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Optimised integral continuation methods
671~
I
Nuclear Physics
B41 (1972) 272-284.
North-Holland Publishing Company
B.11
OPTIMISED
INTEGRAL C O N T I N U A T I O N
METHODS
G. G. ROSS
Rutherford High Energy Laboratory, Chilton, Didcol, Berkshire, England Received 25 October 1971 Abstract: We consider the problem of finding the analytic function which m i n i m i ses the difference between it and our information about the function on the boundary, weighted according to the accuracy to which the function is known. A r e p r e s e n t a tion of the function is found in the form of an integral over the boundary involving the function boundary values and e r r o r s . Applications involving continuation throughout the analyticity region and, in p a r t i c u l a r , to the boundary are i n v e s tigated. Optimised relations are derived for the determination of Regge-pole p a r a m e t e r s from low-energy information and also for the determination of pole r es i d u es . In each case r e a l i s t i e statistical e r r o r s are given.
1. I N T R O D U C T I O N T h e c o n t i n u a t i o n of an a n a l y t i c f u n c t i o n t h r o u g h o u t i t s d o m a i n of a n a ! y t i c i t y h a s c o n v e n t i o n a l l y b e e n p e r f o r m e d u s i n g C a u c h y ' s i n t e g r a l f o r m u l a to e x p r e s s the c o n t i n u e d f u n c t i o n as an i n t e g r a l i n v o l v i n g the f u n c t i o n e v a l u a t e d o v e r the b o u n d a r y of it s a n a l y t i c i t y r e g i o n . S i m i l a r m e t h o d s h a v e b e e n u s e d to r e l a t e the i n t e g r a l s o v e r v a r i o u s r e g i o n s of the b o u n d a r y . F o r e x a m p l e , i n f o r m a t i o n about R e g g e p a r a m e t e r s m a y be o b t a i n e d v i a f i n i t e e n e r g y s u m r u l e s [1] f r o m i n t e g r a l s i n v o l v i n g the s c a t t e r i n g a m p l i t u d e o v e r the l o w e n e r g y p a r t of the b o u n d a r y . I n f i n i t e l y m a n y of s u c h r e l a t i o n s a r e p o s s i b l e in w h i c h the w e i g h t i n g of d i f f e r e n t r e g i o n s of the b o u n d a r y v a r i e s . M o r e o v e r , in the a b s e n c e of a p r e c i s e k n o w l e d g e of the f u n c t i o n , t h e s e d i f f e r e n t relations may yield widely varying results. I d e a l l y only one r e l a t i o n would be a p p l i c a b l e in any p a r t i c u l a r c a s e : that r e l a t i o n b e i n g d e t e r m i n e d in s u c h a m a n n e r a s to w e i g h t the c o n t r i b u t i o n of e a c h p a r t of the b o u n d a r y in p r o p o r t i o n to the a c c u r a c y to w h i c h the f u n c t i o n i s known. T h i s i d e a m a y be m a d e m o r e p r e c i s e if, f o l l o w i n g C u t k o s k y [2], we i n t r o d u c e a p r o b a b i l i t y f u n c t i o n a l P(F) of the f o r m
P(F)
=
exp {- }x2(F)} ,
(1)
where ×2(F)
1
= £ f !F(Z)- (fltZ)I2M-2(Z)idZ! B
.
(2)
G.G.Ross, Optimised integral continuation
273
H e r e L i s the l e n g t h of the b o u n d a r y B, c0(Z) i s the g i v e n i n f o r m a t i o n of the f u n c t i o n on the b o u n d a r y and M(Z) 0 d e s c r i b e s the s t a t i s t i c a l e r r o r in d e t e r m i n i n g ~(Z). T h e p r o b a b i l i t y f u n c t i o n a l d e s c r i b e s the c o n s i s t e n c y of a n y a n a l y t i c f u n c t i o n F(Z) with the d a t a . U s i n g t h i s f o r m f o r the p r o b a b i l i t y f u n c t i o n a l , P r e s n a j d e r and P i s ~ t [3] h a v e found an i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e f u n c t i o n , n a m e l y that f u n c t i o n F(Z) w h i c h m a x i m i s e s
P(F). U n f o r t u n a t e l y the f o r m of eq. (2) p r e s u m e s that the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n a r e d e t e r m i n e d with e q u a l p r e c i s i o n . A m o r e g e n e r a l c h o i c e f o r x 2 ( F ) i s g i v e n by 1 Bf t [ R e F ( Z ) - (pR(Z)]2 ×2(F) = L [Im F ( Z ) - ~ I ( Z ) ] 2 el(Z)2 + c2(Z) 2 I idZ/ '
(3)
H e r e e l ( Z ) and e2(Z) 0 a r e the u n c e r t a i n t i e s in d e t e r m i n i n g the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n a l o n g the b o u n d a r y , (?R(Z) and (pi(Z) r e s p e c t i v e l y . In r e f . [4] we h a v e found, f o r c o n t i n u a t i o n w i t h i n a s i m p l y c o n n e c t e d d o m a i n , an i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e f u n c t i o n f o r a r e s t r i c t e d c l a s s of e r r o r f u n c t i o n s e l ( Z ) and e2(Z). One of the a i m s of t h i s p a p e r i s to e x t e n d t h i s s o l u t i o n to a m o r e g e n e r a l c l a s s of e r r o r f u n c t i o n s w h i c h s h o u l d e n a b l e the m e t h o d to be u s e d in m o s t c i r c u m s t a n c e s , An i m p o r t a n t r e s u l t of the t e c h n i q u e is that the a v e r a g e v a l u e and s t a n d a r d d e v i a tion of a n y f u n c t i o n a l m a y e a s i l y be found o n c e the m o s t p r o b a b l e f u n c t i o n i s known. It s h o u l d be s t r e s s e d that a r e a l i s t i c e r r o r f o r the c o n t i n u e d v a l u e of a f u n c t i o n is of g r e a t p r a c t i c a l i m p o r t a n c e and h e r e i n l i e s a g r e a t a d v a n t a g e of the m e t h o d o v e r p r e v i o u s d i s p e r s i o n r e l a t i o n a p p l i c a t i o n s . M o r e o v e r , a s we s t a t e d a b o v e , o n l y one r e l a t i o n i s a p p l i c a b l e o n c e the s t a t i s t i c a l e r r o r s of the i n p u t d a t a a r e d e f i n e d . By a s u i t a b l e c h o i c e of f u n c t i o n a l , c o n t i n u a t i o n t h r o u g h o u t the c o m p l e x z p l a n e i s s t r a i g h t f o r w a r d . H o w e v e r c o n t i n u a t i o n to the b o u n d a r y p r e s e n t s s p e c i a l d i f f i c u l t i e s s i n c e the s t a n d a r d d e v i a t i o n b e c o m e s i n f i n i t e f o r s u c h a c o n t i n u a t i o n . W e c o n s i d e r w a y s to o v e r c o m e t h i s p r o b l e m and d e v e l o p a m e t h o d to t e s t any h y p o t h e s i s about the function o v e r a r e g i o n of the b o u n d a r y and to d e t e r m i n e the p a r a m e t e r s i n v o l v e d in t h i s h y p o t h e s i s . As an e x a m p l e of t h i s m e t h o d we d i s c u s s the o p t i m i s e d f i n i t e e n e r g y s u m r u l e s f o r the d e t e r m i n a t i o n of R e g g e - p o l e p a r a m e t e r s , t o g e t h e r with e x p r e s s i o n s f o r the s t a n d a r d d e v i a t i o n of t h e s e p a r a m e t e r s . We a l s o a p p l y the m e t h o d to the p r o b l e m of d e t e r m i n i n g the p o l e r e s i d u e s of a m e r o m o r p h i c function. In s e c t . 2 we i n t r o d u c e the n o t a t i o n and r e v i e w the r e s u l t s of r e f . [4] n e c e s s a r y f o r the s u b s e q u e n t d i s c u s s i o n . S e c t . 3 d e a l s with the d e r i v a t i o n of an i n t e g r a l r e p r e s e n t a t i o n of the m o s t p r o b a b l e f u n c t i o n for the c a s e of a s i m p l y c o n n e c t e d a n a l y t i e i t y r e g i o n and e x t e n d s the r e s u l t s of r e f . [4] to a m o r e g e n e r a l c l a s s of e r r o r f u n c t i o n . F i n a l l y in s e c t . 4 we c o n s i d e r p h y s i c a l a p p l i c a t i o n s of the m e t h o d and in p a r t i c u l a r a p p l y it to t h e d e t e r m i n a t i o n of R e g g e p a r a m e t e r s and p o l e r e s i d u e s .
274
G. G . R o s s , OptinHsed integral continuation
2. T H E P R O B A B I L I T Y F U N C T I O N A L , MOST P R O B A B L E F U N C T I O N , A V E R A G E V A L U E S AND S T A N D A R D D E V I A T I O N S We s u p p o s e we h a v e a f u n c t i o n F ( Z ) a n a l y t i c in a r e g i o n D with b o u n d a r y B. O v e r the r e g i o n of the b o u n d a r y w h e r e e x p e r i m e n t a l i n f o r m a t i o n is a v a i l a b l e we d e f i n e the f u n c t i o n s q)R(Z) and (pI(Z) a s s m o o t h i n t e r p o l a t i o n s to the d a t a f o r the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n r e s p e c t i v e l y : e l ( Z ) and c 2 ( Z ) a r e a l s o s m o o t h f u n c t i o n s w h i c h d e s c r i b e the s t a t i s t i c a l e r r o r s in d e t e r m i n i n g WR(Z) and (pI(Z). In r e f . [4] we d e f i n e d t h e m by t h e conditions 1
e l ( Z / ) : 5RiPR"~(Zi ) , / -.)
