') cat- (f- (1-x) + c2o{'. - - a'') p iy}x aF2/ d x :. : . ..! \i :-. Y ?;,, r ::.::, i , -;1_ , ; . " j
._ "." {
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- 61 ..
~ib. .-.-
Euler integral representa'tions. The hypergeometric functions ~ I
:g.f two variables, except for .F 4 , have Euler integral represen-
. /~ations which are double .~ntegrals but very similar to that of
":;~_t.1'' ~ ~ , i , x > . .. THEOREM 10.1: We have .
' . . ' ' .
.. re~> fJ --1 ~-1 i-~-p'~1 ~ . ~ r(~)rc~) ro-~- ~) . u v (1-u-v) (1-xu-yv) dudv
1~ u, v, 1-u-:v~O I ' i
: '.1,f. Re ( ~ ) '> 0, . Re ( ~ ') "> 0, Re ( 'I - ~ - (3 ') > 0; ~.'' . .
",.j, I '~
' ;'j. .. :, :Jno. 2) F 2 ( a( t p , ~ ' , r ' 't I , x ' y)
::. :1 ... .. ' ~ ' .. J . i; .! :. : .: ; . I "; .. : ...
:,:'if Re ( fl ) > 0' Re ( ~ I ) ') 0 , Re ( l - p ) '> 0, Re ( 1 ' - 13' ) >. o; ' .
:'.{10. 3) . . F 3 (or< ~~ rc~-~-n . H u!-lvP'-1(1-u~v) -~~,~ -l Ct-xu) -~ c'1-~) ~"'dudv ... . . .. ' u,v,1:-u-v_O . ; : . ... , . . ' : .
J' Re ( p ) ') 0 ~ ~e ( ~ 1 ) :> O , Re ( ~ - ~ .. ~ ' ) > O ~ .. . :. ; . ... . . . r .
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. ~ .
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. -o
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= f (oc. m+n) rcp+m>rr F1
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~ ( "' ) r ' I' ' II > ' This completes the prcof of Theorem 10.1.
The function F4 does not have any simple integral represen-
tation such as (10.1), (10.2) and (10.3). Thi~, however, does not . . '.
. . mean that F4 does not h~ve any double integral representation.
For example, the following formula is kn.own:
, r re 11 > F4 (c
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From (10.4) we can conclude that F1 is analytically con-
tinuable and holomorphic for [C - [l, C)O)] x [C - [l, oo)]. Similarly
(10.3) implies that F3 is holomorphic for [- [l;oo)] x [C- (l,oo)].
On the other hand> we can prove the following theorem by using (10.4).
THEOREM 10. 3: We have
(10.5) Fl (OC, ~, ~', 'lf, x, y)
(1-x)-p(l-y)-/J' F.1
( -o< > p, p ', '(, x/(x-1), y/(y-1)) -ct
.. (1-x) F1 (ct, J- ~- ~, p', 'l, x/(x-1), (~-y)/(x-1)) ~ .
.. (1-y) r1 (Q(,~,Y-(3-8',r, (y-x)/(y-1),y/(y-l})
..., (1-x) ~-g( -p (1-y) - ~IF 1 ( t - 0( ' r- p - p I ' fl' , "( , x, (y-x) I (y-1) ) -ta l--13' (1-x) (1-y) F1 (r-oc,~, ~-~-{J',, {x-y)/(x-1),y).
Proof: This theorem can be proved by changing variable u
respectively by
u = 1-v, u "" v/[ (1-x)+vxJ, u ... v/[ (1-y)+vy] . ,
u c (l-v)/(1-vx) and u = (l-v)/(1-vy) .
Exercise 1. Prove Theorems 10.2 and 10.3.
It is easily verified that the following six changes of
variables:
u D 1-u' , v ... v' , u u' , v . 1-v' , u 1-u' '
v *" 1-v' , u zo 1-v' , v ""'u' ,
u ... v' ' v -1-u' I u c lv' , v -lu'
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map the square . 0 " u, v ~ l . onto itself and do not change the
form of the integral (10 .2). By using these transformations we ca ri"' :;~~
. _pr.ove the :following theorem.
: THEOREM 10.4: We have . ' , ,. -- .
(10. 6 > r 2 < , ~, 13 ' , r , r' , x, y > (1 ... x)-"'F2 ("', r-p, f, "I, 'I', x/(x-lL y/(1-x)) (l-y)~F2 (C(~ p, 1'-p',r, 't',x/(l~y),y/(y-1)) : (1-x..:y):.-F
2(0c, y ;..13, y ! - ~, '/, 't!, x/(x+y-1.), y/(x+y-1) )
- (1-y) -~If 2 (: > .. r .-p' ,-p, i' >:'( ''y /(yl)-; x/ (1-y)) . (l-x} ~4 ~'-2 ( .2 P', 'j- P; '/ 1, '(. > y'/(1.:.x) > X/ (K-1))
. . -Cl( I I t ' .. . ., :. . ... .. . - .(1-x-.y) . F2 (ot, -(3 . > r-_ ~> .'t ,1,y/(x+y ... l),x/(x+y-l)).
It can be,_, shq~n that there .is no change of variable's of the
form:
A '.u' + B 'v ,. + C u = Au'+ Bv' + C
A"u I + B"v I + c" . . v ' . = Au' + Bv ' + C
which maps the triangie u; v~ .. 1-u-v ~ 0 into itself and which
does not change the f~rm' of the .. int:egral (10.3). The~efore, it is
difficult to find any rtransformati on .. formulas" for F 3 '. which are similar to (10.5) and (10 . 6).
.. ; .
Let us nex't find some relations between contiguous functions.
To do this, consider the . representation (10.4) of F 1
Raisi ng cc , by one we obtain
rc'+n re y -~-1> r F1 (((+1, p, P', "(, x, Y> 1 . .
J ot f -2 - A - ~ t a u (1-u) (1-xu) t' (l ~yv) du 0 . .
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On the other hand, lowering ~ by one we get
::'., r c ' > r < t - o( - i > , . rci-1> Fi
f 1 .
~-1 t--2 - - I '"" u (1-u) (1-xu) f. (1-yu) /3 du
o If we put
we have
r 1 . > (IX ) r (1 - ~) F ( t:< (.f ' /.t I ) a f u (u) du ' . r( ) 1 ' r' IJ ' ~ ' X' y
0
1 f(ij.+l) r((--l) F (-'+1 f'. a.' " . ) "" [ u(l-u)-lU(u) du r < ~ > i .... . , ,... , ,~ , 11 , x , y o
':}. ;
.: . . : ' 1 ' f 1 ' -Io U(u)du+
0 (1-u)-
1U(u)du
f(ot) CCX-ot.-1) F (,..., A a' " 1 ) ( 1(1-u):-lU(u)J.u r < J -1 > 1 V' ' , ... ' j.J ' u - , x ' Y = J, 0
and hence 'i:
1 olF1 (ot+l, ~ ~ ~, 'J, x, y) ~ (~ --l)F1 (O{, p, ~', i, ~, y ) - ( ~ -1) Fl ( o< , p , ~ ' , ;r -1, _x , y) = 0
In this computation, we used the. identity r(x+l) = x r l. Then
lim U(u) -= 0 , . lim U(u) 0 u,+0 U' 1-0
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Therefore, integrating both hand members of (10.7) from 0 to 1,
we get
(ci -l)J 1
u-1U(u)du+ (ot.+1-~)J. 1 (l-u)-1U(u)du 0 0 . .
r1 . 1 p 'y Jo (1-yu) - U(u)du '"" (1 . . l
+ px Ji (1-xu)- U(u)du+ 0 .
Thia. means that
gral representations . In the proof of Theorem 10.1, we expanded
-oc O( -ot' the kernels of integrals (1-xu-yv) and (1-xu) (1-yv) into
the corresponding double power series in u and v. If we expand '
these kernels into series of different types, . we shall obtain some
other expressions of Fp F2, F3 in fonns of certain series. We
shall present such an expression for Fl. In (10.1), change
variables by
u ... s(l-t), v "" st . This maps the square 0 ~ t, s ' 1 into the triangle u, v, 1-u-v ~ o. If s .. O, then u ~ 0 arid v ; 0 for ' , .' . ' . 0 ~ t ' 1. Hence the side: s = O, 0 ~- t ' 1 of t:he square is mapped onto a point': u ... 0, v = 0 . Otherwise. the mapping is one-to:--one. (Se~. Fig~
I , ,I ; , '
10.1.)
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t
u
Fig. 10.L ;i, ..
. t
rep' >r
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f(p)r(p')r([-(3-p') F = i (o
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;?{.. .: Barnes integral ~.:::;:;. ~ .;>tlLl..aLlOClS . In Section 4 (Chapter I) we
iti~e,ved that the function F( o< , p, ~, x) admits an integral repre -;:)~~q,~ation of the form
?\Hr:; . ,'.;if; 't .; 0, -1, -2, , where B is the path given by . ');:~!\
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- 1 r . r.rrc~'+t> re >< >t 21ti BF\+t,p,~.x) r(t'+t) -t -y dt' ) rC')f(6'> ( , , ) (11. 4 r
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. r(oc+t)C(~) F(o
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(11. 5')
( 1 )
2 s s r(ot+s+t) r o(' , p, t1 t )x O( y -o< 'F 2 (o
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(11. 7) F(o
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In this manner, we obtain
. Res +cs> = r 1 a nd larg(-x)I < 7t . ' . . ( ' To prove this, it is sufficient I ."._!
to estimate .. .. ' ' ' ' : :' ' . . -
:_ rclJ(~(~r!)+s) rs~.
on OR. We can not use ' stirli~g Is formula dir~,ctiy~ because
Stirling's formula is not ,valid in a sector 'larg{~)-~l < f. . To , 1 ' ; ~ ' ; , l_ .' , , , ,',; :' I
1
~ , .!
~void this difficulty, ' iet us .use the formula .. :1 ! .. _ .. , . . - .. ' . ' ~~- : . ._.~ .. . .
... "IC . r
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of 'f'(s) on DR.
Exercise 1. Complete the proof of Theorem 11.4. . ;
At :.the:. ~nd of Se~tion 5, we mentioned that t:Wo functions
(-x)F(' ,9'+1-()' ,o t c -p -n, t - -o('-n, t - -p' -n. For example,
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R f(ot+s)C(oc:'+t)r(G+s)C([i'+t) f(- )r(-t)(- )s(- )t es rci+s+t) s x y s = o( -m {( ~ . . ta-~1 -n ~
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J~;~ ;. Systems of partial differential equations.. The function . .. .. . . . . . ' - ... .. .. . .
:'.rt~: , ~, r. x) satisfies the differential ~quation
x (1 - x)y" +: [ t .- ~o( + ~ + 1):2C.l.Y ' ; .:. .ot..p y .. ~ 0 .; ' ' ' '.... " " ,. I ; - ., ' . ,. ' ' ~ : _,
11; ~.I!: . ... ' , , ' ': : : I , , , '
ft.)}~ hypergeometric functions Fj (j ' = 1, 2, 3, 4) also satisfy
j1y13_tems of pa,~tial . diffe.rentia.l eq.up:;:ipns . ,,,._., ... .- : .: It ' . . . .
