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On Representation and Series Expansions ofF(z)
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
211, 527]544 Ž1997.
AY975496
On Representation and Series Expansions of F Ž z . David S. Tselnik 2416 18th Street South, Apartment 204, Fargo, North Dakota 58103 Submitted by H. M. Sri¨ asta¨ a Received May 20, 1996
The paper is a sequel to the author’s pervious works Ž Complex Variables 25, 1994, 159]171; J. Math. Anal. Appl. 193, 1995, 522]542.. Q 1997 Academic Press
1. INTRODUCTION In this paper, the author combines results which are important and useful additions to the results of his works w3, 4x. Almost all of the results given here are new Žnot contained in w3, 4x., but in a couple of cases, where it is important, certain results of w3, 4x are given here in refined forms, too.
2. REPRESENTATION OF F Ž z . WITHIN A CURVE We now state w3, Theorem 3x in the following refined form: THEOREM 1. Let D be a domain in the complex z-plane, and let F Ž z . be a single-¨ alued analytic function on D, with exception of poles. Let C be a closed rectifiable Jordan cur¨ e contained in D together with its interior I Ž C ., and such that C passes through no poles of F Ž z .. Let the poles of F Ž z . inside C be located at Ž distinct . points b1 , b 2 , . . . , bp and b 1 , b 2 , . . . , bm , and let the principal parts of the corresponding Laurent expansions of F Ž z . be ˜ gn Ž z . Ž n s 1, 2, . . . , p . and gk
gk Ž z . s
Ý AŽysk . Ž z y bk .
ys
Ž k s 1, 2, 3, . . . , m . ,
Ž 1.
ss1
527 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
528
DAVID S. TSELNIK
respecti¨ ely. Let z 0 be any regular point of F Ž z . inside C or any of the points b1 , b 2 , . . . , bp , and let the coefficients of the Taylor series of p
f˜Ž z . s F Ž z . y
Ý ˜gn Ž z .
Ž 2.
ns1
at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Then for any regular point z of F Ž z . inside C, F Ž z . is represented by w4, Eq. Ž97.x p
FŽ z. s
N
Ý ˜gn Ž z . q Ý a˜n Ž z y z 0 .
ns1
?
n
m
ns0
ks1
FŽ z .
1
H 2p i C Ž z y z .
Ý Gk Ž z, z 0 ; N . q Ž z y z 0 . Nq1
q
dz Nq 1
0
Ž 3.
zyz
Ž N s 0, 1, 2, . . . .; all the notations here are as in w3, 4x. Remark. Note that we state here this theorem for a function F Ž z . analytic on D with the exception of poles rather than for a meromorphic function as in w3, Theorem 3x: the condition of meromorphicity of F Ž z . Žin the entire finite z-plane, that is. is not used in the proof of w3, Theorem 3x. Remark. If F Ž z . does not have poles at b 1 , b 2 , . . . , bm , then the sum of Gk in Ž3. vanishes. The following statement is a special case of Theorem 1: THEOREM 1*. Let D be a domain in the complex z-plane, and let F Ž z . be single-¨ alued analytic function on D with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , b 2 , . . . , bm g D. Let C be a closed rectifiable Jordan cur¨ e contained in D together with its interior I Ž C . and such that all the poles b 0 , b 1 , . . . , bm g I Ž C .. Let the principal parts of F Ž z . at its poles b 0 , b 1 , . . . , bm be g k Ž z . by Ž1. Ž with k s 0, 1, 2, . . . , m., and let the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 s b 0 be a0 , a1 , a2 , . . . . Then for any regular point z of F Ž z . inside C N
F Ž z . s g0 Ž z . q
Ý an Ž z y z0 .
n
m
q
ns0
?
1
Ý Gk Ž z, z 0 ; N . q Ž z y z 0 . Nq1 ks1
FŽ z .
H 2p i C Ž z y z . 0
dz Nq 1
zyz
Ž N s 0, 1, 2, . . . .. Ž See w4, Eq. Ž94. and p. 542x.. Remark. If F Ž z . is analytic at z 0 , then g 0 Ž z . s 0 in Ž4..
Ž 4.
REPRESENTATION AND EXPANSIONS OF F Ž z .
529
It does not require any work to obtain Ž4. from Ž3.. However, the representation Ž3. can be obtained from Ž4., too. This requires lengthy derivations Žwith usage of w3, Eq. Ž4.; 4, Eqs. Ž85. ] Ž89.x.; these derivations are not included here Žas a space-saving measure.. In practical usage, Ž3. has advantage over Ž4.. In Ž3., the sum of principal parts of F Ž z . at its poles b1 , b 2 , . . . , bp is extracted Žby the sum in n ., and that can be of use. For example, from Ž4. we obtain Žvia w4, Eq. Ž95.x. expansion w3, Eq. Ž38.x for tan z. At the same time, Ž3. leads Žvia w3, Eq. Ž49.x. to expansions w3, Eqs. Ž38. and Ž60.x for tan z.
3. REPRESENTATION OF F Ž z . ON A DISK If F Ž z . is analytic on a disk with the exception of the poles, it can be represented on the disk as in w4, Eq. Ž36.x. However, it follows from Ž3. that F Ž z . can be represented on that disk in different Žfrom w4, Eq. Ž36.x. forms, too. Namely, the following theorem is true: THEOREM 2. Let DR be an open disk of radius R ) 0 with its center at z 0 , and let F Ž z . be a single-¨ alued analytic function on DR , with the exception of poles at Ž distinct . points b1 , b 2 , . . . , bp g DR Ž one of these points may coincide with the center of the disk, z 0 . and also at points b 1 , b 2 , . . . , bm g DR Ž no one of these points coincides with z 0 .. Let the principal parts of F Ž z . at its poles bn , b k be ˜ gn Ž z . and g k Ž z . w by Ž1.x, respecti¨ ely, and let the Taylor coefficients of f˜Ž z . by Ž2. at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Then for any regular point z of F Ž z . on DR p
FŽ z. s
N
lim Ý a ˜n Ž z y z 0 . Ý ˜gn Ž z . q Nª`
n s1
n
m
q
ns0
Ý Gk Ž z, z 0 ; N .
. Ž 5.
ks1
If instead of Ž3. one uses Ž4., then the following theorem is obtained THEOREM 2*. Let DR be an open disk of radius R ) 0 with its center at z 0 and let F Ž z . be a single-¨ alued analytic function on DR with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , . . . , bm g DR . Let the principal parts of F Ž z . at b k Ž k s 0, 1, . . . , m. be g k Ž z . by Ž1. Ž with k s 0, 1, . . . , m., and let the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 s b 0 be a0 , a1 , a2 , . . . . Then for any regular point z of F Ž z . on DR N
F Ž z . s g 0 Ž z . q lim
Nª`
Ý an Ž z y z0 . ns0
n
m
q
Ý Gk Ž z, z 0 ; N . ks1
.
Ž 6.
530
DAVID S. TSELNIK
It is possible to show that representations Ž5., Ž6., and expansion w4, Eq. Ž36.x can be reduced to one another. EXAMPLE. Let F Ž z . s cot z, z 0 s 0, and R s 2p . The representation Ž6. for F Ž z . on DR Ž< z < - 2p , z / 0, " p . can be found to be cot z s
1 z
M
q lim y Mª`
Ý
z
xm z 2 my1 q
žp/
ms1
2p
2 Mq1
z yp2 2
,
Ž 7.
and the expansion w4, Eq. Ž36.x for the case in question is cot z s
1
q
z
2z z2 y p 2
`
y
Ý
x 1 m z 2 my1
Ž 8.
ms1
Ž< z < - 2p , z / 0, " p . w4, Eq. Ž71.x, where
xm s 2 2 m Bm r Ž 2 m . !,
x 1 m s xm y Ž 2rp 2 m .