E2(Z i) = 5 I i P I - ( Z i) ,
(4)
w h e r e 5Ri, 0 R ( Z i ) and 6it, p I ( Z i ) a r e the e x p e r i m e n t a l e r r o r at Z = Z i and d e n s i t y of e x p e r i m e n t a l p o i n t s n e a r Z - Z i f o r the r e a l a n d i m a g i n a r y p a r t s of the f u n c t i o n r e s p e c t i v e l y . T h i s d e f i n i t i o n h a s the a d v a n t a g e of m a k i n g the p r o b a b i l i t y f u n c t i o n a l d e f i n e d in e q s . (1) a n d (3) c o n f o r m a t l y i n v a r i a n t . H o w e v e r , in p r a c t i c e it m a y be m o r e a p p r o p r i a t e to u s e f o r ~ I ( Z ) and ~2(Z) the e r r o r s o b t a i n e d when d e t e r m i n i n g the s m o o t h e d i n t e r p o l a t i o n s to the d a t a q)R(Z) and (pi(Z), T h i s w i l l c o n t a i n i n f o r m a t i o n a b o u t the d e n s i t y of d a t a p o i n t s and i s p e r h a p s m o r e n a t u r a l than eq. (4). H o w e v e r a s t h i s c h o i c e w i l t not n e c e s s a r i l y e n s u r e the c o n f o r m a l i n v a r i a n c e of the p r o b a b i l i t y f u n c t i o n a l c a r e m u s t be t a k e n to input e 1 and ~2 in the v a r i a b l e f o r w h i c h the p r o b a b i l i t y f u n c t i o n a l i s d e f i n e d . On t h e p a r t of the b o u n d a r y w h e r e no e x p e r i m e n t a l i n f o r m a t i o n i s a v a i l a b l e cp(Z) w i l l be g i v e n by the t h e o r e t i c a l e s t i m a t e of the f u n c t i o n and t h e e r r o r f u n c t i o n s m a y be c h o s e n to r e f l e c t o u r faith in t h e s e e s t i m a t e s . We c o n s i d e r the H i l b e r t s p a c e H b e i n g the r e a l v e c t o r s p a c e of a l l f u n c t i o n s a n a l y t i c in D with i n n e r p r o d u c t d e f i n e d by (F, G) H
(Re F, Re (;)1 + (Ira F , I m G) 2 ,
(5)
where
(/' ~)i
i ff(z)~(z)ei2(z) idZ ~' •
= Z
B
oO
W e choose a basis {pn(Z)}o of H so that any function F(Z) in H m a y be expressed in the form co
F(Z) = ~
bnPn(Z),
(6)
n:0 w h e r e the b n a r e r e a l . It i s a l s o c o n v e n i e n t to i n t r o d u c e two m o r e H i l b e r t s p a c e s H 1 and H 2 on the b o u n d a r y B, b e i n g the s p a c e s of r e a l f u n c t i o n s on B with i n n e r p r o d u c t s
G.G.Ross, optimised integral continuation
275
oo
(f, g)l and (f, g)2 respectively. Let {~l(Z)}o and {7;2(Z)}o be real orthogonal bases for H 1 and H 2 respectively. On B RePn(Z) and ImPn(Z) may be written
RePn(Z)= ~ clnk~(Z), k=O ImPn(Z) = ~ k=0
C~ten2k(Z).
(7)
Similarly ,~R(Z) and cpI(Z) may be expanded in t e r m of this basis , ~R(Z)
~oi(Z)
=
=
2_5
~
1
(z),
~ 992DZ ~ 2~q~/( z ) . m=0
(8)
It was found in ref. [4] that x2(F) could be written in the form x2(F) =
(bX- Y)(bX- Y ) T + w ,
where
XX T = C 1C 1 T + C 2C 2T -: U, y x T = ~ I c 1T+cy2C 2T_= V, W= ~ lq~lT+cp2cy 2T-
yyT,
(9)
and the m a t r i c e s C I, q~I and b are defined by
(C I)ij = c iIj . (qI)i = ~p[ ,
(b).= b.. $ $ nal
The most probable function if(Z) which m i n i m i s e s the probability functioP(F) defined in eqs. (1) and (3) is given by
Y(zl = ~ Bnpn(z), n=0 where
B n =-(yx-1)n = (VU-1)n. Finally if R is a linear functional in H we may write
(lo)
276
G.G.Ross, Optimised integral continuation
R(F) = (F, r) H ,
(11)
f o r s o m e function r(Z) in H. The a v e r a g e value of a functional R in H m a y be defined a s
f d2F
P(F) R(F)
f d2F
(12)
P(F)
w h e r e the f d 2 F r u n s o v e r all the v a r i a b l e s being (- ~, ~). It was shown in ref. [4] that
bn the l i m i t s of i n t e g r a t i o n
(R) R(~V), (AR) 2 = (r. r)H .
(13)
It is t h e r e f o r e e a s y to e v a l u a t e the a v e r a g e value and s t a n d a r d d e v i a t i o n of any l i n e a r functional once the m o s t p r o b a b l e function is known.
3. C O N T I N U A T I O N W I T H I N A S I M P L E C O N N E C T E D A N A L Y T I C I T Y REGION F o r s i m p l i c i t y we m a p the a n a l y t i c i t y d o m a i n into the unit c i r c l e in the c o m p l e x Z - p l a n e . The e x p l i c i t f o r m of the m a p p i n g is given in ref. [4]. If we confine the a n a l y s i s to hermitiantanalytic_ functions we m a y c h o o s e the given by are 27; C!. 1 f zy 2~ 0
cos ii3 7;) (t3) Cl 2 (/~) dfi ' 27r
C2..= 1 U
f
sini/3 ,v~ t3)E;2(i3)d~ ,
(14)
2~ 0
w h e r e the i n t e g r a l s a r e round the unit c i r c l e Z - e i13, T h u s U d e f i n e d in eq. (9) h a s the f o r m c o s n$ c o s mfi + sin
oo
Using this e x p r e s s i o n we m a y e v a l u a t e ~ n = 0
a m Zm
1
}cos
where
y = ei$
nfi sin rnfi
UnrnZm to obtain
nil+ sin n~? I dy
(16)
G.G.Rvss, optimised integral continuation
277
If we write the left-hand side in the f o r m of an integral over the boundary. and r e - a r r a n g e the t e r m s on the r i g h t - h a n d side. we obtain the equation
f ~ ~;~m - - - yrr' dy 2~i B n 0 y - Z 1
~
{ {[ci2(y)+E
;2(y)] n
_,
)~v-n~7 vdYz . +[e12(y)- - 22, ~y]~'
(17)
T-1l and sum over the index n to obtain the relation Finally we multipty by gk~ defining U- I
~'= 1
n 0 (18)
~/=0 where
the a r are constants.
o
W e will solve e q (18) for U-I for the ease in which c~(y) and c~(y) m a y
be expressed as the quotient of finite order polynomials in cosfi. W e write ~12(y) and ~ 2 ( y ) i n the form
c1 (y) : - - ,
c
y) -
,
(19)
w h e r e 721(5,) and ~72(y) are p o l y n o m i a l s in cos t3 of d e g r e e M and 52(y) and 522(y) a r e p o l y n o m i a l s in cos/3 of d e g r e e N. Eq. (18) b e c o m e s 2r12(y) ~yk + r =1
at
L'k~z Y + A2(Y)
n :0
n :0
kn y
(20)
where 2, ,) ~721(y ) 2 77 (5 = U2(y) ,
A 21(y)
=
521(y) 2 522(y) 2 ~2(y) + ~I(Y) ,
A22(y ) : 521(y) 2q2(y) - 5~(y) 2771(y)
(21)
Eq. (20) r e q u i r e s that ~ k ~, n=O ZkUkln yn be of the f o r m (for N >~ M) (22)
k,.:0
k:0 r:-N-M
G.G.Ross, Optimised integral continuation
278
w h e r e the kkr a r e c o n s t a n t s to be d e t e r m i n e d . We now define the functions ~'I(Y) and ~(y) a n a l y t i c and without z e r o s within the unit c i r c l e . On B t h e s e f u n c t i o n s a r e c h o s e n such that 2 X l ( 3 ; ) a 1 (1) : A l ( Y ) , 1
~(3') r/(~) = ~72(3,) ,
(23)
AI(Y) and ~(3') will be p o l y n o m i a l s in v of d e g r e e N+M and 2 M r e s p e c t i v e l y . U s i n g t h e s e functions and eq. (22) we m a y w r i t e n=0 Z yn in the form k -1 ~z
2
k,n 0 Z UknY = X l ( y ) ~ I ( Z ) ( 1 - y Z ) ×
~[=1~ l(y)
~ ( 1 - yZ)
~Xl(1)
~ k=0 r---N-M
'
~tkrY
rzk
, (24)
t
w h e r e ~tky a r e c o n s t a n t s . We o b s e r v e that - ~ ( y ) [ ~ ( 1 / y ) ~-AI(Z)- ~ ( Z ) ~" A l"( 1 / y -~ ) ] / ( 1 - ya) m a y be w r i t t e n in the f o r m
N+M
M ~ akr y r z k k=0 r=-N- M
and t h e r e f o r e (24) m a y be r e w r i t t e n as
k,n:O
Z
-i n 2 UknY : ~ I ( ~ A ~ )
t~(y)~(Z ) N+M ,, r I t 1 - y Z +k,r:O ~ XkrY Zk "
(25)
H e r e we have used the fact that U is s y m m e t r i c to r e s t r i c t the n u m b e r of k r e n t e r i n g the equation. T h e s e c o n s t a n t s m a y be e v a l u a t e d by c o n s t a n t s ~t" s u b s t i t u t i n g eq. (25) in eq. (20) for e a c h p o w e r of Z to give
N+M
,,
1
r
}
rO +~(y) 7~2(Y)~2(1)
k ~ 7?ryk_r = ~(1_) ~ akryr, (26) r=O r--i
where
2M ~(y)= ~ ,lry r . r=0
G. G. Ross, Optimised integral continuation
279
D e m a n d i n g that the l e f t - h a n d side of eq. (26) has the 2M z e r o s of ~(1/y) TT g i v e s 2 M e q u a t i o n s i n v o l v i n g the N+M+I c o n s t a n t s X k r , k fixed. A f u r t h e r N - M + 1 e q u a t i o n s a r e o b t a i n e d f r o m the condition that the l e f t - h a n d side c o n t a i n s no p o w e r o f y l e s s than ( - 2 M + l ) . T h e s e N + M + l equations uniquely d e t e r m i n e the N+M+ 1 c o n s t a n t s X~r, and thus we m a y d e t e r m i n e all the TT c o n s t a n t s ~'kr i n v o l v e d in eq. (25). We a r e now in a p o s i t i o n to e v a l u a t e the m o s t p r o b a b l e function F(Z). We have ((plc1T+ (p2c2T)n
i 2~
:
}7' {WR(~)c°~sn[~ c2(2) +(PI(i3) sei 2n ~n ~
0
1
~ (PR(Y)
iq°I(Y)
ldi?