. i :
: THEOREM 12.1: F1
, F2 , F3 and F4 satisfy respectiyely : . .'. . j . : ~. I .' , ,. . ~ . ., : . .
:,'following systems of partial differential equatfons ': . ~ . :: .: . . :
'.~c:i2 ~ 2 > ,.:::\:,;,_.'-' .
{ : ~~ ~: ~= ~~ :;; :: ; : ~::~:::: ~ ~ ~ ~~~ ~:~i~~ ~\.'o . { x (1-x)r~~ys+( 1,-(ct+~~l)x] ~ .. -~yq : .. ~~z";.: '({ < .~."_
y(l-y)txys+[ t-(6'+p'+l)y']q-~xp-'~' z = o ~,
{
x(l-x)r+ys~i'-
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) . (oCm+n+l>(~.m+l) r2 o, .
. .. . , .~2 .. - 1ce+r+ o() F 2 "'.' .o ,
{
8(B+t+1-l}F3 - x(0+oe)(8+p)F3 = 0 ,
. ,,. ,(8+t+r.-1)F3 - :y(g>+oe')('+ 1J'F3 = O ..
ece+r-1>F4 -xCB+,+Q(}(8+,+ tJ>F4 ... o ,
'fCCf+ '-l)F4 -y(8+'+~)(8+t+P>F4 ""o . From these the systems (12.2), (12.3) and (12~4f can be easily
derived. ~ .. . : I. "
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81 . -.
~;: : . Systems of total differential equations. Before '~tudylng the
;~i: :. hsystems of . par~ial ~~ff~;enti~l. equafions which . Fj :: satisfy, : ~e
[:~h~ll explain some aspe_cts of total differential equations. ;~{ ~ .. [.~i\,; ~i' Let us co_nsider. a .system of tota~ ,differential eqli.ations : of
;t~~~ form:
~zj = .~j '.~'.~ ,z1 ' . ~- ,zn>,~ + gj (x,y ,~1 , . ,~zn~~-! . . . . . ... . . . . - . . .. . .,.. (j i,l,,n),
,.
,~~'.~~ere f 1 ; .. , f 0 , g1 , , g0 are functions of x, y, z1 , ;~~0::~ ~hich ai;e .. define,d .and co~t;inuously ._differentiable in a domain ::i~: ' n+2 ~~;.~- contained in ,R _If w~ .. C?OQ.sider ; x,:_: Y - ~s independent ~#"~ \t:.~riabl~-~: and ,~ 1 , : , zn as : dependent variables (i. e ; unknown
~'.:tti~ceicms of. ;_ ~, y)., ' then the system. (13 .1) is equivalent to the . . f,~~~: :t . :system . -. :, .:: r= ..
.:- . ~ . ~ , - _i :. ~ ....
. . , lzj/ax == . fj (x, y, z1 ,_ . ~, z0 ),
~zj/'t>y ,;;. ; gj(x,y~ z1; ~ zn)
. ;, ; . . : ... .. _i, ,: .. :
... ~. ; : ! zj .. 'j (x,. y) .... (j .a:; l ~ . i . :. , ' n)
I 0)
. ( . ~ : ' ,. . l 1" ,. l '
:~.~hich is defined and twice co~tintlousiy differentiable in a domain' ;;,,; . '' 2 . :,D c R are independent ilii~j; :of the order of i .differ~nti,.ation with : respect to x . arid "; O
y: i.e. ;
. i).2 'f ~ ~ : ' ~2 'f j .~ x ~ y .. a y a x (j - 1, n)
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From (13.2) we derive
(j = 1, , n).
Hence
and hence n ._, n
(13.3) ~fj/"dy + Ei_gk 'd fj/dzk "" ;>gj/ax + ~ fk dgj/dzk
(j = 1, , n).
This condition holds for (x, y, ".
there exists a unique solution of (13.1) which is defined and
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.' .twice contiriuously differentiable. in a neighborhood of (x0
, y0
)
~nd satisfies the initial condition
.(13 .4) (j 1 > , n)
DEFINITION 13.1: A system of total differential equations of
the form (13.1) is called completely integrable if (13 . 3) is satis-
. tied. .. :~
!..
We shall now consider the case when f. and g. are all J J
holomorphic in x, y, z1 , , z . n
THEOREM 13.2: Suppose that fj and gj are holomorphic at
,
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Moreover if we put
w ... (.() 11' , ln [~.11 z = . Q = ....... . z n
we can write the system (13.5) as
dZ = J2. Z
The condition of integrability becomes
.: zk[-af.k/jy+ ~ f 'kghk] = t zklag .k/ax+ t_g.hfhkl. ks:1 J h=l J k=l l: J h=l J J
j = 1, n
Since these are identities, we obtain
(13.6) n
~fjk/;,y+ ). f.h~k = h=i J
n ()g .. ktax+ _Lg.hfhk
J h=l J
j, k = 1, n .
As it is well known, solutions of linear ordinary differential
equations exist in an interval or a domain where coeff.icie nts of
the equations are continuous or holomorphic. The same fact holds
for a system of linear total differential equations. We shall
state such a result for the holomo;rphic case.
THEOREM 13.3: Suppose that fjk and gjk are all holomorphic
in D and satisfy the condition (13.6) there. Let l
zj .. 'f j (x, y) (j "" 1, ... n) , be a solution of (13.5) which is holomorphic at (xo, Yo). Then
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>.1'f111 ..... ,'i'n starting at
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are analyticatty continuable along any Eath in D
It.is clear that the set of all solutions of a system of
linear total differential equations is a vector space. Since
solutions are uniquely determined by their initial values, the
dimension of this vector space is n
iHEOREM 13 . 4: The solution space of (13.5) is an n-dimensional
vector space.
Consider n ~olutions of (13.5): ~.'I .; ' , -1 J. : ' : .. ' - ~ . '
(j , k "" 1, p n)
and the determinant obtained from these solutions:
,, 11 'f ln
..
In the sawe way as for linear ordinary differential equation~) we
derive
n . n dA ss ( L_fjjdx + ~g . dy)d
jml j""l JJ
or n n
d(log A) ... ,L.fjj dx + L,gj. dy jml j..:l J
THEOREM 13. 5: We have
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. .
. (13. 7) J(x,y) n n
A(x ,y) ... '1(x0,y 0) exp[ L fj. (s, t)ds+ 2:.g .. (s, t}d:t]; 0
,y0
) J=l J J=l JJ
We have completely similar theorems for the general system of
total differential equations
m ~zj ~ : 2',fjk(x1 , ~ ,x ,z1 ,. ,z )dxk ' j l, ,n . . . . kl .. m n .
Exercise 13 . 1: Show th~t, if we write' ' system (13 . 5) in the
. form
dZ "" Jl Z ,
then the integrability condition (13 .. 6) can be written as
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Transformation of the differential equations for F. J
into -systems of total differential equations. We saw that the systems
!: .
:,~f partial differential equations satisfied by F j are ~ritten in
;the form
{
Al r + A2s + A3t + A4p + Asq + A6z
Blr + B2S + B3t + B4P + Bsq -i- B6z
= 0'
.where Aj and Bj are holomorphic in . x and y. A3 --= B1 = 0 for
F1 , F2 and F3 . These equations are solv~ble with respect to r
' and t so that we obtain
where a. and b . are rationa~ functions in x . and y. J 3 -
Let us differentiate-- the first equation with respect to -y : ' -
;J r I";) y . = a 1 J s I ~ y + _s ~a 1/ ~ y + a 2 d p I 'd y + p d a 2 / 'd y I I
Thus we derive . ' ; ' .. :(14.2) ,-s/';)x ;1- a 1ds/'dy ... . (da1/;;)y+.a2)s + a 3 t + . C>a2t~y)p
+ (aa3/dy+ a 4)q + (7'a-,/'dy)z -'
Similarly, by differentiating the second equation of (14.1) with
respect to x, we obtain
(14. 3) b1 ~s/"ax+ ~s/~y = (()b1/dx+b3)s+b2r+(db2/~x +b4 )p + (3b
3/ax)q+ (;)b4/ax)z.
-
. the left-ltand members .. o{ (14 .2) .and (14. 3) are linear forms in
.} _s/":i!' . and . "'ds/''dy ~ The determinant obtained from the coef-
.ficients of these linear forms is . . .
We distinguish two cases:
Case I : 1 - a1b1 0 ,
Case II: l a1b1 ~ 0
Case t: tri th~s case, we can eliminate dS/';>x and ' 'ds/"'Jy from
(14.2) and (14.3) at the same time to obtain a linear relation
between r, s, t, p, q, z :
(14.4) c1r + _c2s + c3t + c4p + c5q + c6z O. Suppose that from (14.1) and (14.4) we derive
whet~ p and olj' . . J definition w~ have
r O(lp + cC.2q + 3Z '
8 ~1P ~ ~2q + .P3z ,
t f 1p + 12q + Y3z ,
Yj are rational functions in x and y. By
~~/ax P. , ~ .z/3y ~ ,
and herice
Ctp/t)x .
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Thus we obtain a system of total differential equations:
(14. s) { :: : dq =
p dx + q dy ,
(ot.lp+ ol.2q+oc.3z)dx + (~lp+ ~2q+ e3z)dy,
( ~ 1 p + $ 2 q + /3 3 z ) dx + ( ~ 1 P + ;r 2 q + ;y-3 z ) d y
In vectorial notations, we can write this system as
z 0 J.x dy z
d p ~ .x3dx+~3dy ~1dx:+P1 dy ~dx+P2dy p q p3dx+t3dY ~1dx+a1dy ~2c1x+12dy q
In general, this system may not be completely integrable. If this
system is completely integrable, then the solution space is a
three-dimensional vector space.
Case II: In this case, we can express -as/ ~x and ()s/6y as
linear forms in r, s, t, p, q, z:
c1r+c
2s+c
3t+c
4p+c
5q+c6z,
d1
r + d2s + d3 t + d4p + d5q + d6z
where coefficients are rational functions of x and y. Inserting
: (14 . 1) into the right-hand members of these equations, we 6t;-L
f . ~StdX = o
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dq/C>x = s ,
as/'Jx = 0(.1 s +otzP +
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System for F2: The function F2 satisfies (12.2). Hence
a = 1
y/(1-x) , b1 = x/(1-y) ' . 1 - a 1 b1 -/J 0 . .
System for F3: In this case, from (12 . 3) we derive
al ... -y/x(l-x) ' b = l. -x/y(l-y) ' l - a 1b1 t 0 . System for F4: From (12.4) we derive
a1 = 2y/(l-x-y) , b1 "" 2x/(l-x-y) '...,. 1 - ~1b1 1= 0
This shows that (12.2), (12.3) and (12.4) belong to Case II.
Sy using the procedure of deriving systems of total differential
equations, we obtain such systems for F1 , F2 , F3 . and F4 . It can
be verified also that these systems for F. are all comp.letely . . J
integrable. The proof of this fact will be left to the readers .
. :?hus we come. to. th.~ fo11owin~ conc1usio~.