Ž m s 1, 2, 3, . . . . Ž 9 .
w4, Eqs. Ž72. ] Ž74.x, and Bm are the Bernoulli numbers B1 s 1r6, B2 s 1r30, B3 s 1r42, . . . . Clearly, Ž7. and Ž8. are two different forms of representation of cot z on the disk in question. At the same time, Ž7. can be obtained from Ž8.: for that one needs to write the infinite sum of Ž8. as M
lim Mª`
M
xm z 2 my1 y Ž 2rz .
Ý ms1
Ý Ž zrp . 2 m
Ž 10 .
ms1
and then to sum the second sum in Ž10..
4. BEHAVIOUR OF THE TAYLOR SERIES ON THE CIRCLE OF CONVERGENCE THEOREM 3. Let DR be an open disk of the radius R ) 0 with its center at z 0 and let F Ž z . be a single-¨ alued analytic function on DR with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , . . . , bm y1; bm , . . . , bm g DR . Let the poles b 0 , b 1 , . . . , bm y1 Ž m F m. be located within a circle Cm of radius 0 - Rm - R with its center at z 0 , and let poles bm , . . . , bm be located on Cm . Let also g k Ž z . by Ž1. Ž with k s 0, 1, . . . , m. be the principal parts of F Ž z . at its poles b k . Consider the expansion my1
FŽ z. y
Ý ks0
gk Ž z . s
`
Ý amy1, n Ž z y z 0 . ns0
n
,
Ž 11 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
531
where the amy1, n are the Taylor coefficients of the left-hand side of Ž11. at z 0 ; the expansion is ¨ alid inside Cm Ž with points b 0 , b 1 , . . . , bm y1 deleted . and is not ¨ alid outside Cm , of course. The Taylor series on the right-hand side of Ž11. will: Ži. Con¨ erge at a regular point z of F Ž z . located on Cm if there exists m
lim
Ý
Nª` ks m
Gk Ž z, z 0 ; N .
Ž 12 .
at that point. And, if this limit is equal to zero, then Ž11. is ¨ alid at the point in question; if the limit is equal to A, say, then at the point in question my1
FŽ z. s
Ý
gk Ž z . q
ks0
`
Ý amy1, n Ž z y z 0 .
n
q A.
Ž 13 .
ns0
Žii. Di¨ erge at regular points of F Ž z . on C where the limit Ž12. does not exist. Proof. From Ž3., it follows that for regular z of F Ž z . on DR , the function F Ž z . Žof Theorem 3. can be represented as my1
FŽ z. s
Ý ks0
N
g k Ž z . q lim
Nª`
Ý amy1, n Ž z y z 0 . ns0
n
m
q
Ý
ks m
Gk Ž z, z 0 ; N . ;
Ž 14 . the sum of Gk in Ž14. pertains to all the poles of F Ž z . located on Cm . Now, compare Ž11. to Ž14.. Remark. Unlike in Tauber’s and Fatou’s theorems w2, Vol. 1, Sect. 85x, our result on convergence Žor divergence. of the power series considered Žthe Taylor series, about z 0 , of Ž11.. does not involve any conditions on coefficients of the series. Rather, the sum of the magnitudes Gk at all the poles of F Ž z . on the circle of convergence of the series in question is involved. Special Case. Suppose that all the poles of F Ž z . Žof Theorem 3. on Cm are simple. In this case Gk Ž z, z 0 ; N . s
ž
z y z0
bk y z0
Nq 1
/
k. AŽy1
z y bk
Ž 15 .
532
DAVID S. TSELNIK
Ž N s 0, 1, 2, . . . . w3, Eq. Ž15.x. For regular z g Cm
Ž z y z 0 . r Ž b k y z 0 . s exp Ž i u k .
Ž 0 - u k - 2p .
Ž 16 .
Ž k s m , . . . , m., and we can also write k. AŽy1 r Ž z y b k . s r k exp Ž i v k .
Ž r k ) 0, 0 F v k - 2p .
Ž 17 .
Ž k s m , . . . , m. so that Ž12. can be written as m
lim
Ý
m
Nª` ks m
Gk Ž z, z 0 ; N . s lim
Ý
Nª` ks m
r k e iwŽ Nq1. u kq v k x
Ž 18 .
in this case. If F Ž z . has only one Žsimple. pole on Cm , then it is evident that limit Ž18. does not exist. If F Ž z . has two Žsimple. poles on Cm , the limit Ž18. does not exist, too. ŽTo arrive at that conclusion, it is useful to interpret the sum on the right-hand side of Ž18. as the sum of two planar vectors.. In the two cases just described, the power ŽTaylor. series of Ž11. will diverge at any regular point z of F Ž z . on Cm . One more case: if F Ž z . has any finite number of simple poles on Cm , but one of the magnitudes r k Žof Ž18.. is larger than the sum of all other magnitudes r k Žof Ž18.., then the limit Ž18., even if it exists, cannot be zero. In this case, Ž11. is not valid for any regular point z of F Ž z . on Cm . EXAMPLE. Let Žin Theorem 3. R s 2p , z 0 s 0, so that DR is open disk < z < - 2p . Let F Ž z . s cot z. On DR , cot z has poles z 0 s b 0 s 0, b 1 s p , and b 2 s yp . Let Cm be circle < z < s p , so that Rm s p , and pole z 0 s b 0 lies inside Cm Ž m s 1., and poles b 1 and b 2 lie on Cm . The expansion Ž11. in this case is cot z y
1 z
sy
`
Ý
xn z 2 ny1
Ž < z < - p , z / 0. .
Ž 19 .
ns1
Since cot z has two poles on Cm , both simple, we immediately conclude that the infinite series of Ž19. is divergent at any regular point of cot z on its circle of convergence < z < s p Žthat is, that series is divergent for < z < s p , z / "p .. This fact is most definitely known, but we established it in a different Žin a new. way}without taking into consideration coefficients of the series Ž19.! Conjecture 1. When all the poles of F Ž z . Žof Theorem 3. on Cm are simple, the limit Ž12. does not exist and the power ŽTaylor. series of Ž11. is divergent at any regular point of F Ž z . on the circle of its convergence Cm .
REPRESENTATION AND EXPANSIONS OF F Ž z .
533
Conjecture 2. For F Ž z . of Theorem 3, the power series ŽTaylor series. of Ž11. is divergent at all regular points of F Ž z . on the circle of convergence of that series Cm . Remark. At regular points of F Ž z . Žof Theorem 3. on Cm where the Taylor series of Ž11. is divergent, the representation Ž11. of F Ž z . is not valid, of course. But Ž14. is valid, and can be used to calculate values of F Ž z . at those points. For example, the point z s ip is a regular point of F Ž z . s cot z on the circle < z < s p , and the series Ž19. is divergent at z s ip , so that cotŽ ip . cannot be calculated as 1 ip
N
y lim
Ý xs
Nª` ss1
Ž y1. i
s
p 2 sy1
Ž 20 .
However, we can use Ž7. and calculate cotŽ ip . as M
cot Ž ip . s y
lim
Ý
Ms1, 3, 5, . . . ª` ms1
xm
Ž y1. i
m
p 2 my1 ,
Ž 21 .
for example}an interesting fact!