{yn dy
T h e n f r o m eqs. (9), (10) and (24)
f(Z)=
1
f __)WR(Y)
iWI(Y) t
-rriTxl(z) C)q°R(Y)ll
~ ynu-1 n,m=0 nm
iq)I(Y) I t
Zm dy
(Z) + 1
-yZ
" v k} 1 N+M xkry z al(rj k,r=O
(27)
T h i s g i v e s the i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e function in t e r m s of the b o u n d a r y v a l u e s of the function and the e r r o r s . It is v a l i d t h r o u g h o u t tile whole d o m a i n D. H a v i n g found the m o s t p r o b a b l e function it is e a s y to find the m e a n v a l u e of the function at any point Z o within D. To do this we c h o o s e the function R(F) to be given by
R(F) -
ReF(Zo) .
T h e n by eq. (13) (R(F)) - ReF(Zo) and Re/F(Z o) Is given by eq. (27). The standard deviation is calculated
n =0
Yn = 5
Then
,
via eq. (13). We
n =0
(Z 2+ (Z;) rn) (U-1)mn . m=O
(28) have
G. G.Ross, Optimised integral continuation
280
(AR) 2 -
rUt T
n, m=0 U s i n g eq. (25) t h i s b e c o m e s (AR) 2 : 2 R e I
~l(-Zo ))
1
A I ( Z o ) A I ( Z o ) (1
1
I Zol 2 )
N+ M
+ ~ Lr,)t" kz (Z~)r) : I kr (zk+r+ o
(29)
k,r:O S i m i l a r l y we m a y c a l c u l a t e the m e a n v a l u e and s t a n d a r d d e v i a t i o n of the i m a g i n a r y p a r t of the function.
4. C O N T I N U A T I O N TO THE B O U N D A R Y AND O T H E R A P P L I C A T I O N S In the p r e v i o u s s e c t i o n we c o n s i d e r e d the p r o b l e m of c o n t i n u a t i o n w i t h i n the a n a l y t i c i t y r e g i o n . C o n t i n u a t i o n to the b o u n d a r y i t s e l f i s an u n s t a b l e p r o c e d u r e f o r , a s m a y be s e e n f r o m eq. (29), t h e s t a n d a r d d e v i a t i o n b e c o m e s i n f i n i t e due to the v a n i s h i n g of the d e n o m i n a t o r (1 - iZo !2) in the s e c o n d t e r m . T h i s i n s t a b i l i t y w a s n o t e d by B e r t e r o a n d V i a n o [5] and a l s o by P r e s n a j d e r and P i s u t [6] who g a v e a s i m p l e p r o o f of the r e s u l t . A s d i s c u s s e d by the l a t t e r a u t h o r s t h e r e a r e two w a y s of o v e r c o m i n g the d i f f i c u l t y . T h e f i r s t , s u g g e s t e d by C i u l l i , r e m o v e s the i n s t a b i l i t y by i m p o s i n g a d d i t i o n a l s m o o t h n e s s r e q u i r e m e n t s on the b e h a v i o u r of the f u n c t i o n on the b o u n d a r y . W e w i l l not c o n s i d e r t h i s m e t h o d f u r t h e r h e r e . T h e s e c o n d , s u g g e s t e d by P r e s n a j d e r and P i s u t [6], e v a d e s the d i f f i c u l t y by c a l c u l a t i n g a v e r a g e v a l u e s of the f u n c t i o n o v e r p r e - c h o s e n a r c s r a t h e r than t h e v a l u e of the f u n c t i o n in a p o i n t of the b o u n d a r y . F o r i n s t a n c e , we d e f i n e the f u n c t i o n a l R ( F ) to be
R(F) = f
{Re F(Z)} F(Z) I dZ
I,
(30)
Y w h e r e the i n t e g r a l is o v e r s o m e a r e 7 of the b o u n d a r y and F ( Z ) i s an a r b i t r a r y r e a l f u n c t i o n . T h e m e a n v a l u e of R(F) w i l l , a s u s u a l , be g i v e n by R(F) and u s i n g eq. (13) we m a y e a s i l y find the f o l l o w i n g e x p r e s s i o n for (AR) 2. (AR) 2
= 2 f ]dZ I f ]dy' I Re Y
N+M
y
,,
+ ~ k . r : O '~kr
t ~(Z)~(y)
1 ~ 1 (Z)/~ l ( y ) (1_ :~2-) +
~(Z) ~'(y*)
1
(1~
}
(ZkYr+ZkY*r) I
F(y)F(Z)
(31)
G.G.Ross, optimised integral continuation
281
It is evident that this will not, in general, be infinite and thus continuation to average values over parts of the boundary may be made a stable procedure. However, in many applications in particle physics, continuation to the boundary is necessary not to evaluate the function at, or near, apoint, but to test some hypothesis about the behaviour of the function over a particular region of the boundary and to obtain predictions about the parameters involved in the hypothesis. For example [i] in finite energy sum rules a Regge hypothes;.s is made about the scattering amplitude over some high-energy region and the Regge parameters then obtained via integrals over the lowenergy regions. The formalism developed above lends itself readily to such problems and, moreover, statistically meaningful errors are obtained for the predictions of the parameters. We suppose that we have an hypothesis, involving the N parameters gi, about the behaviour of the function over some part of the boundary, K. go(Z) is then given on K by the hypothesis and tile error functions el(Z) and E2(Z) are chosen inversely proportional to the accuracy to which the hypothesis is expected to be true. For example, in the case of a Regge-pole hypothesis, q(Z) and e2(Z) would be of the order of the lower pole terms ignored in the Regge expansion and any cut contributions expected to be present. We again consider the probability functional P(F) defined in eqs. (1) and (3). This will now involve the additional parameters gi' From eq. (9) we see that P(F) will be a maximum when W(g) is a minimum and bn =Bn(g) where Bn is defined in eq. (10) and is now a function of the additional parameters gi" The function W(g) is of the form
1 f)go2(y)
1 (27r) 2 Bf dy
go2(y)tdy
d Z )goR (y)
i goI(Y) __ _
~
7nu-1z
rn IgoR t ~ ) (Z)
igoI (Z) 1' c2(Z) (32)
and
7nu -1 Z m n, wI-O
nrtz
is given by eq. (24). The m o s t p r o b a b l e p a r a m e t e r s equations ~W(g)[
Ogi ~g:g
= 0 ,
gi
i = 1 .... N,
a r e given by the (33)
t o g e t h e r with the condition that this should be a m i n i m u m of W(g). H a v i n g found t h e s e p a r a m e t e r s the m o s t p r o b a b l e function F(Z) is a g a i n given by eq. (27) u s i n g gi = g'i in e v a l u a t i n g go(Z). T h e a v e r a g e v a l u e of a functional in H m a y be defined as in eq. (12), but
G.G.Ross, Optimised integral continuation
282
now the f u n c t i o n a l i n t e g r a t i o n r u n s a l s o o v e r the a l l o w e d v a l u e s of the gi" D e f i n i n g G = b X - Y we note that dG n and db n d i f f e r only by the J a c o b i a n of the t r a n s f o r m a t i o n a n d we m a y r e d e f i n e the f u n c t i o n a l i n t e g r a t i o n in eq. (12) to be o v e r the G n and g n w i t h o u t c h a n g i n g the p r o b a b i l i t y f u n c t i o n a l . A l i n e a r f u n c t i o n a l in H m a y , by eqs. (11) and (5), be w r i t t e n in the f o r m
R(F) - r U b T , w h e r e (r) n = r n and we h a v e e x p a n d e d r(Z) d e f i n e d in eq. (11) in the f o r m oO
r(Z) : ~
rnPn(Z).
n=0 F r o m eq. (11), a n d the r e d e f i n i t i o n of the i n n e r p r o d u c t we h a v e
(R) = (ru[GX -1 + B(g)]W) , (R) = (~'uBT(g)>g, w h e r e ( }g d e n o t e s the a v e r a g e i s only o v e r the N p a r a m e t e r s s t a n d a r d d e v i a t i o n AR of a f u n c t i o n a l i s g i v e n by
(34)
gi. The
(An) 2 : (R 2)- ( n ) 2 . We
have
B(g)]T}2) ({ruBT(g)) 2)g.
(I{2) : ( { r U [ G X -I + = rurT+ Thus
(AR)2 = r u r T + ( { r u B T (g)} 2 )g_ ( r u B T (g))~ .