THEOREM 14.1: The dimension of the sol~tion space of the .: ::_ .
l_,::: . . system for F1 is three, while the dimensions of the s~lution
. ,: .. . _.,.:, .. . . ' spaces of the syste~s for F2 ,.F3 . . =- .. . .
and '
are all four~ .
Let us examine the system for , . F. in more detai,1. - Differen-. . . . . . .:. l. '.>
tiating the first equation of (12.1) with respect to y we obtain
. ,or r I ~ :
.x(l-K)cls/C>x ,+ y(l~~). c}s/Jy! LY+l~(o(+p~2)x}s .-~yt - (oC+.l)~q = 0 . ,' I '. . '. , . - - ' ' ' '
Similarly, differenti~ting the second equation of (12.1) we obtain
.y(l-y)~s/~y+x(l.-y)as/ax+ [+1-(0(.+~'+2)y]s - ~'xr - (+l)p'p = o.
To eliminate as/~x and ds/~y from these two equations, we
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multiply .the fir~t equation by 1-y and th~ second equation by
1-x and we subtract one from the ocher. Then we get
[ (+ 1) (1-y )- ( +e+2 )x (1-y) -
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(yx)s by ~q-~'p, we obtain the third equation of (i4.7).
Exercise 1. Prove that
(x-y)s ~'p +pq = 0
has a solution of the form
where f is an arbitrary function. In particular, if we take '1-( ) = (cx,m) l m
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- 94 -
This implies that any solution of the system for F1 is regular
analytic in
c ~ c-{x. = o~ u {x = 1} u/y = o~ v{y = qu{x = Y) However, solutions are in general multiple-valued functions .
y
x /
In the same way, if we write the systems for F2 , F3 and F4 in
the form
r = a 1s + a2p + a3q + a4z ,
t = b1s + b2p + b3q + h4z
~s/ ~x = '1 s + o
-
- 95 ..
y . y . y
I\_
Remark 1. A natural compactification of the complex plane C . .
is the Riemann sphere which is the unit sphere of real dimension
two, i.e. s2 This can be regarded as the complex projective line
P1 . On the other hand, C x ( has two natural compactifications.
The projective plane IP2 is one of them. The other is e1 x 1P1
Both of them are compact c.omplex manifolds, but they are not bi-
holomorphically equivalent. This means .that there is no biholomor-
phic mapping from one into the other. They have, however, a common
modification and hence they are equivalent in a certain sense.
Since coefficients of the systems for F. are rational in x and J
y, these systems are well d~fined in the compactifications of
( ~ . In various cases, a choice between two compactifications of
C l(. C is not a serious problem.
Remark 2. The Gauss differential equation
x(l-y')y" + [-(+~+l)x}y' - 'Jlpy = 0
is ~educed to a system
-
- 96
( 14. 8) . f yz'' z/x , l :c _ xi3 Y + (-1_-_r + r-1)(-8-1] z
x-1 x x-1 '
if we put
z = xy' .
Note that
z ' xy" + y '
and hence
x(l-x)z' = x[x(l-x)y'I + (1-x)xy'
a:o(~xy - LY-(o
-
- 97 -
+cyaz/dy] = yt + q . . *Y
Therefore, in a way similar to that in the case of the Gauss equa-
tions, we derive from (14.7) the following system:
z z
d ~)z/)x = {A~x + Bgy_ +C dx +ok-+Ed(x-y)) x';>z/Clx , A y . x-1 y-1 x-y
where
y ~z/ ~y y ;Jz/ 'dy
A
0
0
0
O
0
0
E m 0
0
1
1-1+13'
-(3'
0
0
0
-~ . (!>'
0
0
0
0
0
0
, B
D =
0
0
0
0
0
0
0
0
0
0
l
-~
1-)'+ ~
0
0
-p i-oc-jl' -
,
,
Note that the coefficients of this system have simple poles at
x = 0 , x = 1 , y a 0 , y ... 1 and y .. x
-
98
CHAPTER III
Monodromy Group of the Gauss Differential Equation
15. Euler transform. In Section 3 (Chapter I) we derived the
Euler integral representation of the function F(oc , p, y, x):
(3 .1) 1 _ rcr> f "-1 _ r-~-1 _ --cc
F(o(,e, '(,x) - r(~)r(l-~) Ou (1 u) (1 xu) du.
The kernel of the integral is -ar
(1-xu) Dropping the constant
factor r
-
- 99 -
i . 4'-1 L(y(x)) = L((l - xu) ) f(u) du, c where
L((l~~m)~-l) = x(L-x)[(,\-l)(A-2)u2 (1-xu).\-J] . ,\ 2 + rr-(O(+~+l)xJ[-()\-l)u(l-xu) - ]
' .' , ~ ' A.-1
.. al~(l-xu) .
Let us write L((l-xu)A-l) in the form
.\-1 A-3 L((l-xu) ) "' (1-xu) H(x, u, ...\),
where
H(x,u, ~) 2 2 = (~-1) (~-2)u x(l-x} - (A-l)u[ t-(oe.+p+l)x] (1-xu) - 0
-
- 100 -
L(y(x)) = o(l (l-xu)-2 [-(cc.+l)xu(l-u)+ (1-xu)(~u-~)} j(u) du c
i --2 4-l o( [-(oc+l)x(l-xu) u(l-u)f(u) + (1-xu) (tu-f1)
-
- 101 -
(i) the path joining 0 to 1 if Re~> 0, Re( - {J) :> O;
(ii) the path joining -oo to 0 if Re~> 0, Re(o O;
(iii) the path joining 1 to +oo if Re( -(J)> 0, Re(o0 .
On the. other hand, Jacobi showed that the following three
curves satisfy our requirements under certain conditions on the
parameters:
(iv) the path joining 0 to 1/x :
(v) the path joining 1 to l/x ;
(vi) the path joining oo to l/x .
In order to find the conditions on the parameters that these three
curves satisfy our requirements, we must examine not only the
cor.dition (15.4'), but also the assumption that
y'(x) =Jc
y"(x) = L ~ ~-1
ax (1-xu) '(u) du '
. In deriving the conditions on ~ , l and C, we actually assumed .that the order of integration and differentiation can be inter-
changed. This requirement must be satisfied by the three curves
(iv), (v) and (vi). Note that we have
d ff (x) J f '(x) ...L d; ja . F(x,u)du = . a i)x F(x ,u)du+ f' (x)F(x,f(x))
Therefore, if F(x,f(x)) ~ 0, we get
-
- 102 -
. .
d Jf(x) Jf (x) ~ d; . F(x,u)du = ~x F(x,u)du
a . a-
If we have F(x,f(x)) = 0 with f(x) ~ l/x and F(x,u) = . ~-1
(1-xu) (u) , then the formula for y' (x) is verified .. On the
other hand, if we have F(x, f(x)) = O for f(x) = 1/x and 3 . ~-1 .
F(x,u) = ~ (1-xu)
-
- 103 ...
proved the following result:
THEOREM 15 .1: If we put
(15.10) U(x,u) u 6-1 Cl-u) 1-~-l(l-xu)-
then the following integrals are solutions of trw hypf'rgeometr i. c
differential equation (15.2):
(15.11)
FOl(x) t
= i U(x,u)du 0
. so F~0 (x) = 00 U(x,u)du
F100(x) = J1tA U(x,u)du fl/x
F 1 (x) = J. U(x,u)du .Ox O
~1/x
F i(x) = U(x,u)du tx-
if Rep >0, Re(!-/3) > 0 ,
if Re,,>O,Re(oO '
if Re('(-{0>0, Re(oO,
if Re~ > 0, Re Q( < 1 ,
F1 (x) = J: 00 U(x,u)du if Re(c.X+l-,r) > O, Re O. Then x is in
the upper half-plane and l{x is in the lower half-plane. More
precisely, we make the convention:
-
0
-
- 105 -
-- branch of U(x,u), it is sufficient to det~rmine the argument of
~ each factor u, 1-u and 1-xu of the function U on th~ paths
of integration. We shall. fix those arguments in the following
manner:
{i) For F 01' arg u = o, arg(l-u) = O, - 7t ~ a rg ( 1-xu) ~ 0 ; "
(ii) for FooO' arg u ,., 'lt', arg(l-u) = 0, 0 ~ arg(l-xu) ~ 7t ;
(iii) for F loo' arg. u = 0, arg (1-u) :.: -'It", -7t ~arg(l-xu) ~ 0;
(iv) for F ol' x
-7t ~ arg u ~ 0, o-a arg(l-u) "'1t , arg(l-xu) = O; (v) for F 1, -1t ~ arg u "' 0, 0 ~ arg(l-u) ' 1l7, -lt ~ arg(l-xu) ~ O; . 1-
x
(vi) for Fl , -1t ~ arg u ~ 0, O" arg(l-u) ~ 1'C, arg(l-xu) = - 7t . -oo x
As we have shown before, the integral F01
(x) is the Euler
integral representation of the function F(, p, r, x) multiplied
by a constant:
rep> r< r-A > re r>
In other words, we have
F () = rcl}>rF 01 x r < "> - , ,.. , 1 , x If we make the change of variable
u .. v(v-1) -l (1 ~ v ~ O) ,
the second integral F~o becomes
So . -1 ~-1 - , -1 r-~-1 -1 :..' -~v
F040 (x) = [v(v-1) ] [l-v(v-1) J [l-xv(v-1) ] 2 1 (v~l)
-
- 106 -
Let us suppose that arg v = 0 and arg(l-v) = 0 as v goes
from 0 to 1. Then, since arg u =7t, we must have
-1 . 1ti. -1 v(v -1) = e v(l-v)
and hence
[v(y-l)-ll ~-1 ... e1d..((J-1) "~-1 (i-v)-S+l .
On the other hand, arg(l-u) = O, -1 arg(l-v) = 0 and 1 - v(v-1) a (1-v)-l imply that
.Cl - v(v-1)-11 cr-s-1 "" (1-v) r+p+1.
If we suppose that
(15.13) 0 ~ atg[l - (lx)v] ~ 7C,
then we get
-1 -0(. -oe. 0( (1 - xv(v-1) ] = [1 - (1-x}v] (1-v) ,
since
-1 . . ._l l - xv(v-1) = (1 - (1-x)v] (1-v)
and
0 ~ arg(l-xu) ~ 1c
The assumption (15.13) is justified by the convention: -1t. <
arg(l-x) < O. Note that we assumed at the beginning that 0 <
arg x < TI:. Thus we have
J 1 1Ci
-
.. 107 -
1d(~-1> rca > r F000 (x) = e r(Q(+p+l-) F(ot, (3,o rn-"'~ -1& =e . rp-+l) (-x) F(~-)'+1 , ~,/J-.+1,1/x)
. . 0
) -1 ~-1 -1 Y-~-1 -ct. 1-x F 1 (x) = [x (1-(1-x)v)] [1-x (1~(1-x)v)] [l - (1 - (1-x)v)] (-7)dv 1- 1 . x .