5. PARTIAL FRACTION EXPANSIONS OF F Ž z . THEOREM 4. Let G be a closed rectifiable Jordan cur¨ e and let F Ž z . be a meromorphic function which has poles at distinct points b1 , b 2 , . . . , bp inside G and b 1 , b 2 , . . . on G or outside G, with principal parts ˜ gn Ž z . Ž n s 1, 2, . . . , p . and g k Ž z . by Ž1. with k s 1, 2, . . . , respecti¨ ely. Let z 0 be any point inside G, and let the coefficients of the Taylor series of f˜Ž z . by Ž2. at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Let Cm 4 be a sequence of closed rectifiable Jordan cur¨ es with the following properties: Ž1. None of the cur¨ es passes through poles of F Ž z .. Ž2. Cur¨ e G is contained in C1 , and each of the cur¨ es in the sequence is contained in the next, that is, G ; I Ž C1 ., Cm ; I Ž Cm q1 . Ž m s 1, 2, 3, . . . .. Ž3. If rm s r Ž z 0 , Cm . is the distance from z 0 to Cm , then rm ª ` as m ª `. Let the poles b 1 , b 2 , b 3 , . . . be ordered in such a way that Cm contains the first mm of these poles, b 1 , b 2 , . . . , bmm, so that mm F mm q1 Ž m s 1, 2, 3, . . . ., and let us denote hm s
HC < F Ž z . < ds m
Ž s is arc-length . .
Ž 22 .
534
DAVID S. TSELNIK
Suppose that for integers Nm G 0 Nmq1
Ž hmrrm .Ž rrrm .
as m ª `
ª0
Ž 23 .
for r ) 0. Then w see Ž3.x Ns
p
FŽ z. s
Ý ˜gn Ž z . q mlim Ý a˜n Ž z y z 0 . ª`
n s1
ns0
n
mm
q
Ý Gk Ž z, z 0 ; Ns .
, Ž 24 .
ks1
where Ns s Nm , Nm q 1, . . . , for any finite regular point z of F Ž z .. The representation Ž24. is more general than the one given in w3, Eq. Ž48.; 4, Eq. Ž4.x. ŽBut of course, the condition Ž23. is different from the condition w4, Eq. Ž3.x Žwith f˜Ž z . in it replaced by F Ž z .. Žsee w4, p. 542x, too... The remainder of the approximation of F Ž z . Žfor regular z g I Ž Cm .. by the expression on the right-hand side of Ž24. with ‘‘lim as m ª `’’ omitted is given by the term with integral of Ž3., with C being Cm and N s Ns in it. The absolute value of the remainder is bounded from above by
Ž hmr2p rm .Ž rrrm .
Nsq1
1 y Ž rrrm .
y1
for < z y z 0 < s r - rm . Ž 25 .
Remark. Instead of the condition Ž23., we could use the condition r Nmq1 Hm Ž Nm . rrm ª 0
as m ª `,
Ž 26 .
< F Ž z .
Ž 27 .
where Hm Ž N . s
HC
m
We can use the bound Hm Ž Ns . r2p rm r Nsq1 1 y Ž rrrm . as an alternative to Ž25..
y1
for < z y z 0 < s r - rm Ž 28 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
535
Written differently Žwe used w3, Eq. Ž4.x., Ž24. is p
FŽ z. s
Ý ˜gn Ž z .
ns1
½
ns0
mm
Ns
q lim
mª`
y
Ns
Ý a˜n Ž z y z 0 .
mm
q
Ý gk Ž z . ks1
gk
z y z0
Žk.
Ý Ý Ý ks1 ns0
n
ss1
ž
Ays nqsy1 s n Ž z0 y bk .
/
ž
bk y z0
n
/5
. Ž 29 .
Suppose now}instead of Ž23. }that for F Ž z . of Theorem 4 and for some integer N s N# G y1 lim
H mª` C
F Ž z . rŽ z y z0 .
Nq 1
Ž z y z . dz s 0
Ž 30 .
m
for any finite regular point z of F Ž z .. Then if N# s y1, F Ž z . can be represented as mm
p
FŽ z. s
Ý ˜gn Ž z . q mlim Ý gk Ž z . , ª`
n s1
Ž 31.
ks1
while if N# G 0, then F Ž z . can be represented as p
FŽ z. s
Ý
n s1
N
˜gn Ž z . q
Ý
n
mm
a ˜n Ž z y z 0 . q lim
ns0
Ý Gk Ž z, z 0 ; N . Ž 32.
mª` ks1
with N s N# for any finite regular point z of F Ž z .. In particular, if lim
H m ª` C
< F Ž z .
Ž 33 .
m
for an integer N s N# G y1, then Ži. if N# s y1, then F Ž z . can be represented both as in Ž31. and also as in Ž32. with N s 0, 1, 2, . . . for any finite regular point z of F Ž z ., while Žii. if N# G 0, then F Ž z . can be represented as in Ž32. with N s N#, N# q 1, . . . for any finite regular point z of F Ž z .. Both in cases Ži. and Žii., the convergence in Ž31., Ž32. is uniform on every compact set containing no poles of F Ž z ..
536
DAVID S. TSELNIK
Remark. Making statements Ži. and Žii., we, in particular, combined together reformulated Cauchy theorem on partial fraction expansions w4, Theorem B ŽReformulated.x and w4, Theorem Ax Žsee also w3, Theorem 4x., with modifications given in w4, pp. 541]542x incorporated. Similarly to Theorems 1 and 1*, we shall state, along with Theorem 4, THEOREM 4*. Let F Ž z . be a meromorphic function which has poles at distinct points b 0 s z 0 , b 1 , . . . , b k , . . . with principal parts g k Ž z . Ž k s 0, 1, 2, . . . . by Ž1.. Let Cm4 be a sequence of closed rectifiable Jordan cur¨ es with the following properties: Ž1. Ž2. I Ž Cmq1 . Ž3.
None of the cur¨ es passes through poles of F Ž z .. Each cur¨ e in the sequence is contained in the next, that is, Cm ; Ž m s 1, 2, 3, . . . .; the point z 0 lies inside C1. The distance from z 0 to Cm , rm ª ` as m ª `.
Let the poles of F Ž z . be ordered in such a way that Cm contains the first mm q 1 of the poles, b 0 s z 0 , b 1 , . . . , bmm, so that mm F mm q1 Ž m s 1, 2, 3, . . . .. Then if condition Ž23. is satisfied, F Ž z . can be represented as F Ž z . s g 0 Ž z . q lim
mª`
Ns
Ý
mm
n
an Ž z y z0 . q
ns0
Ý Gk Ž z, z 0 ; Ns . Ž 34. ks1
Ž Ns s Nm , Nm q 1, . . . . for any finite regular z of F Ž z .. Now, suppose}instead of Ž23. }that for F Ž z . of Theorem 4* and for some integer N s N# G y1 the condition Ž30. is satisfied for any finite regular point z of F Ž z .. Then if N# s y1, F Ž z . can be represented as mm
F Ž z . s lim
Ý gk Ž z . ,
Ž 35 .
mª` ks0
while if N# ) y1, F Ž z . can be represented as N
F Ž z . s g0 Ž z . q
Ý an Ž z y z0 . ns0
n
mm
q lim
Ý Gk Ž z, z 0 ; N .
mª` ks1
Ž 36 .
with N s N# for any finite regular point z of F Ž z .; the a n in Ž36. are the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 . In particular, if the condition Ž33. is satisfied for some integer N s N# G y1, then Ži. if N# s y1, F Ž z . can be represented both as in Ž35. and as in Ž36. with N s 0, 1, 2, . . . for any finite regular point z of F Ž z ., while
REPRESENTATION AND EXPANSIONS OF F Ž z .