(35)
We c o n s i d e r the a p p l i c a t i o n of t h e s e r e s u l t s to the e x a m p l e of a f u n c t i o n
f ( s , t) which by h y p o t h e s i s , m a y be e x p a n d e d in a r e g i o n s I ~< s ~< s 2 in t e r m s of a f i n i t e n u m b e r of Regge p o l e s .
f ( s , t ) = ~N g i ( t ) [ ± l - e - i ~ i ( t ) ] (~s-)cq(t)/sin ~ i ( t ) i=1 ~s°/
(36)
We w i s h , u s i n g f i x e d - t r e l a t i o n s , to e x t r a c t i n f o r m a t i o n f r o m the data a b o u t the r e s i d u e f u n c t i o n s gi(t). Note that a p a r t i c u l a r a d v a n t a g e of this m e t h o d o v e r c o n v e n t i o n a l f i n i t e e n e r g y s u m r u l e s is that by a s u i t a b l e c h o i c e of ¢1(Z) and ¢2(Z) c o n s i s t e n c y i s only r e q u i r e d with the Regge h y p o t h e s i s o v e r a f i n i t e r a n g e of s a n d not to i n f i n i t y o r into the c o m p l e x s - p l a n e . T h e m o s t p r o b a b l e v a l u e s of gi(t) a r e then found by s o l v i n g the eqs. (33), g i v e n the f o r m of W(g), eq. (32), and the d e p e n d e n c e on the p a r a m e t e r s gi(t) o v e r the r e g i o n K : ( Z ( S l ) , Z (s2)) s p e c i f i e d by eq. (36). T h e a v e r a g e v a l u e and s t a n d a r d d e v i a t i o n of a n y l i n e a r f u n c t i o n in H a r e , in this c a s e , e a s i l y o b t a i n e d f r o m the m o s t p r o b a b l e f u n c t i o n for, e x c e p t for an e l a s t i c a m p l i t u d e in the f o r w a r d d i r e c t i o n , the p a r a m e t e r s g~(t) m a y r a n g e f r o m (_ o% oo) and t h i s , t o g e t h e r with the b i l i n e a r f o r m of ×~'(g) g i v e n by eq. (9) a n d eq. (32) e n a b l e s u s to w r i t e
G. G. Ross, Oplimised inlegral continuation
283
(R) : @uBT(g)}g : rUB(g) N (AR) 2 = r U t T + ~
(37)
R(~),
- 1 ( r UB T (2)) , (ruBT(g)) Eij
(38 )
i, j l where
aB(g) Bi(g) = ~gi- g : ~ ' a n d the e r r o r m a t r i x Eij i s d e f i n e d by ~,2W(g)
2Eij-
Ogi~gj •
In the f o r w a r d d i r e c t i o n the i m a g i n a r y p a r t of the a m p l i t u d e g i v e n by eq. (36) w i l l h a v e to s a t i s f y a p o s i t i v i t y c o n s t r a i n t but, p r o v i d e d the p r o b a b i l i t y i s s m a l l f o r v a l u e s v i o l a t i n g t h i s c o n s t r a i n t the e r r o r s i n c u r r e d by i n t e g r a t i n g o v e r t h e s e u n p h y s i c a l v a l u e s w i l l be n e g l i g i b l e and e q s . (37) and (38) will remain valid. T h e m e a n v a l u e s and s t a n d a r d d e v i a t i o n of the R e g g e p a r a m e t e r s g i a r e g i v e n s i m p l y by
(gi) : F'i, (Ag/)2 = R:I~z "
(39)
F i n a l l y we c o n s i d e r the c a s e in w h i c h we a r e g i v e n i n f o r m a t i o n q~M(Z), ~ I ( Z ) and ¢2(Z) a b o u t the b o u n d a r y v a l u e s and e r r o r s of a f u n c t i o n F M ( Z ) , m e r o m o r p h i c w i t h i n D. If the f u n c t i o n h a s N s i m p l e p o l e s w i t h i n D at p o s i t i o n s Z i w e m a y c o n s t r u c t a f u n c t i o n F(Z), a n a l y t i c w i t h i n D, by e x p l i c i t l y subtracting these poles N
F(Z) : F M ( Z ) - ~
z
gi_ z
i
•
i:1
(38)
T h e b o u n d a r y v a l u e s of F(Z) w i l l now b e g i v e n by q~(Z) = gOM(Z ) -
N gi i=l Z - Z i "
(39)
If, u s i n g t h i s f o r m f o r go(Z), t o g e t h e r w i t h ¢ l ( Z ) a n d e 2 ( Z ) , we c o n s t r u c t the p r o b a b i l i t y f u n c t i o n a l P(F) a s d e f i n e d in e q s . (1) a n d (3), it w i l l be a f u n c t i o n of t h e r e s i d u e p a r a m e t e r s gi a s w e l l a s the b n. As in the p r e v i o u s c a s e we find the m o s t p r o b a b l e v a l u e s ~ a r e g i v e n by s o l v i n g e q s . (33) and the m e a n v a l u e and s t a n d a r d d e v i a t i o n of the p a r a m e t e r s eq. (39). O n c e a g a i n c a r e m u s t b e t a k e n in u s i n g eq. (39) if the gi h a v e a positivity requirement.
284
G. G. R o s s , Oplimised inleg~'al conlinuation
In all the a p p l i c a t i o n s c o n s i d e r e d a b o v e , h a v i n g found the m o s t f u n c t i o n and, if n e c e s s a r y , the m o s t p r o b a b l e v a l u e s for the o t h e r e t e r s i n v o l v e d , we m a y e v a l u a t e the v a l u e of X2 f o r t h i s s o l u t i o n . (9) we s e e that this i s g i v e n by ×2(~) W(g) w h e r e W(g) i s d e f i n e d (32). We now d e f i n e the q u a n t i t y C by the e q u a t i o n
probable paramF r o m eq. in eq.
cO
C : ,,J,2 - P(x2) dx2 ' (40) X (f) C is the p r o b a b i l i t y that X2 e x c e e d s the o b s e r v e d v a l u e of X2(F). We m a y u s e C as d e f i n e d a b o v e a s a m e a s u r e of the c o m p a t i b i l i t y of an a n a l y t i c f u n c t i o n , t o g e t h e r with any o t h e r h y p o t h e s e s m a d e , with the data i n p u t a s b o u n d a r y v a l u e s f o r the f u n c t i o n .
5. CONCLUSIONS In the m e t h o d d e s c r i b e d a b o v e we have w o r k e d in the s p a c e of all a n a l y t i c f u n c t i o n s . T h e a d v a n t a g e of t h i s l i e s in the fact that it r e m o v e s a n y b i a s i n h e r e n t in a p a r t i c u l a r c h o i c e of p a r a m e t e r i s a t i o n for a n a n a l y t i c f u n c t i o n . O n e s t i l l h a s the p r o b l e m of d e f i n i n g the p r o b a b i l i t y f u n c t i o n a l and, in p a r t i c u l a r , the e r r o r f u n c t i o n s El(Z) a n d E2(Z). In r e g i o n s w h e r e d a t a i s a v a i l a b l e t h i s is s t r a i g h t f o r w a r d but o v e r r e g i o n s of the b o u n d a r y w h e r e s o m e h y p o t h e s i s is m a d e t h e i r c h o i c e r e q u i r e s s o m e p h y s i c a l j u d g e m e n t . T h i s of c o u r s e j u s t s t a t e s the n e c e s s i t y c l e a r l y to d e f i n e the p r o b l e m to be solved. The m a i n a d v a n t a g e of the t e c h n i q u e is that, o n c e the e r r o r f u n c t i o n s a r e g i v e n , the b e s t c o n t i n u a t i o n p o s s i b l e i s u n a m b i g u o u s l y d e t e r m i n e d . M o r e o v e r r e a l i s t i c e x p r e s s i o n s for the s t a t i s t i c a l e r r o r s a r e o b t a i n e d , t o g e t h e r with a m e a s u r e of the c o n s i s t e n c y of the f i n a l r e s u l t with the data. Although the s o l u t i o n p r e s e n t e d a p p l i e s to a r e s t r i c t e d c l a s s of e r r o r f u n c t i o n s if i s hoped that it will be a p p l i c a b l e to m o s t p h y s i c a l p r o b l e m s of i n t e r e s t . A p r a c t i c a l a p p l i c a t i o n of o p t i m i s e d i n t e g r a l c o n t i n u a t i o n m e t h o d s to the p r o b l e m of the d e t e r m i n a t i o n of the p i o n - n u c l e o n c o u p l i n g c o n s t a n t , u s i n g e q u a l e r r o r f u n c t i o n s for the r e a l and i m a g i n a r y p a r t s , h a s a l r e a d y a p p e a r e d [8]. F u r t h e r a p p l i c a t i o n s , u s i n g d i f f e r e n t e r r o r s for the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n , a r e at p r e s e n t b e i n g p u r s u e d . RE F E R E N C ES [1] A.A.Logunov, L.D. SoIoviev and A.N.Tabkhelidze, Phys. Letters 24B (1967) 181; K.Igi and S.Matsuda, Phys. Bey. Letters 18 (1967) 625; P.Dolen, D.Horn and C.Sehmid, Phys. Rev. 166 (1968) 1768. [21 P . E . Cutkosky, Ann. of Phys. 54 (1969) 350. [31 P . P r e ~ n a j d e r and J.Pi~fit, Nuovo Cimento 3A (1971) 603. [4] G . G . R o s s , Nuel. Phys. B31 (1971) 113. [5] M . B e r t e r o and G.A.Viano, Nuovo Cimento 38 (1965) 1915. [6] P . P r e ~ n a j d e r and J . P i ~ f t , Nuel. Phys. B22 0970) 365. [7] S.Cuilli, Nuovo Cimento 61A (1969) 786; 628 (1969) 301. [8] P . L i e h a r d and P. Pre~najder, Bratislava preprint.
I
Nuclear Physics
B41 (1972) 272-284.