Jl -1 . /3-1 ri -1 'r-P- 1 -CK _:1
-= 0
[x (1-(1-x)v)] [e x (1-x)(l-v)] [(1-x)v] x (1-x)dv .
e id. f 1 xl-Y (1-x) r-o
-
- 108 -
e7ti.(-p-l) f(l-~)r(r-O)(l-x)r-~-~F(Y-~ -~ 1-~-~1 1-x) rcr--~+_1> . , ' ,
5 -1 -1 (3-1 -1 -1 (-~-1 -1 -o( -2 -1 . F 1 (x) = (v x ) (1-v x ) (1-v ) (-v x )dv -~ . 1 . x f,
1 -1 -1 (3-1 Jti -1 -1 r-p-i -ti . -1 -at -2 -1 a: (v x ) [e (1-vx)v x ] [e (1-v)v ] v x dv 0 . .
f,1 t-~ ~~ 1 - t-~-1 u(t+oe-p-1)
x v ("v) (1-xv) e dv 0
~ eiti.(r+Dl.-~-l) C(Q{+~(I~r51 -0(> x1-~F(p-r+l,+1.;.,2-t,x).
We shall summarize these results in Table 15.1.
The integral of the form (15.1) is transformed by the change
of variable
u = l/f
into
(15 .14)
where
'l'< 5) .= -(~, ~ )1-A t
-
109 -
is called the Euler transform.
The integral b . . .
J 8-1 ~-(1-1 '-ot a u (1-u) . . (1-xu) du , \ where a, b = 0, 1, 1/x, &0 , is transformed by u = 1/5 into
d . .
J ~-r r-~-1 -~
const. c s (1- s) (x- s) ds where c, d = 0, 1, x, oa . From this we derive the following
theorem.
THEOREM 15.2: If we put
then
(15.15-1) f: ECx, S )d5 = const. (-x)-"F(o
-
.. 110 -
are all solutions of the hypergeometric differential equation
(15.2).
Supposing that Im x > O, taking the paths shown in the
figure given below and fixing branches of E(x, 5), we can verify
the formulas (15 . 15).
-
.. 111 -
16. Connection formulas and monodromy group of the Gauss dif-
ferential equati~ Suppose that
(16.1) 0
-
R
We sha~l fix a
this branch is
- 112 -
/ /
C1
branch
-r 0
ul (x.,.., u)
single-valued .on
r
of
CRr
' ' '
1-r
the function U(x,
and its interior.
u) so that
This
condition is equivalent to the condition that arg u, arg(l-u)
and arg(l-xu) change cbntinuously on the curve ~r Hence u1 (x, u) is uniquely determined by the conditions:
(16.4) arg u =it' , arg(l-u) = 0 , 0 ~ arg(l-xu) ~7t
on the segment c1 Under the assumptions (16.4), we observe that
(i) on ~l' arg u decreases from 7C to 0,
(ii) on c2'
arg (1-u) starts from 0 and comes back to 0
after taking negative values,
arg(l-xu} changes continuously from a positive
value to a negative value;
arg u = 0, arg(l-u) = 0, and arg(l-xu) varies
between -1t and 0 and takes a negative value
at u 1-r ;
(iii) on 12, arg u = 0 at u = 1-r and l+r, arg(l-u) changes from 0 to -:rt
-1t: :.;; arg (1-xu) ' 0 ;
-
. 113 -
(iv) on C3, arg u ::: 0, arg(l-u) = -'7t, - 7t 1i: arg (1-xu) ' 0 ; (v) on r-3, arg u changes from 0 to 7t and comes back to
the initial value (i.e. 7t. ),
arg(l-u) changes from -11: to 0,
arg(l-xu) increases and arrives at the initial
positive value.
By virtue of Cauchy's theorem, we have
i ul (x, u) du = 0 . CRr
Let r tend to zero and let R tend to infinity, then
and
and
lim J . u1(x,u)du = 0
1j (j = 1, 2 , 3)
lim f u1 (x, u) du = F000 (x) , cl
lim I ul (x, u) du = F 01 (x) , . c2
. lim ( u1
(x , u) du = F ~ (x) le . J.O> . 3 .
Thus .we obtain the following relation:
F000 Cx) + F01 (x) +. F100 (x) = o
Consider next three integrals F 000 , F 1 ox case, we shall use the following curves :
In this
-
cl :
-r-1 : c2 :
~2
C3
lJ :
- 114 -
a line -R ' u ~ -r, where R > r > 0 , i9
a circular arc u = re , where - 1t~ 6 ~ arg(l/x) < 0 ,
a line u :;;; f exp(i arg(l/x)) , r ~ f ~ lxt-1-r , .J -1 i6 -1 -1 .
a semi-circle u = x +re , -n:+ arg(x ) ~ e ~ arg(x ) >: . -1
a line u = f exp(i .arg(l/x)) , tx l + r ~ f" R , -i0 -1
a circular arc u = Re , -arg (x ) ~ 6 ~ 7t:.
Determine u1 (x, u) uniquely by taking
-1C < arg u,.. arg(x-1) < O ,
0 < arg(l-u) < n:..
arg(l-xu) . = 0
on the line segment c2
. Letting r -> 0 and R --> +oo , we
obtain
and
-2rcift F + F + -2.1tiotF 0 e ooO .l. e 1 = . ox xi)()
-R
>,, 1/x . "" ...
1"3 "-
Similarly we derive'---.._. ____ _____ ,.....
F 01 + F 1 - F 1 = 0 lx- 0-
1r
Rlxl/x
:).
-
Tab
le 1
5.1
inte
gra
l arg
u arg
(l-u) arg
(l-xu
) tran
sform
ation
id
en
tificatio
n
---------1
. F 01 (x
)'"' s 0
0 [ .;c, OJ
r(l3)r(
-ft) F(ae ~
y x
) r
' , ,
0 0
F.,0 (x
) ~ 7
t 0
[ 0' 1t]
um
v/(v
l) e'1
ti(i'l) f
} (-x)-oCF(ot,ot-1+1,_~+l,1/x) -
7t
1 -
---
1/x
..
.. F l (x)=
= ~
[-7c:,OJ
[ 0 ;n:]
0 u
=v
/x
-7ti~ r ro-ocL rrn+1f-r+1) , , -= e rroc+~+fl'-r+1) Fl(.,~' f3 ,ot+~+ ~ -r+1, 1-x, 1-y).
As the first integral (18.8-1) has six expressions, (Theorem 10.3,
p.65), each integral has also six expressions. Therefore, alto-
gether, we have sixty expressions of solutions of (12.1).
Remark 1: The integral (18.3) is transformed by the change
of variable
u = 1/v
-
?able 18.1
integral arg u arg(l-u) arg(l-xu) arg(l-yu) transformation
1
~ 0
0 0 [ -7r",0] [-7t,O] .
0
~ 1t 0 [0,7r] [ 0 .7t] uv/(v-1) IO
00
) 0 -~ [-7t, OJ [-7(,0) u l/v 1
1/x
t [ -'Jt, O] [ 0 ,7r] 0 [0,1t] uv/x l/y
~ 0
[-7,0) [ 0 ,1t] . [-7t,O] 0 uv/y
l/x
~ [-7t,O] [ 0, 7t] [-1t,0] [0,1t] u = l/[x+(l-x)v] 1
l/y
5. [-Jt,0] [ 0 ;rr.] [ -1t,-O] [ -lr, OJ u = l/[y+(l-y)v] 1 00
} [-1t,O] [0,7tl -1t [O,ir] u l/(xv) l/x co
~ [-'JC, O] [ 0 ;re] [ -1[,0] -1t u l/(yv) 1/y l/y .
it [ -7[, 7r] [ 0 ,iC] [ -l:', 0] [0,1r] u = 1/[y+(x-y)v]
-
Table 18.2
integral arg v arg(v-1) arg(v-x) arg(v-y) restrictions
1
~ 0 -TC [-7t,O] I . [ -1r, O] Re (p+f3' -r) > -1 ! Re(t - cx) > 0 0 0 I
l -1t -7C [-'It:~ O] ,.
[ -7C,O] Re(~+~' -1) > -1 I ! Re> 0
60 ! 00 I
~ . 0 0 [-1t,O} [ -7t,OJ Re(1-) > 0 Re I( > 0
I I . ~ x I I Re(/3+~' -)') > -1 ) [ 0, n:J I [ 0, 1C} (-7,0] [O ,211:] i I ! Re /3 < 1
0 !
y j
~ [0,7&] I [ 0 ;n:] [ -lt, 1C] [-7t,0] Re(~+~ -r) > -1 Re /!,' < 1
0 i
x . I ~ [ 0, 11:] [ 0, 7t} [-'it,O] [ 0,27ti
Re (If-)> 0 I
I Re fJ < 1 1 I
I y ! Re(Y-ot) >0 ~ [0,Jr] [0,7t'] [ -1t,lt] (-7C,O] Re~'< 1 ' 1 I i 60 I ! I Re~< 1 i [0,'!r] ' [ 0, 1t] [0,1t] [ 0 '211:} Re Pi. > 0 x
. co Re /3' < 1 I [ 0, 1t] i (0,7['] [ -'7t, 1t] [ -7r ,O] Re ot. > 0 y
y Re ~
-
- 133 -
into r V(x, y, v}dv , where
~ B' -i Y-cc-1 -/3 - a V(x, y, v) vt'' (v-1) (v-x) . (v-y) P ,
and
c l/b , d l/a
Thus we derive ten integrals of V(x, y,v) from (18.8-1}""(18.8-
10). These ten integrals are solutions of (F1).
THEOREM 18.2: The integrals
. fd ,....'2'-J J~Ol~l -A --' (18.15) c v P (v-1) , (v-x) l"'(v-y) dv are solutions of the system (12.2) under the respective restric-
tions given in Table 18.2, where c and d are any two points of
O, 1, x, y and oo The paths of integration are taken as
indicated by Fig. 18.4 and branches of integrand are determined
as indicated by Table 18.2.
Fig. 18.~
In Fig. 18.4 and Table 18.2, we assumed that
{ 0 < arg y < arg x < 1C , 0 < arg(y-1) < arg(x-1) < 7t:
-
- 134 -
Remark 2. In order to guarantee the convergence of the
integrals (18.3) and (18.15) we must assume certain conditions
on the parameters orcoe>r F (a(~ A' > . . r 1. ,,..,,.. ,y,x,y
if none of ae. , X-o
-
- 135 -
I divergent integrals is essentially based on. the concept of
analytic continuation. For example 1 f 0 U(x, y, u)du
is originally defined for 0
-
- 136 - .
19. Connection formula and monodromy representations for the
system (12.1) satisfied by F1 . In this section, we shall briefly
explain connection formulas amcng solutions of the system (12.1)
and monodromy representations for the system (12.1).
To begin with, let us consider the ten integrals (18.15). We
suppose that these ten integrals are well defined at the same time.
This is possible under a suitable assumption on the parameters ,
p , ~ ' and 't The paths of integration for these integrals are shown by Fig. 18.4. (Cf. Theorem 18.2, p.133.) We shall use
the same idea as in Section 16 (Chapter III) to find connection
formulas among solutions of (12.1). The basic idea is
(i} to take a closed curve consisting of parts of these paths
of integration and small circular arcs and large circular arcs sc
that Cauchy's integral theorem can be applied, and then
(ii) to let the radius of small circular arcs tend to zero and to
let the radius of large circular arcs tend to infinity.