537
Žii. if N# G 0, then F Ž z . can be represented as in Ž36. with N s N#, N# q 1, N# q 2, . . . for any finite regular point z of F Ž z .. EXAMPLE. Consider F Ž z . s tan z with z 0 s 0 and Cm described in w3, p. 164, Examplex. Using w3, Eqs. Ž35., Ž36.x, one finds that Ž23. is satisfied with Nm s 0 in this case, so that we can choose Ns G 0. Let us choose Ns s 2 m y 1. Then from Ž34., and using Ž15. for Gk , we find tan z s lim
mª`
½
m
Ý a n z 2 ny1 ns1
y2 z
2m
z
ž /
m
Ý
pr2
ks1
1
Ž 2 k y 1.
2m
?
1 z y Ž 2 k y 1 . pr2 2
2
5
Ž 37 .
5
Ž 38 .
and Ž29. yields tan z s lim
mª`
½
m
m
a n z 2 ny1 y
Ý ns1
m
y
2z
Ý
z y Ž 2 k y 1 . pr2 2
ks1
m
z
2 ny1
Ý Ý ks1 ns1 Ž 2 k y 1 . pr2
2
2
Ž 2 k y 1 . pr2
in this case; a n are as in w3, Eq. Ž40.x. Both in Ž37. and Ž38., < z < - `, and poles of tan z are deleted. Remark. Equation Ž37. can be obtained from w3, Eq. Ž38.x. ŽRedenote N by m , and then write the sum in k of w3, Eq. Ž38.x as two sums: one from k s 1 to k s m , and another from k s m q 1 to k s `.. Equation Ž38. can be reduced to Ž37.. Remark. Integers Nm G 0 to satisfy condition Ž23. can always be found. Namely, let «m ) 0 Ž m s 1, 2, . . . ., «m ª 0 as m ª `. Choose t # g Ž0, 1. Žt # s 1r2, for example., and for each curve Cm Ž m s 1, 2, . . . . Žof Theorems 4 or 4*. determine Nm as the smallest nonnegative integer such that
Ž hmrrm . Ž t#. N q1 - «m . m
Ž 39 .
Thus, F Ž z . Žof Theorem 4 or 4*. can always be represented as in Ž24., Ž29., Ž34. Žcondition Ž33. or Ž30. does not need to be satisfied for that..1 1
That is to say that any meromorphic Žin the entire finite z-plane. function F Ž z . can always be represented in the form Ž24., Ž29., or Ž34. wsince it is evident that the system of curves Cm4 Žof Theorems 4 or 4*. can always be constructed Žfor any meromorphic function F Ž z ...
538
DAVID S. TSELNIK
In conclusion of this section, we shall give an example showing how slowly convergent partial fraction expansions obtained by Cauchy can be. EXAMPLE. Consider for tan z its partial fraction expansion w3, Eq. Ž37.x, which for z s x is tan x s Ž 8 xrp 2 . S Ž x . ,
Ž 40 .
where SŽ x . s
`
Ý f Ž t , x . < tsk ,
2
f Ž t , x . s Ž 2 t y 1 . y Ž 2 xrp .
2 y1
. Ž 41 .
ks1
For x g Ž0, pr2. we use the Maclaurin]Cauchy estimate for the remainder of the series S after its K th Ž K s 1, 2, 3, . . . . term D SŽ K, x . w1, pp. 281]284x to find yF Ž t s K q 1, x . F D S Ž K , x . F yF Ž t s K , x . ,
Ž 42 .
where F Ž t , x . s Ž pr8 x . ln Ž 1 y 2 xrp Ž 2 t y 1 . . r Ž 1 q 2 xrp Ž 2 t y 1 .
.4. Ž 43 .
For the corresponding error in the value of tan x, calculated by Ž40. with 1 F k F K in Ž41., denoted D tanŽ x, K ., we find for K large D tan Ž x s pr4, K . ( 1r2p K .
Ž 44 .
Using Ž40., Ž41., and a calculator with the eight-digit-of-mantissa display, to obtain the mantissa of tanŽpr4. on the display as 1, we need to take into account such number of terms Ž K . of the series Ž41., that the corresponding D tanŽ x s pr4, K . would be less than 5 ? 10y8 . For that we must have K ( 10 7rp ( 3.18 ? 10 6 . In other words, using Cauchy’s partial fraction expansion w3, Eq. Ž37.x for tan z, we need to take into account more than three million Ž!. terms of that expansion to get tanŽpr4., in the display of our calculator, as 1. This is an enormously slow convergence, of course. For comparison, it takes five terms Ž N s 5. of the sum in n and one term of the sum in k of w3, Eq. Ž39.x to get tanŽpr4. s 1 in the display of our calculator!
6. OBTAINING ESTIMATES FOR REMAINDERS OF TAYLOR SERIES Expansions w3, Eq. Ž18.x are good instruments for obtaining estimates for remainders of Taylor series.
REPRESENTATION AND EXPANSIONS OF F Ž z .
539
EXAMPLE. Let us use w3, Eq. Ž39.x to obtain an estimate for the remainder of the Taylor series `
Ý Ž y1. ny 1 a n z 2 ny1 Ž < z < - pr2.
tanh z s
Ž 45 .
ns1
on the interval 0 - z s x - pr2 from below. For a n of Ž45. we have
a n s 2 2 n Ž 2 2 n y 1 . Bnr Ž 2 n . !
Ž n s 1, 2, 3, . . . .
Ž 46 .
w4, Eq. Ž44.x, where the Bn are the Bernoulli numbers, B1 s 1r6, B2 s 1r30, B3 s 1r42, . . . . From w3, Eq. Ž39.x, for the remainder of Ž45. after its Nth term Ž N s 1, 2, 3, . . . . we get rN Ž x . s Ž y1 .
N
2N
x
`
2x
ž / pr2
x 2 q Ž pr2 .
2
Ý ks1
1
Ž 2 k y 1.
2N
n k Ž x . , Ž 47 .
where 2
n k Ž x . s x 2 q Ž pr2 . r x 2 q Ž 2 k y 1 . pr2
2
4
Ž k s 1, 2, 3, . . . . , Ž 48 .
and then we find < rN Ž x . < )
x
2N
ž / pr2
`
2x x 2 q Ž pr2 .
2
Ý ms0
1
Ž 2 m q 1.
2 Nq2
Ž 49 .
Ž0 - x - pr2, N s 1, 2, 3, . . . .. Using that `
Ý Ž 2 m q 1. y2 Ny2 s Ž 1 y 2y2 Ny2 . z Ž 2 N q 2.
Ž 50 .
ms0
Ž N s 0, 1, 2, . . . ., where z is the Riemann zeta-function w12, Sect. 13.3x, and also that
z Ž 2 N q 2 . s Ž 2p .
2 Nq2
BNq 1r2 Ž 2 N q 2 . !
Ž N s 0, 1, 2, . . . . Ž 51 .
w4, Eq. Ž65.x, we find < rN Ž x . < )
a Nq1 x 2 Nq1 1 q xr Ž pr2 .
2
Ž 52 .
for x g Ž0, pr2., N s 1, 2, 3, . . . . This is the estimate for the absolute value of the remainder of the Taylor series Ž45. after its Nth term on the interval x g Ž0, pr2. from below, expressed in terms of the absolute value of the first neglected Žthe Ž N q 1.st. term of that series.
540
DAVID S. TSELNIK
Now, on the circle of convergence of Ž45., < z < s pr2, tanh z has two poles, z s "p ir2, both simple. Thus, Ž45. will diverge at z s x s pr2 Žsee Special Case in Section 4 above.. Therefore, one cannot calculate tanhŽpr2. as N
lim
Ý Ž y1. ny 1 a n Ž pr2. 2 ny1 .
Ž 53 .
Nª` ns1
From w3, Eq. Ž39.x it follows, however, that tanhŽpr2. can be calculated as Žfor example. tanh
p 2
2M
s lim
2
Ý Ž y1. ny 1 a n Ž pr2. 2 ny1 q p
Ž 54 .