North-Holland Publishing Company
B.11
OPTIMISED
INTEGRAL C O N T I N U A T I O N
METHODS
G. G. ROSS
Rutherford High Energy Laboratory, Chilton, Didcol, Berkshire, England Received 25 October 1971 Abstract: We consider the problem of finding the analytic function which m i n i m i ses the difference between it and our information about the function on the boundary, weighted according to the accuracy to which the function is known. A r e p r e s e n t a tion of the function is found in the form of an integral over the boundary involving the function boundary values and e r r o r s . Applications involving continuation throughout the analyticity region and, in p a r t i c u l a r , to the boundary are i n v e s tigated. Optimised relations are derived for the determination of Regge-pole p a r a m e t e r s from low-energy information and also for the determination of pole r es i d u es . In each case r e a l i s t i e statistical e r r o r s are given.
1. I N T R O D U C T I O N T h e c o n t i n u a t i o n of an a n a l y t i c f u n c t i o n t h r o u g h o u t i t s d o m a i n of a n a ! y t i c i t y h a s c o n v e n t i o n a l l y b e e n p e r f o r m e d u s i n g C a u c h y ' s i n t e g r a l f o r m u l a to e x p r e s s the c o n t i n u e d f u n c t i o n as an i n t e g r a l i n v o l v i n g the f u n c t i o n e v a l u a t e d o v e r the b o u n d a r y of it s a n a l y t i c i t y r e g i o n . S i m i l a r m e t h o d s h a v e b e e n u s e d to r e l a t e the i n t e g r a l s o v e r v a r i o u s r e g i o n s of the b o u n d a r y . F o r e x a m p l e , i n f o r m a t i o n about R e g g e p a r a m e t e r s m a y be o b t a i n e d v i a f i n i t e e n e r g y s u m r u l e s [1] f r o m i n t e g r a l s i n v o l v i n g the s c a t t e r i n g a m p l i t u d e o v e r the l o w e n e r g y p a r t of the b o u n d a r y . I n f i n i t e l y m a n y of s u c h r e l a t i o n s a r e p o s s i b l e in w h i c h the w e i g h t i n g of d i f f e r e n t r e g i o n s of the b o u n d a r y v a r i e s . M o r e o v e r , in the a b s e n c e of a p r e c i s e k n o w l e d g e of the f u n c t i o n , t h e s e d i f f e r e n t relations may yield widely varying results. I d e a l l y only one r e l a t i o n would be a p p l i c a b l e in any p a r t i c u l a r c a s e : that r e l a t i o n b e i n g d e t e r m i n e d in s u c h a m a n n e r a s to w e i g h t the c o n t r i b u t i o n of e a c h p a r t of the b o u n d a r y in p r o p o r t i o n to the a c c u r a c y to w h i c h the f u n c t i o n i s known. T h i s i d e a m a y be m a d e m o r e p r e c i s e if, f o l l o w i n g C u t k o s k y [2], we i n t r o d u c e a p r o b a b i l i t y f u n c t i o n a l P(F) of the f o r m
P(F)
=
exp {- }x2(F)} ,
(1)
where ×2(F)
1
= £ f !F(Z)- (fltZ)I2M-2(Z)idZ! B
.
(2)
G.G.Ross, Optimised integral continuation
273
H e r e L i s the l e n g t h of the b o u n d a r y B, c0(Z) i s the g i v e n i n f o r m a t i o n of the f u n c t i o n on the b o u n d a r y and M(Z) 0 d e s c r i b e s the s t a t i s t i c a l e r r o r in d e t e r m i n i n g ~(Z). T h e p r o b a b i l i t y f u n c t i o n a l d e s c r i b e s the c o n s i s t e n c y of a n y a n a l y t i c f u n c t i o n F(Z) with the d a t a . U s i n g t h i s f o r m f o r the p r o b a b i l i t y f u n c t i o n a l , P r e s n a j d e r and P i s ~ t [3] h a v e found an i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e f u n c t i o n , n a m e l y that f u n c t i o n F(Z) w h i c h m a x i m i s e s
P(F). U n f o r t u n a t e l y the f o r m of eq. (2) p r e s u m e s that the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n a r e d e t e r m i n e d with e q u a l p r e c i s i o n . A m o r e g e n e r a l c h o i c e f o r x 2 ( F ) i s g i v e n by 1 Bf t [ R e F ( Z ) - (pR(Z)]2 ×2(F) = L [Im F ( Z ) - ~ I ( Z ) ] 2 el(Z)2 + c2(Z) 2 I idZ/ '
(3)
H e r e e l ( Z ) and e2(Z) 0 a r e the u n c e r t a i n t i e s in d e t e r m i n i n g the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n a l o n g the b o u n d a r y , (?R(Z) and (pi(Z) r e s p e c t i v e l y . In r e f . [4] we h a v e found, f o r c o n t i n u a t i o n w i t h i n a s i m p l y c o n n e c t e d d o m a i n , an i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e f u n c t i o n f o r a r e s t r i c t e d c l a s s of e r r o r f u n c t i o n s e l ( Z ) and e2(Z). One of the a i m s of t h i s p a p e r i s to e x t e n d t h i s s o l u t i o n to a m o r e g e n e r a l c l a s s of e r r o r f u n c t i o n s w h i c h s h o u l d e n a b l e the m e t h o d to be u s e d in m o s t c i r c u m s t a n c e s , An i m p o r t a n t r e s u l t of the t e c h n i q u e is that the a v e r a g e v a l u e and s t a n d a r d d e v i a tion of a n y f u n c t i o n a l m a y e a s i l y be found o n c e the m o s t p r o b a b l e f u n c t i o n i s known. It s h o u l d be s t r e s s e d that a r e a l i s t i c e r r o r f o r the c o n t i n u e d v a l u e of a f u n c t i o n is of g r e a t p r a c t i c a l i m p o r t a n c e and h e r e i n l i e s a g r e a t a d v a n t a g e of the m e t h o d o v e r p r e v i o u s d i s p e r s i o n r e l a t i o n a p p l i c a t i o n s . M o r e o v e r , a s we s t a t e d a b o v e , o n l y one r e l a t i o n i s a p p l i c a b l e o n c e the s t a t i s t i c a l e r r o r s of the i n p u t d a t a a r e d e f i n e d . By a s u i t a b l e c h o i c e of f u n c t i o n a l , c o n t i n u a t i o n t h r o u g h o u t the c o m p l e x z p l a n e i s s t r a i g h t f o r w a r d . H o w e v e r c o n t i n u a t i o n to the b o u n d a r y p r e s e n t s s p e c i a l d i f f i c u l t i e s s i n c e the s t a n d a r d d e v i a t i o n b e c o m e s i n f i n i t e f o r s u c h a c o n t i n u a t i o n . W e c o n s i d e r w a y s to o v e r c o m e t h i s p r o b l e m and d e v e l o p a m e t h o d to t e s t any h y p o t h e s i s about the function o v e r a r e g i o n of the b o u n d a r y and to d e t e r m i n e the p a r a m e t e r s i n v o l v e d in t h i s h y p o t h e s i s . As an e x a m p l e of t h i s m e t h o d we d i s c u s s the o p t i m i s e d f i n i t e e n e r g y s u m r u l e s f o r the d e t e r m i n a t i o n of R e g g e - p o l e p a r a m e t e r s , t o g e t h e r with e x p r e s s i o n s f o r the s t a n d a r d d e v i a t i o n of t h e s e p a r a m e t e r s . We a l s o a p p l y the m e t h o d to the p r o b l e m of d e t e r m i n i n g the p o l e r e s i d u e s of a m e r o m o r p h i c function. In s e c t . 2 we i n t r o d u c e the n o t a t i o n and r e v i e w the r e s u l t s of r e f . [4] n e c e s s a r y f o r the s u b s e q u e n t d i s c u s s i o n . S e c t . 3 d e a l s with the d e r i v a t i o n of an i n t e g r a l r e p r e s e n t a t i o n of the m o s t p r o b a b l e f u n c t i o n for the c a s e of a s i m p l y c o n n e c t e d a n a l y t i e i t y r e g i o n and e x t e n d s the r e s u l t s of r e f . [4] to a m o r e g e n e r a l c l a s s of e r r o r f u n c t i o n . F i n a l l y in s e c t . 4 we c o n s i d e r p h y s i c a l a p p l i c a t i o n s of the m e t h o d and in p a r t i c u l a r a p p l y it to t h e d e t e r m i n a t i o n of R e g g e p a r a m e t e r s and p o l e r e s i d u e s .
274
G. G . R o s s , OptinHsed integral continuation
2. T H E P R O B A B I L I T Y F U N C T I O N A L , MOST P R O B A B L E F U N C T I O N , A V E R A G E V A L U E S AND S T A N D A R D D E V I A T I O N S We s u p p o s e we h a v e a f u n c t i o n F ( Z ) a n a l y t i c in a r e g i o n D with b o u n d a r y B. O v e r the r e g i o n of the b o u n d a r y w h e r e e x p e r i m e n t a l i n f o r m a t i o n is a v a i l a b l e we d e f i n e the f u n c t i o n s q)R(Z) and (pI(Z) a s s m o o t h i n t e r p o l a t i o n s to the d a t a f o r the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n r e s p e c t i v e l y : e l ( Z ) and c 2 ( Z ) a r e a l s o s m o o t h f u n c t i o n s w h i c h d e s c r i b e the s t a t i s t i c a l e r r o r s in d e t e r m i n i n g WR(Z) and (pI(Z). In r e f . [4] we d e f i n e d t h e m by t h e conditions 1
e l ( Z / ) : 5RiPR"~(Zi ) , / -.)