By using various closed curves of this kind, we can find more
than thirty relations among the ten integrals (18.15). For example:
if we use the closed curve given by Fig. 19 .1, we get
JO V dv + 1 ~ V dv + f, 00 V dv = 0 to 0 1
It is known that the system (12.1) has only three linearly inde-
pendent solutions. (Cf. Theorem 14.1> p.91.) This means that
there are only seven independent relations. As these relations
-
- 137 -
show, any three of the ten integrals (18.15) are not necessarily
linearly independent. It can be shown that three integrals
(vdv, f: Vdv and are linearly independent.
0 1
~ .
\
Fig. 19 .1
We shall now proceed to the monodromy for the system (12. :_).
In Section 14, it was explained that the system (12.1) has the
singular set which is the union of the five lines-:
(19 .1)
and that any solution of the system (12.1) is regular analytic
in the domain
(19 .2)
(Cf. Section 14, p.94.) In the same. way as we defined the funda-
mental group 1tl (D, x0) of D C - { 0> 1 J with the base point x0 in Section 6 (Chapter I), we can define the fundamental group
TC l (J) , {x0 , y 0)) of ~- with a base point (x0 , y 0). Then
taking a fundamental system of solutions of (12.1), we can define
the monodromy representation with respect to this fundamental
system in the same way as we defined monodromy representations
-
- 138 -
of the Gauss differential equation (6.1). (Cf. Section 6,
Chapter I.) The monodromy representation thus defined is a homo-
morphism of 7t 1
(JJ , (x0 , y 0)) into GL (3, C).
It is known that 7t1(D, x
0) is a free group generated by
two elements. (See Section 17, Chapter III.) However,
1t"1 (~, (x0 , y0)) is more complicated. It is true that 7r1 (oi@, (x0
, y0)) is generated by five elements. To find these
~ive elements, consider a complex line in C x IC which passes
through the base point (x0 , y0) and intersects with the five
singular lines (19.1). This means that this line is not parallel
to any of these five singular lines (19.1). Furthermore assume
that this line does not go through four points (0, 0), (0, 1),
(1, O) and (1, 1). These four points are. intersection-points
between the singular lines (19.1). This complex line can be
identified with the complex plane C. Then the intersection-
points with five singular lines (19.1) are represented by five ..
points A1 , A2 ; A3 , . A4 and AS on th.is complex plane. The base
point (x0 , y0) is also represented by a point B on this .com-
plex plane . The intersection of ~ and this complex line is
identified with C. - {A1 , ~ ", A5}. The fundamental group
1t.1 (C -,{A1 , ,As}, B) of _c - {Al' , Asl is a free group generated by five elements. Those generators are represented by
five loops surrounding . A1
, A ' 5
respectively. Let us denote
these loops by .l 1 , , ./. 5 . (See Fig. 19.2~) . It can be proved
-
- 139 -
that '7r 1 (~, (x0 , y0)) is generated. by five elements correspond-ing to these five loops.
Fig. 19.2
Although ' n:1 (C - {A1 , , A5}, B) is a free group,
1ti ( .!; , Cxo, y 0)) . is not a free group. - In other words, there
are relation$ among the five generators. To see this, let us
consider i. 2 and l. 3 . Note that the complex line intersects
with the singular lines x = l and y - 0 at A2 and A3 re-
spectively. Suppose that the base point (xo, Yo) is in a
neighborhood of (1, O). The point (1, 0) is the intersection-
point of the two singular lines x = 1 and y "" 0. More precise-
ly speaking, suppose that (x0 ~ y0) is in the domain
(19.3) 0
-
- 140
loop can be considered as a circle s2 defined by
i6 Sz X = 1 + (x0 -1) e > y "'" y O .
The loop .I, 3
can be deformed into a loop lying in the line
x = x 0 , and this deformed loop can be considered as a circle
83 defined by
83 = x - XO , y Yoe
18
The product of s2 and s3 is a torus contained in ~ ' and
82 and S3 are two circles on this torus. It is well known that
52 and S3 are commutative on the torus. In other words, s2s3
is homo topic to S3S2 on th~ to~s. Therefore, S2S3 is homo-
topic to S3S2 in ~ This means that l.2 L3 is homo topic
to l3 l.2 in . ~ Thus we conclude that c i 21 and [ .l 31 are commutative in the group 7r
1 (,J, (x
0, y
0)). This shows that
this group is not a free group. Similarly, . [ 1.1
] and [ .i 4]
are commutative in 7t 1 (~, (x
0, y 0)) . . There are other relations.
For example, there are. _ relations among
These relations arise from the fact that three singular lines
x ... 0, y = 0 and x = y intersect at one point (0, 0).
In order to compute the monodromy representation with respect
to the fundamental system
Jx V(x, y, v)dv , 0 . J y V (x , y, v) dv , 0 . .. .
1 f V(x, y, v)dv O
of the system (12 .1), let us first consider three loops l l' i 2 and l 5. The line y = Yo intersects with three singular lines
-
- 141 -
x ... 0, x = 1 and y = x at . (0, y 0r, (l, Yo) and
-
- 142 -
This is due to the fact that arg v and arg(v-x) change to
arg v+ 27ri and arg(v-x) + 27t'i, but other arguments do not
change after the analytic continuation along .li. Let us next continue the integral
(19.4) f: V(x, y, v)dV along the loop li We must deform the path of integration so that x does not go across the path. Hence at the end of analy-
tic continuation the path of integration becomes a path as shown
by Fig. 19.4.
0
Fig. 19.4
Note that the path shown by 19.4 can be further deformed to a
path as shown by Fig. 19.5~
y
0 Fig. 19 . 5
Taking the change of arg(v-x) into consideration, we conclude
that the integral (19.4) becomes
-
- 143 -
21ti~ 5x f y {e- -1) Vdv + Vdv . 0 0
Similarly the integral
50
1 V(x, y, v)dv
becomes
x 1 . (e2~ -1) J Vdv + J Vdv
0 0
after the analytic continuation along ./. l. Therefore the circuit matrix along li is
e 27t1cs-n 0 0 -2~ e . -1 l 0
e-~-1 0 1
In the same way we get other four matrices corresponding to the
other four genera tors of 7rl ~ , (x0 , y 0)). By these ma trices
the monodromy representation with respect to the fund~mental
system
f x Vdv, . 0 ...
is completely determined.
~ y
J Vdv, . 0
-
- 14!~
20 . General solutions of the systems (12.2)~ (12.3) and (12.4).
In this section, we shall derive general solutions of the systems
of partial differential equations which are satisfied by F2, F3
and F4
respectively. We shall first consider the system (12.2)
which is satisfied by F 2 :
(12.2) {
x(l-x)r - xys + [t c (ot+~+l)x] p - ~yq -Ill.~ z - 0 ~
y(l-y)t -xys+ ri'-(oc.+p'+l)y]q-{3'xp -p'z 0 . Let us make the change of variable
(20 .1)
Then the system (12.2) is transformed into
{20.2)
x(l-x)r' -xys '+[ 2A+1-(2A+fl+O{+p+l)x] p' -().+Myq'
-[(.A+p+oc.)(A.+~)-A(A+Y-l)x-1]z' = 0,
y (1-y) t I -xys '+ [ 2fl+ )'' - (7p.+A+~+ ~'+ 1) y] q I - ~+~' )xp I
-[ (.\.+f+o
-
- 145 -
z z'
The second case of (20.4) yields the transformation
(20.5) t-r z x z'
and the transformed system (20.2) is obtained from (12.2) by
replacing Q(. ' ~ ' -~'. '( , . l ' by oC+l-t " f3 +1- r , ~., 2- '( , Y' respectively. Therefore, this system admits a solution
(20.6) z' F2 (0(+1-l'_,. ~+1-~, Ji', 2-r, y', x, y).
From this we obta~n a_ solution of (12 .2): 1-J . .
(2 0. 7) z = x F 2 ( oc. + 1- , ~ + 1-t , f , 2 - l , l ' , x, y) Similarly, the third case of (20.4) yields a solution of (12.2):
(20.8)
and the four-th case of {20.4) yields a solution of (12.2): 1-1 1-r' ,
c20.9) z = x y F2(+2-r-1, ~+1-r, 1J'+1-y', 2-r, 2-r, x, y) . . Thus we obtain the following theorem.
THEOREM 20.1: A general solution of -the system (12.2) is
given by
(20.10) z = AF2 (Q(, p. f, r, J',x,y) + Bx1 -)'F2 (o+1-l,~+l-t,~' ,2-1, ' ,x,y) + cy1-r'F2
-
- 146 -
(12.2)" by the change of variables:
(20.11) x - 1/ ~ ' y = 1/71' ' Z = s~ 'ld. Z I
In fact, by this transformation, the syscem
(12.3) .. { x_
-
- 147 -
representation in Section 11 (Chapter II). (See Theorem 11.3,
p.74.)
Let us now consider the system (12.4) which is satisfied by
F4 :
{
x(l-x)r - y2t - 2xys + [r-(+~+l)x]p - (D(+(i+l)yq -lz = 0 , (12 .4) 2
y(l-y)t -x .~ - 2xys + [ r' - ~+~+l)y]q - (a(+~+l)xp - o. and }J-- by
(20~16) A. (X + r -1) = O, )A(p.+ t'-1) = 0
The equations (20.16) yield
. { A.= 0 ,
f- = 0,
{A=l-t, {.A=O, \ ~- o , fo = 1- r', {
~-1-1,
f'-= i- r'.
Hence in the same way as we - derive~ Theorem 20.1, we obtain the
following theorem:
THEOREM 20.3: A general solution of the system (12.4) is
given by
-
- 148 -
(20.17) z AF4 (ot, p, Y, 't'~x,y) 1-1. R. I +Bx F4 ((l(-i+1,,..-r+l,2-~1 ,x,y)
+ cy1- r' F 4
(.- J'+l, 13- r'+1,1,2-r' ,x,y)
+ Dxl-yl-t'F4 (o
-
- 149 -
21. Euler transform in double integral. In the previous sec-
tion, we found four linearly independent solutions for each of
the systems (12.2), (12.3), and (12.4). However, we would be
unable to calculate the monodromy representations of those systems
with respect to these fundamental systems of solutions. In
Section 19, we explained how to use the simple integral represen-
tation of Euler for computing circuit matrices of the system
(12 .1) which is satisfied by F 1
A similar method based on the
double integral representations of Euler may yield monodromy
representations for the systems (12.2) and (12.3) which are
satisfied respectively by F2 and F3, although, to the author's
knowledge, nobody has ever tried to compute the monodromy repre-
sentations for the system (12.2) and (12.3).
First of all, we must generalize Cauchy's integral theorem.
Such a generalization was given by H. Poincar~ as follows: Let
f (x~ y) be a holomorphic function of x and y in a domain D.
Let S (CD) be .a closed smooth surface of real dimension two.