Mª` ns1
an interesting fact. Finally, for < x < g Žpr2, `., when Ž45. diverges, w3, Eq. Ž39.x is still valid, of course, and for N large we find the following approximation for the segment of Ž45. N
Ý Ž y1. ny 1 a n x 2 ny1 ( tanh
N
x y Ž y1 . 2 x
ns1
x
2N
ž / pr2
1 x q Ž pr2 . 2
2
Ž 55 . also an interesting fact.
7. FOUR COMMENTS ON MITTAG-LEFFLER’S THEOREM In this section, we give concrete expressions for series which Mittag-Leffler uses in the course of proof of his theorem on partial fraction expansions. These expressions are given in the comments to follow. But first we need to state the theorem itself. MITTAG-LEFFLER’S THEOREM w2, Vol. 2, pp. 299]301x.
Let
b 0 s z0 , b1 , . . . , bk , . . .
Ž 56 .
be a sequence of distinct complex numbers such that < b 1 y z 0 < F < b 2 y z 0 < F ??? F < b k y z 0 < F ???
Ž 57 .
and < bk y z0 < ª `
as k ª `,
Ž 58 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
541
and let g 0 Ž z ., g 1Ž z ., . . . , g k Ž z ., . . . be a sequence of rational functions of the form Ž1.. Then there exists a meromorphic function F Ž z . whose poles coincide with the points Ž56. and whose principal part at the pole b k equals g k Ž z . for each k s 0, 1, 2, . . . . ŽThe special case when g 0 Ž z . s 0 is included in the theorem.. Remark. We used notations of this paper in writing out this theorem. Moreover, in Mittag-Leffler’s theorem as it is given in w2x, z 0 s 0. We have chosen to set b 0 s z 0 Žnot zero necessarily . for more generality and to be able to use expressions for Gk Ž z, z 0 ; N . of w3, 4x below. Mittag-Leffler starts the proof of the theorem from the Taylor series w2, Vol. 2, p. 300x n
g k Ž z . s aŽ0k . q a1Ž k . Ž z y z 0 . q ??? qaŽnk . Ž z y z 0 . q ???
Ž 59 .
Ž k s 1, 2, 3, . . . .. Our Comment. Written in concrete form, series Ž59. is gk Ž z . s
gk
`
Ý Ý ns0
ss1
Žk.
ž
Ays nqsy1 s n Ž z0 y bk .
/
ž
n
z y z0
bk y z0
/
Ž 60 .
Ž k s 1, 2, 3, . . . . w4, Eq. Ž25.x. Then Mittag-Leffler introduces a sequence « k 4 of positive numbers such that `
Ý «k - `
Ž 61 .
ks1
w2, Vol. 2, Eq. Ž10.35.x, and he chooses integers n1 , n 2 , n 3 , . . . such that on the disk D k : < z y z 0 < - 12 < b k y z 0 <, < g k Ž z . q pk Ž z . < - « k
Ž k s 1, 2, 3, . . . . ,
Ž 62 .
where pk Ž z . s yaŽ0k . y a1Ž k . Ž z y z 0 . y ??? yaŽnkk. Ž z y z 0 .
nk
Ž 63 .
Žsee w2, Vol. 2, Eqs. Ž10.36., Ž10.37.x.. Our Comment. From Ž59., Ž60., Ž63., and w4, Eq. Ž26.x, we find that g k Ž z . q pk Ž z . s Gk Ž z, z 0 ; n k .
Ž 64 .
542
DAVID S. TSELNIK
Ž k s 1, 2, 3, . . . .. Thus, Mittag-Leffler’s condition Ž62. can be written as < Gk Ž z, z 0 ; n k . < - « k
Ž 65 .
Ž k s 1, 2, 3, . . . . for all z g D k ; concrete expressions for Gk are given in w3, 4x Žreplace N of these expressions by n k .. With « k already chosen, Ž65. can be used to find the corresponding n k . For example, if the pole b k is simple, then using Ž15. one finds from Ž65. that it is sufficient to have
Ž 12 .
nk
k. < < < AŽy1 r bk y z0 < F « k
Ž 66 .
for Ž65. to be satisfied on D k . Which leads to n k G y Ž ln 2 .
y1
k. < ln Ž « k < b k y z 0
Ž 67 .
in the case in question. Finally, Mittag-Leffler finds F Ž z . in the form FŽ z. s
`
Ý
g k Ž z . q pk Ž z . ,
Ž 68.
ks0
where p 0 Ž z . is an arbitrary polynomial w2, Vol. 2, p. 300x. Our Comment. Expression Ž68. for F Ž z . in its concrete form is F Ž z . s g 0 Ž z . q p0 Ž z . q
`
Ý Gk Ž z, z 0 ; n k . ,
Ž 69 .
ks1
where the Gk are as described above. Also, according to the corollary to Mittag-Leffler’s Theorem w2, Vol. 2, p. 301x, a meromorphic function F Ž z . with poles Ž56. and the corresponding principal parts g k Ž z . can be represented by expansion FŽ z. s vŽ z. q
`
Ý
g k Ž z . q pk Ž z . ,
Ž 70.
ks0
where v Ž z . is an entire function and p 0 Ž z . is an arbitrary polynomial. Our Comment. In a more concrete form, Ž70. is Ž p 0 is omitted in Ž71.. F Ž z . s v Ž z . q g0 Ž z . q
`
Ý Gk Ž z, z 0 ; n k . . ks1
Ž 71 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
543
Remark. For the case b 0 s z 0 s 0 Žconsidered in w2, pp. 299]301x. one should use Gk Ž z, 0; n k . instead of Gk Ž z, z 0 ; n k .. The concrete expressions for g k Ž z . Ž60., for g k Ž z . q pk Ž z . Ž64., and for F Ž z . Ž69., Ž71. given above Žwith concrete expressions for Gk Ž z, z 0 ; n k . referred to. will be of help if one wants to use Mittag-Leffler’s theorem to actually construct expansions Ž68. or Ž70. in any concrete cases.
8. CONCLUDING REMARKS This paper originated from the paper ‘‘A Note on Representation and Series Expansions of F Ž z . in Complex Domains’’ Žits contents reflected Žto a very large extent. by abstracts w8]10x. the present author submitted to J. Math. Anal. Appl. in July 1995. With some material deleted and new material added, that paper evolved into the paper entitled ‘‘On Representation and Series Expansions of F Ž z .’’ submitted to J. Math. Anal. Appl. in May 1996. The present paper is a compression Žto approximately 60% of its original length. of the latter manuscript. Some related Žto the material of w3, 4x, and of this paper. developments are described in the present author’s works w5, 7x Žfor linear operators, for Fredholm integral equations, in particular. and in w6x Žfor Jacobian elliptic functions.. On January 8, 1997, the present author made a presentation entitled ‘‘On Representation of F Ž z . in multiply-connected domains Žin the complex z-plane.’’ w11x at the 103rd Annual Meeting of the AMS in San Diego, California. It was expedient, from the author’s point of view, to include the essence of the Theorems 4 and 4*, and of the Remark with Eq. Ž39. of the present paper with that presentation. And they were included, with indication that these results are from a different paper by the author. From the Žnumbered. equations of the present paper, Eqs. Ž22. ] Ž24., Ž34., and Ž39. were presented Žwith pertinent explanations. in San Diego.