E2(Z i) = 5 I i P I - ( Z i) ,
(4)
w h e r e 5Ri, 0 R ( Z i ) and 6it, p I ( Z i ) a r e the e x p e r i m e n t a l e r r o r at Z = Z i and d e n s i t y of e x p e r i m e n t a l p o i n t s n e a r Z - Z i f o r the r e a l a n d i m a g i n a r y p a r t s of the f u n c t i o n r e s p e c t i v e l y . T h i s d e f i n i t i o n h a s the a d v a n t a g e of m a k i n g the p r o b a b i l i t y f u n c t i o n a l d e f i n e d in e q s . (1) a n d (3) c o n f o r m a t l y i n v a r i a n t . H o w e v e r , in p r a c t i c e it m a y be m o r e a p p r o p r i a t e to u s e f o r ~ I ( Z ) and ~2(Z) the e r r o r s o b t a i n e d when d e t e r m i n i n g the s m o o t h e d i n t e r p o l a t i o n s to the d a t a q)R(Z) and (pi(Z), T h i s w i l l c o n t a i n i n f o r m a t i o n a b o u t the d e n s i t y of d a t a p o i n t s and i s p e r h a p s m o r e n a t u r a l than eq. (4). H o w e v e r a s t h i s c h o i c e w i l t not n e c e s s a r i l y e n s u r e the c o n f o r m a l i n v a r i a n c e of the p r o b a b i l i t y f u n c t i o n a l c a r e m u s t be t a k e n to input e 1 and ~2 in the v a r i a b l e f o r w h i c h the p r o b a b i l i t y f u n c t i o n a l i s d e f i n e d . On t h e p a r t of the b o u n d a r y w h e r e no e x p e r i m e n t a l i n f o r m a t i o n i s a v a i l a b l e cp(Z) w i l l be g i v e n by the t h e o r e t i c a l e s t i m a t e of the f u n c t i o n and t h e e r r o r f u n c t i o n s m a y be c h o s e n to r e f l e c t o u r faith in t h e s e e s t i m a t e s . We c o n s i d e r the H i l b e r t s p a c e H b e i n g the r e a l v e c t o r s p a c e of a l l f u n c t i o n s a n a l y t i c in D with i n n e r p r o d u c t d e f i n e d by (F, G) H
(Re F, Re (;)1 + (Ira F , I m G) 2 ,
(5)
where
(/' ~)i
i ff(z)~(z)ei2(z) idZ ~' •
= Z
B
oO
W e choose a basis {pn(Z)}o of H so that any function F(Z) in H m a y be expressed in the form co
F(Z) = ~
bnPn(Z),
(6)
n:0 w h e r e the b n a r e r e a l . It i s a l s o c o n v e n i e n t to i n t r o d u c e two m o r e H i l b e r t s p a c e s H 1 and H 2 on the b o u n d a r y B, b e i n g the s p a c e s of r e a l f u n c t i o n s on B with i n n e r p r o d u c t s
G.G.Ross, optimised integral continuation
275
oo
(f, g)l and (f, g)2 respectively. Let {~l(Z)}o and {7;2(Z)}o be real orthogonal bases for H 1 and H 2 respectively. On B RePn(Z) and ImPn(Z) may be written
RePn(Z)= ~ clnk~(Z), k=O ImPn(Z) = ~ k=0
C~ten2k(Z).
(7)
Similarly ,~R(Z) and cpI(Z) may be expanded in t e r m of this basis , ~R(Z)
~oi(Z)
=
=
2_5
~
1
(z),
~ 992DZ ~ 2~q~/( z ) . m=0
(8)
It was found in ref. [4] that x2(F) could be written in the form x2(F) =
(bX- Y)(bX- Y ) T + w ,
where
XX T = C 1C 1 T + C 2C 2T -: U, y x T = ~ I c 1T+cy2C 2T_= V, W= ~ lq~lT+cp2cy 2T-
yyT,
(9)
and the m a t r i c e s C I, q~I and b are defined by
(C I)ij = c iIj . (qI)i = ~p[ ,
(b).= b.. $ $ nal
The most probable function if(Z) which m i n i m i s e s the probability functioP(F) defined in eqs. (1) and (3) is given by
Y(zl = ~ Bnpn(z), n=0 where
B n =-(yx-1)n = (VU-1)n. Finally if R is a linear functional in H we may write
(lo)
276
G.G.Ross, Optimised integral continuation
R(F) = (F, r) H ,
(11)
f o r s o m e function r(Z) in H. The a v e r a g e value of a functional R in H m a y be defined a s
f d2F
P(F) R(F)
f d2F
(12)
P(F)
w h e r e the f d 2 F r u n s o v e r all the v a r i a b l e s being (- ~, ~). It was shown in ref. [4] that
bn the l i m i t s of i n t e g r a t i o n
(R) R(~V), (AR) 2 = (r. r)H .
(13)
It is t h e r e f o r e e a s y to e v a l u a t e the a v e r a g e value and s t a n d a r d d e v i a t i o n of any l i n e a r functional once the m o s t p r o b a b l e function is known.
3. C O N T I N U A T I O N W I T H I N A S I M P L E C O N N E C T E D A N A L Y T I C I T Y REGION F o r s i m p l i c i t y we m a p the a n a l y t i c i t y d o m a i n into the unit c i r c l e in the c o m p l e x Z - p l a n e . The e x p l i c i t f o r m of the m a p p i n g is given in ref. [4]. If we confine the a n a l y s i s to hermitiantanalytic_ functions we m a y c h o o s e the given by are 27; C!. 1 f zy 2~ 0
cos ii3 7;) (t3) Cl 2 (/~) dfi ' 27r
C2..= 1 U
f
sini/3 ,v~ t3)E;2(i3)d~ ,
(14)
2~ 0
w h e r e the i n t e g r a l s a r e round the unit c i r c l e Z - e i13, T h u s U d e f i n e d in eq. (9) h a s the f o r m c o s n$ c o s mfi + sin
oo
Using this e x p r e s s i o n we m a y e v a l u a t e ~ n = 0
a m Zm
1
}cos
where
y = ei$
nfi sin rnfi
UnrnZm to obtain
nil+ sin n~? I dy
(16)
G.G.Rvss, optimised integral continuation
277
If we write the left-hand side in the f o r m of an integral over the boundary. and r e - a r r a n g e the t e r m s on the r i g h t - h a n d side. we obtain the equation
f ~ ~;~m - - - yrr' dy 2~i B n 0 y - Z 1
~
{ {[ci2(y)+E
;2(y)] n
_,
)~v-n~7 vdYz . +[e12(y)- - 22, ~y]~'
(17)
T-1l and sum over the index n to obtain the relation Finally we multipty by gk~ defining U- I
~'= 1
n 0 (18)
~/=0 where
the a r are constants.
o
W e will solve e q (18) for U-I for the ease in which c~(y) and c~(y) m a y
be expressed as the quotient of finite order polynomials in cosfi. W e write ~12(y) and ~ 2 ( y ) i n the form
c1 (y) : - - ,
c
y) -
,
(19)
w h e r e 721(5,) and ~72(y) are p o l y n o m i a l s in cos t3 of d e g r e e M and 52(y) and 522(y) a r e p o l y n o m i a l s in cos/3 of d e g r e e N. Eq. (18) b e c o m e s 2r12(y) ~yk + r =1
at
L'k~z Y + A2(Y)
n :0
n :0
kn y
(20)
where 2, ,) ~721(y ) 2 77 (5 = U2(y) ,
A 21(y)
=
521(y) 2 522(y) 2 ~2(y) + ~I(Y) ,
A22(y ) : 521(y) 2q2(y) - 5~(y) 2771(y)
(21)
Eq. (20) r e q u i r e s that ~ k ~, n=O ZkUkln yn be of the f o r m (for N >~ M) (22)
k,.:0
k:0 r:-N-M
G.G.Ross, Optimised integral continuation
278
w h e r e the kkr a r e c o n s t a n t s to be d e t e r m i n e d . We now define the functions ~'I(Y) and ~(y) a n a l y t i c and without z e r o s within the unit c i r c l e . On B t h e s e f u n c t i o n s a r e c h o s e n such that 2 X l ( 3 ; ) a 1 (1) : A l ( Y ) , 1
~(3') r/(~) = ~72(3,) ,
(23)
AI(Y) and ~(3') will be p o l y n o m i a l s in v of d e g r e e N+M and 2 M r e s p e c t i v e l y . U s i n g t h e s e functions and eq. (22) we m a y w r i t e n=0 Z yn in the form k -1 ~z
2
k,n 0 Z UknY = X l ( y ) ~ I ( Z ) ( 1 - y Z ) ×
~[=1~ l(y)
~ ( 1 - yZ)
~Xl(1)
~ k=0 r---N-M
'
~tkrY
rzk
, (24)
t
w h e r e ~tky a r e c o n s t a n t s . We o b s e r v e that - ~ ( y ) [ ~ ( 1 / y ) ~-AI(Z)- ~ ( Z ) ~" A l"( 1 / y -~ ) ] / ( 1 - ya) m a y be w r i t t e n in the f o r m
N+M
M ~ akr y r z k k=0 r=-N- M
and t h e r e f o r e (24) m a y be r e w r i t t e n as
k,n:O
Z
-i n 2 UknY : ~ I ( ~ A ~ )
t~(y)~(Z ) N+M ,, r I t 1 - y Z +k,r:O ~ XkrY Zk "
(25)
H e r e we have used the fact that U is s y m m e t r i c to r e s t r i c t the n u m b e r of k r e n t e r i n g the equation. T h e s e c o n s t a n t s m a y be e v a l u a t e d by c o n s t a n t s ~t" s u b s t i t u t i n g eq. (25) in eq. (20) for e a c h p o w e r of Z to give
N+M
,,
1
r
}
rO +~(y) 7~2(Y)~2(1)
k ~ 7?ryk_r = ~(1_) ~ akryr, (26) r=O r--i
where
2M ~(y)= ~ ,lry r . r=0
G. G. Ross, Optimised integral continuation
279
D e m a n d i n g that the l e f t - h a n d side of eq. (26) has the 2M z e r o s of ~(1/y) TT g i v e s 2 M e q u a t i o n s i n v o l v i n g the N+M+I c o n s t a n t s X k r , k fixed. A f u r t h e r N - M + 1 e q u a t i o n s a r e o b t a i n e d f r o m the condition that the l e f t - h a n d side c o n t a i n s no p o w e r o f y l e s s than ( - 2 M + l ) . T h e s e N + M + l equations uniquely d e t e r m i n e the N+M+ 1 c o n s t a n t s X~r, and thus we m a y d e t e r m i n e all the TT c o n s t a n t s ~'kr i n v o l v e d in eq. (25). We a r e now in a p o s i t i o n to e v a l u a t e the m o s t p r o b a b l e function F(Z). We have ((plc1T+ (p2c2T)n
i 2~
:
}7' {WR(~)c°~sn[~ c2(2) +(PI(i3) sei 2n ~n ~
0
1
~ (PR(Y)
iq°I(Y)
ldi?