If there exists a set V of real dimension three such that
(i) V C D ,
(ii) the boundary of V is S ,
then
ff f (x, y )dx dy 0 s
In Section 10 (Chapter II), we derived the double integral
-
- 150 -
representations of Euler for F1 , F 2 and F3 . (S" e Th ''' rcm 10.1
on p.61.) We showed that the three integrals
(21.1) J J u '3-lv 13' -l (1-u-v) r-,-~, -l (1-xu-yv) at du dv , u,v,1-u-v~O
and
(2.13) r r ~1 ~-1 ~-~-$'-1 - -ot' } } u v (1-u-v) (1-xu} (1-yv) du dv u , v , 1-u -v~ 0
are solutions of (12.1), (12.2) and (12.3) respectively, if these
'integrals are convergent. The method of the Euler transform which
was explained in Chapter III suggests, as in Section 18, that we
may replace the domains of integration
u, v, 1-u-v ? 0 and 0 .~ u ~ 1, 0 ~ v . ~ 1
by other suitable domains of integrations so that the integrals
thus obtained also satisfy the systems (12.1) , (12.2) or (12.3).
Actually we cari find .more than four solutions of (12 . 2) and (12 .. 3)
in this manner. However, any new solution of (12 .1). can not . be
obtained by this method.
-
- 151 -
22. Characterization of the systems of partial differential
equations satisfied by F1
, F2 , r 3 and F4 . Consider first a
diferential equation of the second order
(22 . 1) y"+p(x)y'+q(x)y = 0 .
Suppose that x "" a (:f co ) ls a singular point of (22 .1). This
means that x = a is a singular point of p(x) or q(x) or
both. The point x = a is called a regular singular point of
(22.1) if x =a is at most a simple pole of p(x) and at most
a double pole of q(x). Suppose that x ""a is a regular sin-
gular point of (22.1). Then the equation (22.1) can be written
in the form
(22.1 '} 2 (x-a) y" + (x-a)P(x)y' + Q (x)y = 0 ,
where P(x) and Q(x) are halomorphic at x = a. As the general
theory of linear differential equations guarantees, the differen-
tial equation (22.1') admits a solution of the form
y = (x-a/
-
- 152 .
Let f 1 and f 2 be the exponents of (22. l ') at x. = a. If f
1 - f 2 I integer, then (22. l ') admits two solution.:>
(22 .2) - f 1 f 2
(x-a) 'fi(x), - - ex-a) cr 2 Cx) ,
where 0, the constant E may be zero. A regular singular
point x a is called logarithmic if b = 1. A system of
solutions (22.2) or (22.3) forms a fundamental system of solutions
of (22.1'). This fundamental system is called a canonical system
of solutions of (22~1') at x a.
-Suppose next that p(x) and q(x) are holomorphic in a
domain
If we make the change of the independent variable:
x ... its
then the differential equation (22.1) becomes
-
- 153 -
(22 . l") 2 2 2 1 1
d y/d5 + (~--2 p(l/~))dy/d~ +-:;z;-q(l/~)y = 0. ~ l .
We shall call the point at x ~ ~ a regular singular point of
(22.1) if 5 - 0 is a regular singular point of (22 . 1"). The indicial equation of (22. l") at ~ = 0 is called the indicial
equation of (22.1) at x oo The exponents of (22.1") at ~ ,. O
are called the exponents of (22 .1) at x oo If x oo is a
regular singular point of (22.1), then x = ~ is a zero of p(x) of multiplicity at least one and .a zero of q(x) of multiplicity
at least two. Therefore, the differential equation (22.1) can be
written as
(22. l" f) 2 x y"+xP(x)y' +Q(x)y - 0 ,
where P(x) and Q(x) are holomorphic at x = oo The indicial
equation of (22.1"') at x oo is given by
~(A+l) - )..P( oo) + Q{oo) - 0
The differential equation (22.1) is called an equation of
Fuchsian type, if all singularities of (22.1) are regular singular
points including the point at infinity. The Gauss differential
equation:
(22.4) . "+{ ! + +@-1+1} f + ot./J . - 0 Y x x-1 Y x(l-x) y
is an equation of Fuchsian type which has three regular singular
points at x - O, 1 and oo The indicial equations and the
exponents at these three singular points are as follows:
-
- 154 -
singular indicial equation exponents point
0 A_(;t-1) + 't A= 0 0, 1-Y
-
1 ~(~-1) +(cl.+~ -l+l);\ = 0 o, t--P
00 ).(~ +l) - (oc.+~+l)A + o(.~ ... 0 o( , , i
If 1 f: integer, then
F(oc.,~,y,x), 1-l'
x F(cC-f+l, ~-+1, 2-r, x)
form a canonical system of solutions of (22.4) at x "" 0. If
"t-0{-ft:f integer, then '--, .
F(,p, oe.+p+1-r, ~-x), (1-x) F('(-oc, 1-13, d'--~+1, 1-x)
form a canonical system of solutions of (22.4) at x z 1. If
c< - p :/- integer, then x-oeF(, D(-1+1, Ol.-p+l, l/x), x-~F(~, p-t+l, p-oc+l, l/x)
form a canonical system of (22.4) at x = oo . . .
A second order equation of Fuchsian type with three regular
singular points at x = a, b and oo can be written as 2
.,.1 .+ Ax+B , + Cx +Dx+E y = O ' (x-a) (x-b)y { )2 ( b)2 ' x-a x-
(22.5)
where A, B, C, D . and E are constants. The indicial equations
of (22.5) at x = a, b and oo are respectively given by
-
lSS
Let fl' f 2 be the exponents at x a, ~1 , ~ 2 the exponents at x '"' b and 't" 1 , T 2 the exponents at x = oo Then the
relations between roots and coefficients of a quadratic equation
yield
r1 + f2 -1
1 - (Aa+B) (a-b) , f1 r2 - (Ca2+Da+E) (a-b)-2 ,
er l + ~2 -1
= 1 - (Ab+B) (b-a) , dj_ '2 - (Ch2+Db+E) (b-a) - 2 ,
't'l + '(2 .. - l+A , "tl t'2 - c .
From these relations, we obtain
{22.6) r i + r 2 + ~ i + 0 2 + 't' 1 + 't" 2 ... 1 This relation is called the Riemann or Fuchs relation.
Suppost! that two points a and b (a + b, a :;. oo , b :! oo ) and six complex numbers fl' f2' ~1 , ~2 , 'C1 and :i:z satis-fying (22.6) are given. Then a second order differential equation
of Fuchsian type with three regular singular points at x = a, b,
and oo is uniquely determined in such a way that its exponents
at x = a , b, and 00 a re f l, fl and If" l, If' l and L' l , -c
2 respectively. In fact1 such a differential equation is given
by
(22. 7). 11+ 1 2+ 1 2 t (1- f - f 1- 0. - ~ )
Y x-a x-b Y
( f 1 r2 (a-b) is-1 ~2 {a-b)) y
+ "C'l 't2 + x - a ... x - b (x-a) (x-b) = O
The differential equation (22 . 7) is called Riemann's equation, and
the set of all solutions of (22.7} are denoted by
-
~
- 156 -
b
(22. 8) x
The notation (22.8) is due to Riemann and it is called Riemann's
P-function. The set of all solutions of the Gauss differential
equati~n (22 .4) is thus given by
1 ca
(22.9) y ... {o
p 0 0 ct x
1-)' r-- ~ p Now we shall explain the characterization of Riemann's dif~
ferential equation due to Riemann. To do t his, for simplicity, sup -
pose that none of the singular points a, b and oo is logarithmic .
Let j be the set of all solutions of Riemann's differential equa-
tion (22. 7). This means that j denotes Riemann's P-function
(22.8). Let D be the domain D = S - ta, b, co}, where S is the
Riemann sphere . Then .; (i) Every function in
multiple-valued.
~ has the following properties :
~ is analytic in D, but it may be
(ii) For every point x0
in D, there are two branches f 1 and
2
of two functions . or one in g. which are defined in a neighbor-I
hood of x0
and which satisfy the condition
; 0 .
-
.. 157 -
A function f defined i _n a neighborhood of x0
is a branch of a
function in J. if and only if f is a linear combination of f1
and 2 .
(iii) In a neighborhood of a, there are two branches of functions
in ~ which are of the form
where ~ and 11
(x .. a)f1_ t1 (x) '
-
- 158 -
lw = p
An outline of the proof is as follows: Let y(x) be an
arbitrary branch of a function in p which is defined in a
neighborhood of x0
. Then the condition (ii) implies that y, f 1
and f 2 are linearly dependent and hence
y(x) f1
(x) f2
(x)
y' (x) fi(x) fz(x) = 0
. y" (x) f'{-(x) f'2 (x) in a neighborhood of XO. This relation can be written as
y"(x)+p(x)y'(x)+q(x)y = 0
which is a linear differential equation. By using (ii), we can
prove that p(x) and q(x) are holomorphic at x0 and that they
do not depend on the choice of branches f 1 and 2 . This means
that p(x) and q(x) are uniquely determined by $ . On the
other hand, the condition (iii) assures that x = a, b and 00
are regular singular points . of this differential equation. This
proves Theorem 22.1.
Remark. If some of a, b and oo , say a, is logarithmic,
then the first part of condition (iii) should be replaced by the
following condition: .,
In a neighborhood of x = a there are two branches of func-
-
- 159 -
tions in Jr which are of the form:
P1 (x-a) '(
1 (x) ,
where Tl and
-
- 160
f 1 (x,. y), f 2 (x, y) and f 3 (x, y) of functions in j
1 such that
fl f2 f 3
r/: 0 at
and that a function f (x, y) defined in a neighborhood of (x0
, y0
)
is a branch of a function in 31
if and only if f is a linear
combination of 1 , 2 and f 3 ;
(iii-1) for any point (0, y0), where y
0 r/: 0, 1, 60 , there are
three branches of functions in jl which are of the form:
~-1+1 . 'f 1 (x , y) ,
-
- 161 -
three branches of functions in ).1 which are of the form:
t-ct.- ~ -(y-1) . 't' 3
(x, y) , -'f 1 (x, y) ' ,... -
where t 2 and 'f'J are holomorphic at (iii-6) for any point (x0, oo), where x 0 :f 0, 1, oo, there are
three branches of functions in '}1
which are of the form:
-~ ..... . -13' - - -Y. X1 (x, y), . y . X2 Cx, y), y . x.3cx, y), - - -where x1 , x. 2 and x.3 are holomorphic at (x0 , oo) ; {iii-7) for any point (x
0, y
0) , where x
0 y
0 .; 0, 1, oo, there
are three branches of functions in ~ 1
which are of the form:
~l (x, y) , ~2 (x, y) , (x-y) '/-,-~, f 3 (x, y) , where ; 1 , ; 2 and t 3 . are holomorphic at . (x0 , y0)
Picard proved that, . if 1' 1 satisfies (i), (ii) and (iii) and
some additional hypotheses, then ;. 1 is the . set of all solutions
of the system (12 .1). We .. do not know clearly what additional
hypotheses must be required. Picard as well as Appell and Kampe de
Feriet gave the hypotheses in a _very vague fo~ in their works.