REFERENCES 1. G. M. Fikhtengol’ts, ‘‘A Course of the Differential and Integral Calculuses,’’ 6th ed., Vol. 2, Nauka, Moscow, 1966. wIn Russianx 2. A. I. Markushevich, ‘‘Theory of Functions of a Complex Variable,’’ Chelsea, New York, 1977. 3. D. S. Tselnik, Representation and series expansions for meromorphic functions, Complex Variables 25 Ž1994., 159]171. 4. D. S. Tselnik, On series expansions for meromorphic functions, J. Math. Anal. Appl. 193 Ž1995., 522]542.
544
DAVID S. TSELNIK
5. D. S. Tselnik, A bound for the remainder of the Hilbert-Schmidt series and other results on representation of solutions to the functional equation of the second kind with a self-adjoint compact operator as an infinite series, Comput. Math. Appl. 29 Ž1995., 61]68. 6. D. S. Tselnik, On trigonometric series expansions of twelve Jacobian elliptic functions, SIAM J. Math. Anal., 28 Ž1997., 715]730. 7. D. S. Tselnik, A simple bound for the remainder of the Neumann series in the case of a self-adjoint compact operator, Appl. Math. Lett. 7 Ž1994., 71]74. 8. D. S. Tselnik, Four theorems on representation and series expansions of F Ž z . in complex domains, Part 1, Abstracts Amer. Math. Soc. 17 Ž1996., 258. 9. D. S. Tselnik, Four theorems on representation and series expansions of F Ž z . in complex domains, Part 2, Abstracts Amer. Math. Soc. 17 Ž1996., 258]259. 10. D. S. Tselnik, Expressions for functions in Mittag-Leffler’s theorem, Abstracts Amer. Math. Soc. 17 Ž1996., 449. 11. D. S. Tselnik, ‘‘On representation of F Ž z . in multiply-connected domains Žin the complex z-plane.,’’ Abstracts Amer. Math. Soc., 18 Ž1997., 66. 12. E. T. Whittaker and G. N. Watson, ‘‘A Course of Modern Analysis,’’ 4th ed., Cambridge Univ. Press, Cambridge, UK, 1927.
211, 527]544 Ž1997.
AY975496
On Representation and Series Expansions of F Ž z . David S. Tselnik 2416 18th Street South, Apartment 204, Fargo, North Dakota 58103 Submitted by H. M. Sri¨ asta¨ a Received May 20, 1996
The paper is a sequel to the author’s pervious works Ž Complex Variables 25, 1994, 159]171; J. Math. Anal. Appl. 193, 1995, 522]542.. Q 1997 Academic Press
1. INTRODUCTION In this paper, the author combines results which are important and useful additions to the results of his works w3, 4x. Almost all of the results given here are new Žnot contained in w3, 4x., but in a couple of cases, where it is important, certain results of w3, 4x are given here in refined forms, too.
2. REPRESENTATION OF F Ž z . WITHIN A CURVE We now state w3, Theorem 3x in the following refined form: THEOREM 1. Let D be a domain in the complex z-plane, and let F Ž z . be a single-¨ alued analytic function on D, with exception of poles. Let C be a closed rectifiable Jordan cur¨ e contained in D together with its interior I Ž C ., and such that C passes through no poles of F Ž z .. Let the poles of F Ž z . inside C be located at Ž distinct . points b1 , b 2 , . . . , bp and b 1 , b 2 , . . . , bm , and let the principal parts of the corresponding Laurent expansions of F Ž z . be ˜ gn Ž z . Ž n s 1, 2, . . . , p . and gk
gk Ž z . s
Ý AŽysk . Ž z y bk .
ys
Ž k s 1, 2, 3, . . . , m . ,
Ž 1.
ss1
527 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
528
DAVID S. TSELNIK
respecti¨ ely. Let z 0 be any regular point of F Ž z . inside C or any of the points b1 , b 2 , . . . , bp , and let the coefficients of the Taylor series of p
f˜Ž z . s F Ž z . y
Ý ˜gn Ž z .
Ž 2.
ns1
at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Then for any regular point z of F Ž z . inside C, F Ž z . is represented by w4, Eq. Ž97.x p
FŽ z. s
N
Ý ˜gn Ž z . q Ý a˜n Ž z y z 0 .
ns1
?
n
m
ns0
ks1
FŽ z .
1
H 2p i C Ž z y z .
Ý Gk Ž z, z 0 ; N . q Ž z y z 0 . Nq1
q
dz Nq 1
0
Ž 3.
zyz
Ž N s 0, 1, 2, . . . .; all the notations here are as in w3, 4x. Remark. Note that we state here this theorem for a function F Ž z . analytic on D with the exception of poles rather than for a meromorphic function as in w3, Theorem 3x: the condition of meromorphicity of F Ž z . Žin the entire finite z-plane, that is. is not used in the proof of w3, Theorem 3x. Remark. If F Ž z . does not have poles at b 1 , b 2 , . . . , bm , then the sum of Gk in Ž3. vanishes. The following statement is a special case of Theorem 1: THEOREM 1*. Let D be a domain in the complex z-plane, and let F Ž z . be single-¨ alued analytic function on D with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , b 2 , . . . , bm g D. Let C be a closed rectifiable Jordan cur¨ e contained in D together with its interior I Ž C . and such that all the poles b 0 , b 1 , . . . , bm g I Ž C .. Let the principal parts of F Ž z . at its poles b 0 , b 1 , . . . , bm be g k Ž z . by Ž1. Ž with k s 0, 1, 2, . . . , m., and let the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 s b 0 be a0 , a1 , a2 , . . . . Then for any regular point z of F Ž z . inside C N
F Ž z . s g0 Ž z . q
Ý an Ž z y z0 .
n
m
q
ns0
?
1
Ý Gk Ž z, z 0 ; N . q Ž z y z 0 . Nq1 ks1
FŽ z .
H 2p i C Ž z y z . 0
dz Nq 1
zyz
Ž N s 0, 1, 2, . . . .. Ž See w4, Eq. Ž94. and p. 542x.. Remark. If F Ž z . is analytic at z 0 , then g 0 Ž z . s 0 in Ž4..
Ž 4.
REPRESENTATION AND EXPANSIONS OF F Ž z .
529
It does not require any work to obtain Ž4. from Ž3.. However, the representation Ž3. can be obtained from Ž4., too. This requires lengthy derivations Žwith usage of w3, Eq. Ž4.; 4, Eqs. Ž85. ] Ž89.x.; these derivations are not included here Žas a space-saving measure.. In practical usage, Ž3. has advantage over Ž4.. In Ž3., the sum of principal parts of F Ž z . at its poles b1 , b 2 , . . . , bp is extracted Žby the sum in n ., and that can be of use. For example, from Ž4. we obtain Žvia w4, Eq. Ž95.x. expansion w3, Eq. Ž38.x for tan z. At the same time, Ž3. leads Žvia w3, Eq. Ž49.x. to expansions w3, Eqs. Ž38. and Ž60.x for tan z.
3. REPRESENTATION OF F Ž z . ON A DISK If F Ž z . is analytic on a disk with the exception of the poles, it can be represented on the disk as in w4, Eq. Ž36.x. However, it follows from Ž3. that F Ž z . can be represented on that disk in different Žfrom w4, Eq. Ž36.x. forms, too. Namely, the following theorem is true: THEOREM 2. Let DR be an open disk of radius R ) 0 with its center at z 0 , and let F Ž z . be a single-¨ alued analytic function on DR , with the exception of poles at Ž distinct . points b1 , b 2 , . . . , bp g DR Ž one of these points may coincide with the center of the disk, z 0 . and also at points b 1 , b 2 , . . . , bm g DR Ž no one of these points coincides with z 0 .. Let the principal parts of F Ž z . at its poles bn , b k be ˜ gn Ž z . and g k Ž z . w by Ž1.x, respecti¨ ely, and let the Taylor coefficients of f˜Ž z . by Ž2. at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Then for any regular point z of F Ž z . on DR p
FŽ z. s
N
lim Ý a ˜n Ž z y z 0 . Ý ˜gn Ž z . q Nª`
n s1
n
m
q
ns0
Ý Gk Ž z, z 0 ; N .