{yn dy
T h e n f r o m eqs. (9), (10) and (24)
f(Z)=
1
f __)WR(Y)
iWI(Y) t
-rriTxl(z) C)q°R(Y)ll
~ ynu-1 n,m=0 nm
iq)I(Y) I t
Zm dy
(Z) + 1
-yZ
" v k} 1 N+M xkry z al(rj k,r=O
(27)
T h i s g i v e s the i n t e g r a l r e p r e s e n t a t i o n f o r the m o s t p r o b a b l e function in t e r m s of the b o u n d a r y v a l u e s of the function and the e r r o r s . It is v a l i d t h r o u g h o u t tile whole d o m a i n D. H a v i n g found the m o s t p r o b a b l e function it is e a s y to find the m e a n v a l u e of the function at any point Z o within D. To do this we c h o o s e the function R(F) to be given by
R(F) -
ReF(Zo) .
T h e n by eq. (13) (R(F)) - ReF(Zo) and Re/F(Z o) Is given by eq. (27). The standard deviation is calculated
n =0
Yn = 5
Then
,
via eq. (13). We
n =0
(Z 2+ (Z;) rn) (U-1)mn . m=O
(28) have
G. G.Ross, Optimised integral continuation
280
(AR) 2 -
rUt T
n, m=0 U s i n g eq. (25) t h i s b e c o m e s (AR) 2 : 2 R e I
~l(-Zo ))
1
A I ( Z o ) A I ( Z o ) (1
1
I Zol 2 )
N+ M
+ ~ Lr,)t" kz (Z~)r) : I kr (zk+r+ o
(29)
k,r:O S i m i l a r l y we m a y c a l c u l a t e the m e a n v a l u e and s t a n d a r d d e v i a t i o n of the i m a g i n a r y p a r t of the function.
4. C O N T I N U A T I O N TO THE B O U N D A R Y AND O T H E R A P P L I C A T I O N S In the p r e v i o u s s e c t i o n we c o n s i d e r e d the p r o b l e m of c o n t i n u a t i o n w i t h i n the a n a l y t i c i t y r e g i o n . C o n t i n u a t i o n to the b o u n d a r y i t s e l f i s an u n s t a b l e p r o c e d u r e f o r , a s m a y be s e e n f r o m eq. (29), t h e s t a n d a r d d e v i a t i o n b e c o m e s i n f i n i t e due to the v a n i s h i n g of the d e n o m i n a t o r (1 - iZo !2) in the s e c o n d t e r m . T h i s i n s t a b i l i t y w a s n o t e d by B e r t e r o a n d V i a n o [5] and a l s o by P r e s n a j d e r and P i s u t [6] who g a v e a s i m p l e p r o o f of the r e s u l t . A s d i s c u s s e d by the l a t t e r a u t h o r s t h e r e a r e two w a y s of o v e r c o m i n g the d i f f i c u l t y . T h e f i r s t , s u g g e s t e d by C i u l l i , r e m o v e s the i n s t a b i l i t y by i m p o s i n g a d d i t i o n a l s m o o t h n e s s r e q u i r e m e n t s on the b e h a v i o u r of the f u n c t i o n on the b o u n d a r y . W e w i l l not c o n s i d e r t h i s m e t h o d f u r t h e r h e r e . T h e s e c o n d , s u g g e s t e d by P r e s n a j d e r and P i s u t [6], e v a d e s the d i f f i c u l t y by c a l c u l a t i n g a v e r a g e v a l u e s of the f u n c t i o n o v e r p r e - c h o s e n a r c s r a t h e r than t h e v a l u e of the f u n c t i o n in a p o i n t of the b o u n d a r y . F o r i n s t a n c e , we d e f i n e the f u n c t i o n a l R ( F ) to be
R(F) = f
{Re F(Z)} F(Z) I dZ
I,
(30)
Y w h e r e the i n t e g r a l is o v e r s o m e a r e 7 of the b o u n d a r y and F ( Z ) i s an a r b i t r a r y r e a l f u n c t i o n . T h e m e a n v a l u e of R(F) w i l l , a s u s u a l , be g i v e n by R(F) and u s i n g eq. (13) we m a y e a s i l y find the f o l l o w i n g e x p r e s s i o n for (AR) 2. (AR) 2
= 2 f ]dZ I f ]dy' I Re Y
N+M
y
,,
+ ~ k . r : O '~kr
t ~(Z)~(y)
1 ~ 1 (Z)/~ l ( y ) (1_ :~2-) +
~(Z) ~'(y*)
1
(1~
}
(ZkYr+ZkY*r) I
F(y)F(Z)
(31)
G.G.Ross, optimised integral continuation
281
It is evident that this will not, in general, be infinite and thus continuation to average values over parts of the boundary may be made a stable procedure. However, in many applications in particle physics, continuation to the boundary is necessary not to evaluate the function at, or near, apoint, but to test some hypothesis about the behaviour of the function over a particular region of the boundary and to obtain predictions about the parameters involved in the hypothesis. For example [i] in finite energy sum rules a Regge hypothes;.s is made about the scattering amplitude over some high-energy region and the Regge parameters then obtained via integrals over the lowenergy regions. The formalism developed above lends itself readily to such problems and, moreover, statistically meaningful errors are obtained for the predictions of the parameters. We suppose that we have an hypothesis, involving the N parameters gi, about the behaviour of the function over some part of the boundary, K. go(Z) is then given on K by the hypothesis and tile error functions el(Z) and E2(Z) are chosen inversely proportional to the accuracy to which the hypothesis is expected to be true. For example, in the case of a Regge-pole hypothesis, q(Z) and e2(Z) would be of the order of the lower pole terms ignored in the Regge expansion and any cut contributions expected to be present. We again consider the probability functional P(F) defined in eqs. (1) and (3). This will now involve the additional parameters gi' From eq. (9) we see that P(F) will be a maximum when W(g) is a minimum and bn =Bn(g) where Bn is defined in eq. (10) and is now a function of the additional parameters gi" The function W(g) is of the form
1 f)go2(y)
1 (27r) 2 Bf dy
go2(y)tdy
d Z )goR (y)
i goI(Y) __ _
~
7nu-1z
rn IgoR t ~ ) (Z)
igoI (Z) 1' c2(Z) (32)
and
7nu -1 Z m n, wI-O
nrtz
is given by eq. (24). The m o s t p r o b a b l e p a r a m e t e r s equations ~W(g)[
Ogi ~g:g
= 0 ,
gi
i = 1 .... N,
a r e given by the (33)
t o g e t h e r with the condition that this should be a m i n i m u m of W(g). H a v i n g found t h e s e p a r a m e t e r s the m o s t p r o b a b l e function F(Z) is a g a i n given by eq. (27) u s i n g gi = g'i in e v a l u a t i n g go(Z). T h e a v e r a g e v a l u e of a functional in H m a y be defined as in eq. (12), but
G.G.Ross, Optimised integral continuation
282
now the f u n c t i o n a l i n t e g r a t i o n r u n s a l s o o v e r the a l l o w e d v a l u e s of the gi" D e f i n i n g G = b X - Y we note that dG n and db n d i f f e r only by the J a c o b i a n of the t r a n s f o r m a t i o n a n d we m a y r e d e f i n e the f u n c t i o n a l i n t e g r a t i o n in eq. (12) to be o v e r the G n and g n w i t h o u t c h a n g i n g the p r o b a b i l i t y f u n c t i o n a l . A l i n e a r f u n c t i o n a l in H m a y , by eqs. (11) and (5), be w r i t t e n in the f o r m
R(F) - r U b T , w h e r e (r) n = r n and we h a v e e x p a n d e d r(Z) d e f i n e d in eq. (11) in the f o r m oO
r(Z) : ~
rnPn(Z).
n=0 F r o m eq. (11), a n d the r e d e f i n i t i o n of the i n n e r p r o d u c t we h a v e
(R) = (ru[GX -1 + B(g)]W) , (R) = (~'uBT(g)>g, w h e r e ( }g d e n o t e s the a v e r a g e i s only o v e r the N p a r a m e t e r s s t a n d a r d d e v i a t i o n AR of a f u n c t i o n a l i s g i v e n by
(34)
gi. The
(An) 2 : (R 2)- ( n ) 2 . We
have
B(g)]T}2) ({ruBT(g)) 2)g.
(I{2) : ( { r U [ G X -I + = rurT+ Thus
(AR)2 = r u r T + ( { r u B T (g)} 2 )g_ ( r u B T (g))~ .