The principle of the proof is the same as in the case of Riemann's
equations. Let z(x, y) be an arbitrary branch of a function in
3r 1 which is defined in a neighborhood of a point (xo~ _Yo> oB i Then the condition (ii) yields
z zl z2 ZJ z zl z2 Z3 p P1 P2 P3 p P1 p P3
- 0 , 2
- 0, q ql q2
-
- 162 -
z zl z2 z 3
p P1 P2 P3 = 0 ,
q ql q2 q3
t tl t'l t ~ i where
p "'Jz/'"Jx, q ")z/'dy, 2 2 r = ~ z/ ax , ..,. t d2Z/dy2
and
p. a tlz./-ax, . J J
These three relations can be written as
{
r _ ... at1
(x, y)p+ oc2 (x, y)q + at3 (x, y)z,
s a ~1 (x, y)p + ~2(x, y)q + p3 (x, y)z' t =
1 (x, y)p+ 1
2(x, y)q+
3(x, y)z.
After very long calculations, Picard showed that this system coin-
cides with (12.1).
PROBLEM 1. Complete the work of Picard.
Riemann's point of view ca~ be applied to the systems (12.2),
(12.3) and (12.4). Goursat applied the same principle ,~o the
systems -(12.2) and (12.3) . However, the author does not know
whether or not the same principle has ever been applied to th
system (12 .4).
PROBLEM 2. Complete. the work of Goursat . l;
PROBLEM 3. Deri~e th~ system for F4- (i.e. the system (12 .4)) by using Riemann's point of view.
-
- 163 -
CHAPTER V
Automorphic Functions, Reducibility
and
Generalizations
23. Automorphic functions. It is well known that the elliptic
function was discovered by Gauss, although he did not publish his
discovery. Then Abel and Jacobi rediscovered the elliptic function
independently . The discovery of the elliptic function opened an
epoch in the history of mathematics. Indeed, this discovery had
influences not only on the analysis, but also upon all fields of
mathematics . Gauss also found a new function which is related to
the elliptic function, quadratic forms, and arithmetico-geometric
mean. This new function is called the elliptic modular function.
It was Dedekind who rediscovered the elliptic modular function .
This is the earliest example of automorphic functions.
In 1972, Schwarz derived, from the Gauss differential function,
automorphic functions other than the elliptic modular function.
Ihen Fuchs tried to generalize Schwarz's results to more general
linear differential equations of the second order. Stimulated by
the work of Fuchs, Poincare founded the general theory of automor-
~hic functions and called a kind of automorphic functions the
~uchsian function.
We shall now explain how we can deri ve automorphic funct i ons
-
- 164
from a linear ordinary equation of the second order
(23 .1) y"+p(x)y'+q(x)y = 0
Suppose that (23 . 1) is a Fuchsian equation with regular singular
point~ at a1 , a 2 , , an. Set D - S ~ {a1 , , aJ. Let 'fi (x)
and ' 2 (x) be linearly independent solutions of (23.1) and let
/ ., : 7tl (D, x0
) -+ GL(2, C)
be the monodromy representation of (23.1) with respect to the
fundamental system of solutions
(23.2) [ :~ l We denote by T the set of linear fractional transformations
( ) az+ b t z .. cz+ d (ad - be + O) , and define a homomorphism r : GL(2, C) ~ T by
Then the composite map '1: of is a homomorphism of 1Ll (D, x0) into T. Let us denote by G the image of 7r1 {D, x0) under the
map "t 0 r : G "" 1: o r (rr1 (D, x0))
' G is a subgroup of T. Let l be a loop in D '"lt x0
rf
ffLl = [: :].
then fundamental system (23.2) becomes
-
- 165 -
by analytic continuation along .1..
Consider the ratio
f(x) = f 1 (x)/ ~2 (x)
Then f (x) becomes
8 f1(x)+b~2(x) _ af(x)+b C'f'l {x) + d
-
0 166 Q
Example l ~ Let .:1 = t and G ={ tn; t (z) = z + 2n7'Ci~ n
n E. i }, where 'J: is the set of all integers. Then
g(z) = e 2
is an automorphic function relative to G.
Example 2: Let .A = fC and i
G "" ~ t (z) = z + m w + n w " ~ m~n 1 . 2
m. ~ ~ E. ::L where ".!) and U.J r ' 1 2 are complex numbers such that:
Then automorph:i.c funct:Lons relc.t:tve tc G are e lliptic functioru3
u.L and w A typical exampls- i2 t he Weierstr ass .J. 2.
~lliptic function
2: I 1n~nE~ ;..
(lll~n)~(O,O)
1 ~~ . 2
1 ... =mr , =rt'' ) ,... "'"'l ~2
Schwarz cons:i.dered the Gauss differential equation under the
assumption that all of the paramet ers and '1 a.re real
numbers . He determined a.11 casES that the image .4 of D = ~ =
{ oll l \ under the map
is a univalent domain contained in S and hence the inverse func ~
ti on
x = g(z)
of f is single-valued in A We shall summa.rize 'his results<
THEOREM 23.1: Assume that are all r eal.
Then
-
2> 16 7 -
(i) g(z) isl single~valued ,t;; l.J. and only if
.A "" ~ 1 - r I 3IZ 1/1. l :s 1 s 2 ~ ... 00 ~ ~ r= lr~q{-i8l '" l/m, m "' 19 2' 0 0 ' ~ v "" l ~ ~ ~ l "" l/n 9 n """ 1:1 2, 00 ~ ~
where if one of >.. / iA and v is one ll the others a re equal;
(ii) if
then g(z) is a rational function;
(iii) if
J\ + )4+ i.> = l/ l + l/m + l/n = 1 9
then there exists a linear fractional transformation t(z) in T
such that g(t(z)) is either a simply periodic function or a
doubly periodic function;
(iv) if
A+~+ Y"' 1/.1. + 1/m + l/n < 1 9 then there exists a linear fractional transformation t(z) in T
such that g(t(z)) is an aut:omorphic function defined in jz!
-
- 168 -
'f r -oo du ~ ~ f,'OO . . du . 1 Jo. Ju(u-l)(u-x)' .. 2. 1 ~u(u-l)(u-x)
and A is the upper half-plane Im z > 0 ~ and G is the group generated by
t 1 (z) = z + 2 , z
2z+ 1
We remark that
t .. f 1 Ii Z dv z . 2. -1 Cl-v ) (1-xv )
It is known that f(x) =
-
- 169 -
" . L. z
y"' 2 2 - - -p.
Y2
Thus we have
(23. 3)
Differentiating both sides, we get
Yz .. -\yi{p(x)+z"/z'}-;y2 {p'(x)+ [zn/z'] '}
.\ y 2 { p (x) + z" /z '} 2 - ; y2 { P' (x). + z'" /z' - [z" /z' J 2} .a.
and hence
(23.4) yl 0 Ji;y2 {p(x)2 -2p'(x) -2z"'/z'+3[z"/z'] 2 +2p(x)z"/z' }.
Since y2 is a solution of (23.1), substituting (23.3) and (23.4)
into (23.1), we get
\y2
{p(x) 2 -2p'(x) -2zm/z'+_3[z"/z'J2 +2p(x)z"/z'} 2 . .
- \y2 {p(x) +p(x)z"/z'}+q(x)y2 -0.
Dividing both sides by \ y2 , we obtain
(23.5) z"'/z' -~ [z"/z'J 2 -z{!p(x)2 +ip'(x) -q(x)}. The left-hand side of (23.5), i.e.
(23.6) zttt/z' -~ [z"/z 1 J2
is called the Schwarzian derivative of z and often denoted by
{z, x } The right-hand side of (23.4) or the quantity
I(x) = tp(x)2 +\p'(x) -q(x)
is an invariant of (23.1) under a transformation of the form
(23. 7) y a(x)u
In other words, if (23.1) is reduced to
u"+p1 (x)u' +q1 (x)u ., 0
-
170 -
by (23.7), we have 2 2
\ p (x) + 12 p' (x) - q (x) = ~pl (x) + ~ p1 (x) - ql (x) .
In particular, if we take a (x) = exp { - ~ Jx p (t)dt}, then (23 .1) becomes
u" - I (x)u = 0 .
For Riemann's equation with scheme
a b oo
we get
1z, x} = ~[(l-'A.2) a-b+ (l- . 2) b-a +l->'2] l , l x-a /A x-b (x-a) (x-b)
where
In particular, for the Gauss equation, we have
.{z, x l.,. 1 { l-.X2 + 1-..2 - l-X2-.2+v2 } I 2 --;r (x-l)Z x(x-1)
where
~ 2 - ( 1 - '1) 2 ' f-2 = ( - - ~ ) 2 ' }12 - (0( - p ) 2
About at the same time when the general theory of automorphic
functions was founded by Poincar~, Picard discovered an example of
automorphic function of two vili:iables by utilizing the system (12.1} I
w~ich is satisfied by F1 . Ao we explained before, the system
(12 .1) has three linearly ir1deeT1d~ni: solutions. Let us denote
them by
-
- 171
z1
(x,y) , z2(x,y), z
3(x,y)
and let f 1 be the monodromy representation with respect to the fundamental system
(23 .8)
z1(x,y)
z2(x, y)
z3(x, y)
The representation ft is a homomorphism of '7C l (~ , x0
) into
GL(3, G:}. For a loop .2 at x0
, the homotopy class [ J. ] is an
then the fundamental system (23.8) becomes
al bl cl
a2 b2 c2
a 3 bJ CJ
z1
(x, y)
z2(x, y)
z3
(x, y)
by the analytic continuation along l. Set
'
z1
(x, y) s f (x, y) "" , z
3(x, y)
z2
(x, y) t = g(x, y) = z
3 (x, y)
rhen the functions f(x, y) and g(x, y) become
alzl + blz2 + clz3 a 1 + b1g+ c 1 = a)f+b3g+c3 8JZ1 + b)z2 + C3Z3
tnd
a2zl + b2z2 + c2z3 a 2f +- b2g+ c2 =
aJzl + b)z2 + C3Z3 a 3 + b3g + C3
-
- 172 -
respectively by the analytic' continuation along 1 . Let
x = j' (s, t), y = f
-
- 173 -
containing a parameter x or two parameters x, y. An int~gral
of algebraic function is called, an Abelian integral which is: very
important in the theory of algebraic functions and in the theory
of algebraic geometry. The theory of automorphic functions has
become a branch of mathematics related to number theory and alge-
.braic geometry.
Finally we shall state an extension of the Schwarzian
derivative to the case of two variables. _Consider a completely
integrable system of partial differential equations
(23. 9) {
r a ~p+ c(2q + C(3z
8 ~lp+ ~2q+ ~3z
t r1p+ Y2q+. Y3z Let z1 , z2 and z3 be linearly independent solutions of (23.9}
and set
Then
- o( 2 '
-
- 174 -
where
The four differential expressions on the left-hand sides are a
generalization of the Schwarzian derivative to a map of. c2 to 2 C : (x, y) ~ (u> v).
-
- .115 -
24 . Reducibility. We shall .consider again a linear differential
equation
(24.1) . y" + p(x)y' + q(x)y -= 0
We shall write the equation (24.1) in a form
The . expression
(24. 2)
d d . .( 2 .) . dxz+p(x) dx +q(x) Y = o .
d2 ' d -z' + p(x) di"+ q (x) dx
is a differential operator. It happens that the operator (24.2) is
decomposed into a product of two operators:
(24.3) ' ~~ + p(x) ! + q(x) ( ! + s(x)) ( ;-+ r(x)) The right~hand member of (24 .3) means that
( ! +s(x)) ( ! + r(x)) ~ d22 + [r(x) + s (X) I! + [r' (X) + s (x);,(x) 1 . . dx < . . .