. Ž 5.
ks1
If instead of Ž3. one uses Ž4., then the following theorem is obtained THEOREM 2*. Let DR be an open disk of radius R ) 0 with its center at z 0 and let F Ž z . be a single-¨ alued analytic function on DR with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , . . . , bm g DR . Let the principal parts of F Ž z . at b k Ž k s 0, 1, . . . , m. be g k Ž z . by Ž1. Ž with k s 0, 1, . . . , m., and let the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 s b 0 be a0 , a1 , a2 , . . . . Then for any regular point z of F Ž z . on DR N
F Ž z . s g 0 Ž z . q lim
Nª`
Ý an Ž z y z0 . ns0
n
m
q
Ý Gk Ž z, z 0 ; N . ks1
.
Ž 6.
530
DAVID S. TSELNIK
It is possible to show that representations Ž5., Ž6., and expansion w4, Eq. Ž36.x can be reduced to one another. EXAMPLE. Let F Ž z . s cot z, z 0 s 0, and R s 2p . The representation Ž6. for F Ž z . on DR Ž< z < - 2p , z / 0, " p . can be found to be cot z s
1 z
M
q lim y Mª`
Ý
z
xm z 2 my1 q
žp/
ms1
2p
2 Mq1
z yp2 2
,
Ž 7.
and the expansion w4, Eq. Ž36.x for the case in question is cot z s
1
q
z
2z z2 y p 2
`
y
Ý
x 1 m z 2 my1
Ž 8.
ms1
Ž< z < - 2p , z / 0, " p . w4, Eq. Ž71.x, where
xm s 2 2 m Bm r Ž 2 m . !,
x 1 m s xm y Ž 2rp 2 m .
Ž m s 1, 2, 3, . . . . Ž 9 .
w4, Eqs. Ž72. ] Ž74.x, and Bm are the Bernoulli numbers B1 s 1r6, B2 s 1r30, B3 s 1r42, . . . . Clearly, Ž7. and Ž8. are two different forms of representation of cot z on the disk in question. At the same time, Ž7. can be obtained from Ž8.: for that one needs to write the infinite sum of Ž8. as M
lim Mª`
M
xm z 2 my1 y Ž 2rz .
Ý ms1
Ý Ž zrp . 2 m
Ž 10 .
ms1
and then to sum the second sum in Ž10..
4. BEHAVIOUR OF THE TAYLOR SERIES ON THE CIRCLE OF CONVERGENCE THEOREM 3. Let DR be an open disk of the radius R ) 0 with its center at z 0 and let F Ž z . be a single-¨ alued analytic function on DR with the exception of poles at Ž distinct . points z 0 s b 0 , b 1 , . . . , bm y1; bm , . . . , bm g DR . Let the poles b 0 , b 1 , . . . , bm y1 Ž m F m. be located within a circle Cm of radius 0 - Rm - R with its center at z 0 , and let poles bm , . . . , bm be located on Cm . Let also g k Ž z . by Ž1. Ž with k s 0, 1, . . . , m. be the principal parts of F Ž z . at its poles b k . Consider the expansion my1
FŽ z. y
Ý ks0
gk Ž z . s
`
Ý amy1, n Ž z y z 0 . ns0
n
,
Ž 11 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
531
where the amy1, n are the Taylor coefficients of the left-hand side of Ž11. at z 0 ; the expansion is ¨ alid inside Cm Ž with points b 0 , b 1 , . . . , bm y1 deleted . and is not ¨ alid outside Cm , of course. The Taylor series on the right-hand side of Ž11. will: Ži. Con¨ erge at a regular point z of F Ž z . located on Cm if there exists m
lim
Ý
Nª` ks m
Gk Ž z, z 0 ; N .
Ž 12 .
at that point. And, if this limit is equal to zero, then Ž11. is ¨ alid at the point in question; if the limit is equal to A, say, then at the point in question my1
FŽ z. s
Ý
gk Ž z . q
ks0
`
Ý amy1, n Ž z y z 0 .
n
q A.
Ž 13 .
ns0
Žii. Di¨ erge at regular points of F Ž z . on C where the limit Ž12. does not exist. Proof. From Ž3., it follows that for regular z of F Ž z . on DR , the function F Ž z . Žof Theorem 3. can be represented as my1
FŽ z. s
Ý ks0
N
g k Ž z . q lim
Nª`
Ý amy1, n Ž z y z 0 . ns0
n
m
q
Ý
ks m
Gk Ž z, z 0 ; N . ;
Ž 14 . the sum of Gk in Ž14. pertains to all the poles of F Ž z . located on Cm . Now, compare Ž11. to Ž14.. Remark. Unlike in Tauber’s and Fatou’s theorems w2, Vol. 1, Sect. 85x, our result on convergence Žor divergence. of the power series considered Žthe Taylor series, about z 0 , of Ž11.. does not involve any conditions on coefficients of the series. Rather, the sum of the magnitudes Gk at all the poles of F Ž z . on the circle of convergence of the series in question is involved. Special Case. Suppose that all the poles of F Ž z . Žof Theorem 3. on Cm are simple. In this case Gk Ž z, z 0 ; N . s
ž
z y z0
bk y z0
Nq 1
/
k. AŽy1
z y bk
Ž 15 .
532
DAVID S. TSELNIK
Ž N s 0, 1, 2, . . . . w3, Eq. Ž15.x. For regular z g Cm
Ž z y z 0 . r Ž b k y z 0 . s exp Ž i u k .
Ž 0 - u k - 2p .
Ž 16 .
Ž k s m , . . . , m., and we can also write k. AŽy1 r Ž z y b k . s r k exp Ž i v k .
Ž r k ) 0, 0 F v k - 2p .
Ž 17 .
Ž k s m , . . . , m. so that Ž12. can be written as m
lim
Ý
m
Nª` ks m
Gk Ž z, z 0 ; N . s lim
Ý
Nª` ks m
r k e iwŽ Nq1. u kq v k x
Ž 18 .
in this case. If F Ž z . has only one Žsimple. pole on Cm , then it is evident that limit Ž18. does not exist. If F Ž z . has two Žsimple. poles on Cm , the limit Ž18. does not exist, too. ŽTo arrive at that conclusion, it is useful to interpret the sum on the right-hand side of Ž18. as the sum of two planar vectors.. In the two cases just described, the power ŽTaylor. series of Ž11. will diverge at any regular point z of F Ž z . on Cm . One more case: if F Ž z . has any finite number of simple poles on Cm , but one of the magnitudes r k Žof Ž18.. is larger than the sum of all other magnitudes r k Žof Ž18.., then the limit Ž18., even if it exists, cannot be zero. In this case, Ž11. is not valid for any regular point z of F Ž z . on Cm . EXAMPLE. Let Žin Theorem 3. R s 2p , z 0 s 0, so that DR is open disk < z < - 2p . Let F Ž z . s cot z. On DR , cot z has poles z 0 s b 0 s 0, b 1 s p , and b 2 s yp . Let Cm be circle < z < s p , so that Rm s p , and pole z 0 s b 0 lies inside Cm Ž m s 1., and poles b 1 and b 2 lie on Cm . The expansion Ž11. in this case is cot z y
1 z
sy
`
Ý
xn z 2 ny1
Ž < z < - p , z / 0. .