(35)
We c o n s i d e r the a p p l i c a t i o n of t h e s e r e s u l t s to the e x a m p l e of a f u n c t i o n
f ( s , t) which by h y p o t h e s i s , m a y be e x p a n d e d in a r e g i o n s I ~< s ~< s 2 in t e r m s of a f i n i t e n u m b e r of Regge p o l e s .
f ( s , t ) = ~N g i ( t ) [ ± l - e - i ~ i ( t ) ] (~s-)cq(t)/sin ~ i ( t ) i=1 ~s°/
(36)
We w i s h , u s i n g f i x e d - t r e l a t i o n s , to e x t r a c t i n f o r m a t i o n f r o m the data a b o u t the r e s i d u e f u n c t i o n s gi(t). Note that a p a r t i c u l a r a d v a n t a g e of this m e t h o d o v e r c o n v e n t i o n a l f i n i t e e n e r g y s u m r u l e s is that by a s u i t a b l e c h o i c e of ¢1(Z) and ¢2(Z) c o n s i s t e n c y i s only r e q u i r e d with the Regge h y p o t h e s i s o v e r a f i n i t e r a n g e of s a n d not to i n f i n i t y o r into the c o m p l e x s - p l a n e . T h e m o s t p r o b a b l e v a l u e s of gi(t) a r e then found by s o l v i n g the eqs. (33), g i v e n the f o r m of W(g), eq. (32), and the d e p e n d e n c e on the p a r a m e t e r s gi(t) o v e r the r e g i o n K : ( Z ( S l ) , Z (s2)) s p e c i f i e d by eq. (36). T h e a v e r a g e v a l u e and s t a n d a r d d e v i a t i o n of a n y l i n e a r f u n c t i o n in H a r e , in this c a s e , e a s i l y o b t a i n e d f r o m the m o s t p r o b a b l e f u n c t i o n for, e x c e p t for an e l a s t i c a m p l i t u d e in the f o r w a r d d i r e c t i o n , the p a r a m e t e r s g~(t) m a y r a n g e f r o m (_ o% oo) and t h i s , t o g e t h e r with the b i l i n e a r f o r m of ×~'(g) g i v e n by eq. (9) a n d eq. (32) e n a b l e s u s to w r i t e
G. G. Ross, Oplimised inlegral continuation
283
(R) : @uBT(g)}g : rUB(g) N (AR) 2 = r U t T + ~
(37)
R(~),
- 1 ( r UB T (2)) , (ruBT(g)) Eij
(38 )
i, j l where
aB(g) Bi(g) = ~gi- g : ~ ' a n d the e r r o r m a t r i x Eij i s d e f i n e d by ~,2W(g)
2Eij-
Ogi~gj •
In the f o r w a r d d i r e c t i o n the i m a g i n a r y p a r t of the a m p l i t u d e g i v e n by eq. (36) w i l l h a v e to s a t i s f y a p o s i t i v i t y c o n s t r a i n t but, p r o v i d e d the p r o b a b i l i t y i s s m a l l f o r v a l u e s v i o l a t i n g t h i s c o n s t r a i n t the e r r o r s i n c u r r e d by i n t e g r a t i n g o v e r t h e s e u n p h y s i c a l v a l u e s w i l l be n e g l i g i b l e and e q s . (37) and (38) will remain valid. T h e m e a n v a l u e s and s t a n d a r d d e v i a t i o n of the R e g g e p a r a m e t e r s g i a r e g i v e n s i m p l y by
(gi) : F'i, (Ag/)2 = R:I~z "
(39)
F i n a l l y we c o n s i d e r the c a s e in w h i c h we a r e g i v e n i n f o r m a t i o n q~M(Z), ~ I ( Z ) and ¢2(Z) a b o u t the b o u n d a r y v a l u e s and e r r o r s of a f u n c t i o n F M ( Z ) , m e r o m o r p h i c w i t h i n D. If the f u n c t i o n h a s N s i m p l e p o l e s w i t h i n D at p o s i t i o n s Z i w e m a y c o n s t r u c t a f u n c t i o n F(Z), a n a l y t i c w i t h i n D, by e x p l i c i t l y subtracting these poles N
F(Z) : F M ( Z ) - ~
z
gi_ z
i
•
i:1
(38)
T h e b o u n d a r y v a l u e s of F(Z) w i l l now b e g i v e n by q~(Z) = gOM(Z ) -
N gi i=l Z - Z i "
(39)
If, u s i n g t h i s f o r m f o r go(Z), t o g e t h e r w i t h ¢ l ( Z ) a n d e 2 ( Z ) , we c o n s t r u c t the p r o b a b i l i t y f u n c t i o n a l P(F) a s d e f i n e d in e q s . (1) a n d (3), it w i l l be a f u n c t i o n of t h e r e s i d u e p a r a m e t e r s gi a s w e l l a s the b n. As in the p r e v i o u s c a s e we find the m o s t p r o b a b l e v a l u e s ~ a r e g i v e n by s o l v i n g e q s . (33) and the m e a n v a l u e and s t a n d a r d d e v i a t i o n of the p a r a m e t e r s eq. (39). O n c e a g a i n c a r e m u s t b e t a k e n in u s i n g eq. (39) if the gi h a v e a positivity requirement.
284
G. G. R o s s , Oplimised inleg~'al conlinuation
In all the a p p l i c a t i o n s c o n s i d e r e d a b o v e , h a v i n g found the m o s t f u n c t i o n and, if n e c e s s a r y , the m o s t p r o b a b l e v a l u e s for the o t h e r e t e r s i n v o l v e d , we m a y e v a l u a t e the v a l u e of X2 f o r t h i s s o l u t i o n . (9) we s e e that this i s g i v e n by ×2(~) W(g) w h e r e W(g) i s d e f i n e d (32). We now d e f i n e the q u a n t i t y C by the e q u a t i o n
probable paramF r o m eq. in eq.
cO
C : ,,J,2 - P(x2) dx2 ' (40) X (f) C is the p r o b a b i l i t y that X2 e x c e e d s the o b s e r v e d v a l u e of X2(F). We m a y u s e C as d e f i n e d a b o v e a s a m e a s u r e of the c o m p a t i b i l i t y of an a n a l y t i c f u n c t i o n , t o g e t h e r with any o t h e r h y p o t h e s e s m a d e , with the data i n p u t a s b o u n d a r y v a l u e s f o r the f u n c t i o n .
5. CONCLUSIONS In the m e t h o d d e s c r i b e d a b o v e we have w o r k e d in the s p a c e of all a n a l y t i c f u n c t i o n s . T h e a d v a n t a g e of t h i s l i e s in the fact that it r e m o v e s a n y b i a s i n h e r e n t in a p a r t i c u l a r c h o i c e of p a r a m e t e r i s a t i o n for a n a n a l y t i c f u n c t i o n . O n e s t i l l h a s the p r o b l e m of d e f i n i n g the p r o b a b i l i t y f u n c t i o n a l and, in p a r t i c u l a r , the e r r o r f u n c t i o n s El(Z) a n d E2(Z). In r e g i o n s w h e r e d a t a i s a v a i l a b l e t h i s is s t r a i g h t f o r w a r d but o v e r r e g i o n s of the b o u n d a r y w h e r e s o m e h y p o t h e s i s is m a d e t h e i r c h o i c e r e q u i r e s s o m e p h y s i c a l j u d g e m e n t . T h i s of c o u r s e j u s t s t a t e s the n e c e s s i t y c l e a r l y to d e f i n e the p r o b l e m to be solved. The m a i n a d v a n t a g e of the t e c h n i q u e is that, o n c e the e r r o r f u n c t i o n s a r e g i v e n , the b e s t c o n t i n u a t i o n p o s s i b l e i s u n a m b i g u o u s l y d e t e r m i n e d . M o r e o v e r r e a l i s t i c e x p r e s s i o n s for the s t a t i s t i c a l e r r o r s a r e o b t a i n e d , t o g e t h e r with a m e a s u r e of the c o n s i s t e n c y of the f i n a l r e s u l t with the data. Although the s o l u t i o n p r e s e n t e d a p p l i e s to a r e s t r i c t e d c l a s s of e r r o r f u n c t i o n s if i s hoped that it will be a p p l i c a b l e to m o s t p h y s i c a l p r o b l e m s of i n t e r e s t . A p r a c t i c a l a p p l i c a t i o n of o p t i m i s e d i n t e g r a l c o n t i n u a t i o n m e t h o d s to the p r o b l e m of the d e t e r m i n a t i o n of the p i o n - n u c l e o n c o u p l i n g c o n s t a n t , u s i n g e q u a l e r r o r f u n c t i o n s for the r e a l and i m a g i n a r y p a r t s , h a s a l r e a d y a p p e a r e d [8]. F u r t h e r a p p l i c a t i o n s , u s i n g d i f f e r e n t e r r o r s for the r e a l and i m a g i n a r y p a r t s of the f u n c t i o n , a r e at p r e s e n t b e i n g p u r s u e d . RE F E R E N C ES [1] A.A.Logunov, L.D. SoIoviev and A.N.Tabkhelidze, Phys. Letters 24B (1967) 181; K.Igi and S.Matsuda, Phys. Bey. Letters 18 (1967) 625; P.Dolen, D.Horn and C.Sehmid, Phys. Rev. 166 (1968) 1768. [21 P . E . Cutkosky, Ann. of Phys. 54 (1969) 350. [31 P . P r e ~ n a j d e r and J.Pi~fit, Nuovo Cimento 3A (1971) 603. [4] G . G . R o s s , Nuel. Phys. B31 (1971) 113. [5] M . B e r t e r o and G.A.Viano, Nuovo Cimento 38 (1965) 1915. [6] P . P r e ~ n a j d e r and J . P i ~ f t , Nuel. Phys. B22 0970) 365. [7] S.Cuilli, Nuovo Cimento 61A (1969) 786; 628 (1969) 301. [8] P . L i e h a r d and P. Pre~najder, Bratislava preprint.