A polynomial is said to be reducible if it can be decomposed
into a product of two polynomials of lower degrees. In general,
the _r .educibility of polynomia_ls depends on the field to which the
coefficients of polynomials belong. Similarly, for differential ' ' - ' ' I ~ .. .
op.erators, the reducibility depends on the coefficient-field. In
other words, it is necessary to prescribe a field to which .the
-
- 176 -
coefficients p, q, r and s .. belong, if we discuss the decom-
position (24.3) of the operator (24.2). We shall restrict our-
selves to the set of all rational functions C(x). Namely, we
shall suppose that coefficients of differential operators belong
to C(x).
Remark that C(x) is not only closed under the usual addition,
subtraction, multiplication and division, but also closed under the
differentiation with respect to x. Such a set is called a dif-
ferential field. A precise definition of a differential field is
as follows:
A set K is called a differential field if
(i) K is a field in a usual sense;
{ii) there exists a map D from K into K such that, for any
a, b E K, we have
(a) D(a+b) ,.. D(a) +D{b) ,
(b) D(ab) .. D(a)b + aD_(b)
The map is called a differentiation. An element c of K is
called a constant if O(c) = 0.
Let us return to the operator (24.2). The equation (24.1) or
the operator (24.2) is said to be reducible if the operator (24.2) admits a decomposition of the form (24~3) in C(x).
. -Suppose that
(24.2) admits a decomposition (24.3). Then any solution of
{24. 4) y' + r (x)y ... 0 is a solution of (24.1). This implies that there exists a non-
-
.. 177 ..
trivial solution of (24.1) which satisfies a first order linear
differential equation (24.4). We shall show that the converse is
also true. Let . ~(x) be a non-trivial solution of (24.1) which
satisfies (24.4). This means that
{24. 5) ' " + p ' ' + q ' - 0 and
(24 .6) ,. + rcr - 0 Differentiating (24.6), we obtain
(24. 7) 'f" + r 'f' + r 'er - 0 . Subtracting (24.7) from (24.5), we have
(24.8)
niminating
iet
:o derive
(p~r) f' + (q~r')1' .. 0 d . ~. . an . J from (24.6) and (24.8), we get
I . 1 . r I .. q "."' r' - (p-r)r = 0 . . p-r q-r' s .. p - r
r+ s = p ,
r' +rs =- q
his means that (24.2) admits the decomposition (l4.J) .
THEOREM 24.1: The operator (24.2) is reducible if and only if
~ere exists a non-trivial solution of (24 .1) which satisfies a
lrst order linear differential equation (24.4).
Consider the Gauss differential equation
-
- 178
(24.9) x(l-x)y" + [ l'-(cx+~+l)x]y' - ll(p-y: .= 0_. _
Suppose that (24.9) is reducible. -Then there exis~s a non-trivial
solution of.(24.9) which satisfies a linear differential equation
(24.4). Such a solution is given by
(24.10) y f(x) = const. exp_[- )x r(t) dtl Since r(x) is a rational function in x, we have
- ~x r(t)dt = rl (x) +~/Li log(x-ai) ,
where are non-zero constants. Hence
rl (x) P.1 T (x) = const. e TT (x-a1) From the fact that (24.9). is an equation of Fuchsian type, it
follows that rl(x) must be a constant. Therefore, we can write
T (x) in the form (24.10.') -rr P.1 y .. .f (x) = const. 11 (x-a1) On the other hand, the equation (24.9) has its singular points at
x = 0, x = 1 and x = oo . The exponents at x = 0 are 0 and
1- '{ ' while the exponents at x - 1 are 0 and r -fJI.- {J
Therefore, if ai ii 0 and t: l, then P.1 - 1. This implies that
'f (x) is expressed as one of the following forms: P(x)
x 1-r P(x) , (24.11) 'f (x) = 't - -~ (x-1) P(x) ,
x1-
-
- . 179 - .
while P(x) is a polynomial in x. Let P be of degree n ~ 0.
Then r
-
180
(24.13) [t(x)J 'fCx)
is a fundamental system of (24.9). Let f be the monodromy representation of (24.9) with respect to the -fundamental system
(24.13). If J. 0 and l. ' 1 are loops at "xo surrounding x - 0 and x = 1 respectively, then the function '(x) becomes con-st-. T (x> by the analytic continuation :a.iong .t
0 and 1
1.
Hence
[** *oJ ~ f = , Consequently, the monodromy group consists of lower t!iangular
matrices. - Conversely, it is . easily verified that, if- the monodromy
group consists of lower triangular matrices, then f (x) takes one
of the forms (24.11). This means that (24.9) is reducible. Thus
we proved the following theorem.
THEOREM 24.3: The Gauss differential equation (24.9) is
~educible if and only if there exists a fundamental system of (24.9)
with respect to which the monodromy group consists of lower tri
angular matrices.
' .. We shall now explain briefly how to generalize the concept of - ,
reducibility to linear differential equations of higher orde~s. , . ~ .
Consider an n-th order equation
(24.14)
-
- 181
. or the corresponding differential operator
dn dn-1 (24. 15) - + p (x) + +
dxn 1 dxn-1 _ Pn (x) .
The equation (24.14) or the operator (24.15) is said to be reducible
if the operator (24.15) admits a decompos~tion dn - -dn-1
- (24.16) - + p (x) + +p -(x)-dxn 1 c:b:n-1 n -_
-( d :1 + .. +Pa (x)Y( a;, + : + ~n~(x)) ____ _ dx l _ - -_ l - ') -dx 2 - 4
T: ..
where 0 < n1 , n2 < n. The following theorem is a generalization
of Theorem 24.1.
THEOREM 24.4: The operator (24.15) is reducible ' if- and -only
if ther-e exists a non-trivial solution of -(24.14) which satisfies a
linear differential equation of a!l order lower than n.
A ._monodromy representation of (24.14) is a -homomorphism f of
_7rl-_(D, x0
) into GL(n,_C), _where D - i$ the. greatest domain in
whic~ the c~efficients of _(24.14) are holomorphic. A monodromy
represe_ntation
0
* . * }02 ~
n2
I
iihere n1
and n2
_ ~re determined by f and they are independent
-
- 182 '.9
of each matrix of f { 1t1
(D, x0)). The following theorem is a
generalization of Theorem 24.3.
THEOREM 24.5: Supp0se that (24.14) is an equation of Fuchsian . . . . -
type. Then the equation (24.14) is reducible if and only if there
exists a reducible monodromy representation of (24.14).
We shall now proceed to the systems (12.1), (12.2), (12.3) and
(12.4). The system (12.1) has a form
(24.17) { r
Q{1 P + oc.zq + 0(3 z ,
s - P1 P + ~zq + ~3 z, t - "1 p + y 2q + '13z ,
while the systems (12.2), (12.3) and (12.4) .have .the form
(24.18) .{. r ::1s+ a 2p+ a3q + a4z ,
t b1s+b
2p+b
3q+b
4z.
In their book, Appell and Kampe de Feriet gave the following defi-
nition of reducibility. The system (24.17) (or (24.18)) is reducible
if there exists a non-trivial solution of (24.17) (or (24.18)) which
satisfies a system of partial differential equations whose solutions
form a vector space of dimension
-
- 183 -
reducibility. We do not know any necessary and suffi..cient condi-tions for reducibility which are given explicitly in terms of
parameters. Appell and Kampe de Feriet gave only a few examples.
rhe reducibility of moriodromy representations of the systems (12.1)
,,..,.(12.4) can be introduced in a natural way. However, it seems to
is that there is no clear . statement concerning the relation between . . . .
~educibility of systems and that of monodromy representations.
PROBLEM 1: Find a necessary and sufficient condition that the
:ystems (12.1) "'- (12.4) be reducible. Find also relations between
educibility of the systems (12.1) ~ (12.4) and that: of their
onodromy representations.
-
-- 184 -
25. Generalizations. We defined the hype:rgeometric func.tions of
two variables. by.utilizing the Gauss function. In a natural way,
we can int~oduce hypergeometric functions of n variables.
Lauricella. introduced the following f~ur functions:
F .(oc.. R ~ v . y . x . x . ) A ' t' l :, , t' n, ' l, , 0 n, 1, , n
.:>:".:-> .. ( m_) (~ - m -)(A m) ~(A m) 1' l n' n t"l' 1 t'n' n
I. (~ ,~+ +m0
) (l,~) (1,mn)
- (at ,ml+ +mn) (~,ml+ +mn) ~ m n . L (11,ml) ... ((Y'n,mn) (l,~) .. (1,mn) xl . x a
and
(a( m.. + +m ) ( II. m ) ( n. m ) m m ' l n t'l' 1 '"'n' n 1 n
' x x '- ('j ,~+ +m0
)(1,m1) (l,mn) 1 n
In case when n 2, we have
. FA = F 2 , F B = F J , F C = F 4 and FD = Fl .
There are many functions of intermediate types. I
I
On the other hand, many functions of one variable are defined '
as generalization of the Gauss function. The Gauss function has
the three expressions:
(i) .. 2:, (Cl( , m) ( ~ , m) xm
m=O (1,m)(l,m) '
-
- 185
(li) . . f,' d..:...., r'-{J-1 -0( const .. 1
u (u-1) . (u-x) . du ,
(iii) const. r(at+s) r(6+s) r(-s) (-x)s ds. f +ieo
-ita . re t+s) The second expression can be. written also in the .forlJl .
(ii') const. fl -1 .,_ -1 -oc u~, (1-u) ~ (lx\l) du
0 . .
The series
F (ac.l, ... I O(p+l ~11 I ~p
. - 1' "'pl' m x - x
)
., - ( o( m) (..1 m)
f:o (~ 1 ,m) ~ (~p,m) (l,m) is derived from the expression (i) in a natural manner. More
~enerally, we get the following series:
(at ,m) m p x (o
-
. f ~ioo -1.00
- 186 -
p TI rcoc .+s> i=l 1.
Tr rep .+s) il 1.
s r(-s)(-x) ds.
These ideas of generalization of the Gauss function are also
applicable to generalization of Appell's and Lauricella's functions.
Finally, we shall talk about the confluence-principle. To
start with, let us consider the Gauss function F(ac, ~, '(, x).
Introducing a new parameter to define a function by
. F(CI(, 1/E., J, x)
and then taking a limit as e. tends to zero, we obtain a new
.. function
lim F(t
-
- 187 -
regular singular point at x z 0 and an irregular singular point
at x .. oo Such equations form an important class of equations
which contains Bessel equation:
(25.3)
The equation (25.3) has the Bessel function of order n as a
solution.
Humbert obtained from Appell's functions seven functions by
the confluence-principle. For example,
, ~ ( ~ ,m) ( 13', n)xtl)rn i!W F 1 (l/t' p, ~ , ~, .x, t.Y) .. }.;J "t ,m+n) (l,m) (l,n) and the function defined by this series admits an integral represen-
tat ion
C