Ž 19 .
ns1
Since cot z has two poles on Cm , both simple, we immediately conclude that the infinite series of Ž19. is divergent at any regular point of cot z on its circle of convergence < z < s p Žthat is, that series is divergent for < z < s p , z / "p .. This fact is most definitely known, but we established it in a different Žin a new. way}without taking into consideration coefficients of the series Ž19.! Conjecture 1. When all the poles of F Ž z . Žof Theorem 3. on Cm are simple, the limit Ž12. does not exist and the power ŽTaylor. series of Ž11. is divergent at any regular point of F Ž z . on the circle of its convergence Cm .
REPRESENTATION AND EXPANSIONS OF F Ž z .
533
Conjecture 2. For F Ž z . of Theorem 3, the power series ŽTaylor series. of Ž11. is divergent at all regular points of F Ž z . on the circle of convergence of that series Cm . Remark. At regular points of F Ž z . Žof Theorem 3. on Cm where the Taylor series of Ž11. is divergent, the representation Ž11. of F Ž z . is not valid, of course. But Ž14. is valid, and can be used to calculate values of F Ž z . at those points. For example, the point z s ip is a regular point of F Ž z . s cot z on the circle < z < s p , and the series Ž19. is divergent at z s ip , so that cotŽ ip . cannot be calculated as 1 ip
N
y lim
Ý xs
Nª` ss1
Ž y1. i
s
p 2 sy1
Ž 20 .
However, we can use Ž7. and calculate cotŽ ip . as M
cot Ž ip . s y
lim
Ý
Ms1, 3, 5, . . . ª` ms1
xm
Ž y1. i
m
p 2 my1 ,
Ž 21 .
for example}an interesting fact!
5. PARTIAL FRACTION EXPANSIONS OF F Ž z . THEOREM 4. Let G be a closed rectifiable Jordan cur¨ e and let F Ž z . be a meromorphic function which has poles at distinct points b1 , b 2 , . . . , bp inside G and b 1 , b 2 , . . . on G or outside G, with principal parts ˜ gn Ž z . Ž n s 1, 2, . . . , p . and g k Ž z . by Ž1. with k s 1, 2, . . . , respecti¨ ely. Let z 0 be any point inside G, and let the coefficients of the Taylor series of f˜Ž z . by Ž2. at z 0 be a ˜0 , a˜1 , a˜2 , . . . . Let Cm 4 be a sequence of closed rectifiable Jordan cur¨ es with the following properties: Ž1. None of the cur¨ es passes through poles of F Ž z .. Ž2. Cur¨ e G is contained in C1 , and each of the cur¨ es in the sequence is contained in the next, that is, G ; I Ž C1 ., Cm ; I Ž Cm q1 . Ž m s 1, 2, 3, . . . .. Ž3. If rm s r Ž z 0 , Cm . is the distance from z 0 to Cm , then rm ª ` as m ª `. Let the poles b 1 , b 2 , b 3 , . . . be ordered in such a way that Cm contains the first mm of these poles, b 1 , b 2 , . . . , bmm, so that mm F mm q1 Ž m s 1, 2, 3, . . . ., and let us denote hm s
HC < F Ž z . < ds m
Ž s is arc-length . .
Ž 22 .
534
DAVID S. TSELNIK
Suppose that for integers Nm G 0 Nmq1
Ž hmrrm .Ž rrrm .
as m ª `
ª0
Ž 23 .
for r ) 0. Then w see Ž3.x Ns
p
FŽ z. s
Ý ˜gn Ž z . q mlim Ý a˜n Ž z y z 0 . ª`
n s1
ns0
n
mm
q
Ý Gk Ž z, z 0 ; Ns .
, Ž 24 .
ks1
where Ns s Nm , Nm q 1, . . . , for any finite regular point z of F Ž z .. The representation Ž24. is more general than the one given in w3, Eq. Ž48.; 4, Eq. Ž4.x. ŽBut of course, the condition Ž23. is different from the condition w4, Eq. Ž3.x Žwith f˜Ž z . in it replaced by F Ž z .. Žsee w4, p. 542x, too... The remainder of the approximation of F Ž z . Žfor regular z g I Ž Cm .. by the expression on the right-hand side of Ž24. with ‘‘lim as m ª `’’ omitted is given by the term with integral of Ž3., with C being Cm and N s Ns in it. The absolute value of the remainder is bounded from above by
Ž hmr2p rm .Ž rrrm .
Nsq1
1 y Ž rrrm .
y1
for < z y z 0 < s r - rm . Ž 25 .
Remark. Instead of the condition Ž23., we could use the condition r Nmq1 Hm Ž Nm . rrm ª 0
as m ª `,
Ž 26 .
< F Ž z .
Ž 27 .
where Hm Ž N . s
HC
m
We can use the bound Hm Ž Ns . r2p rm r Nsq1 1 y Ž rrrm . as an alternative to Ž25..
y1
for < z y z 0 < s r - rm Ž 28 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
535
Written differently Žwe used w3, Eq. Ž4.x., Ž24. is p
FŽ z. s
Ý ˜gn Ž z .
ns1
½
ns0
mm
Ns
q lim
mª`
y
Ns
Ý a˜n Ž z y z 0 .
mm
q
Ý gk Ž z . ks1
gk
z y z0
Žk.
Ý Ý Ý ks1 ns0
n
ss1
ž
Ays nqsy1 s n Ž z0 y bk .
/
ž
bk y z0
n
/5
. Ž 29 .
Suppose now}instead of Ž23. }that for F Ž z . of Theorem 4 and for some integer N s N# G y1 lim
H mª` C
F Ž z . rŽ z y z0 .
Nq 1
Ž z y z . dz s 0
Ž 30 .
m
for any finite regular point z of F Ž z .. Then if N# s y1, F Ž z . can be represented as mm
p
FŽ z. s
Ý ˜gn Ž z . q mlim Ý gk Ž z . , ª`
n s1
Ž 31.
ks1
while if N# G 0, then F Ž z . can be represented as p
FŽ z. s
Ý
n s1
N
˜gn Ž z . q
Ý
n
mm
a ˜n Ž z y z 0 . q lim
ns0
Ý Gk Ž z, z 0 ; N . Ž 32.
mª` ks1
with N s N# for any finite regular point z of F Ž z .. In particular, if lim
H m ª` C
< F Ž z .
Ž 33 .
m
for an integer N s N# G y1, then Ži. if N# s y1, then F Ž z . can be represented both as in Ž31. and also as in Ž32. with N s 0, 1, 2, . . . for any finite regular point z of F Ž z ., while Žii. if N# G 0, then F Ž z . can be represented as in Ž32. with N s N#, N# q 1, . . . for any finite regular point z of F Ž z .. Both in cases Ži. and Žii., the convergence in Ž31., Ž32. is uniform on every compact set containing no poles of F Ž z ..
536
DAVID S. TSELNIK
Remark. Making statements Ži. and Žii., we, in particular, combined together reformulated Cauchy theorem on partial fraction expansions w4, Theorem B ŽReformulated.x and w4, Theorem Ax Žsee also w3, Theorem 4x., with modifications given in w4, pp. 541]542x incorporated. Similarly to Theorems 1 and 1*, we shall state, along with Theorem 4, THEOREM 4*. Let F Ž z . be a meromorphic function which has poles at distinct points b 0 s z 0 , b 1 , . . . , b k , . . . with principal parts g k Ž z . Ž k s 0, 1, 2, . . . . by Ž1.. Let Cm4 be a sequence of closed rectifiable Jordan cur¨ es with the following properties: Ž1. Ž2. I Ž Cmq1 . Ž3.
None of the cur¨ es passes through poles of F Ž z .. Each cur¨ e in the sequence is contained in the next, that is, Cm ; Ž m s 1, 2, 3, . . . .; the point z 0 lies inside C1. The distance from z 0 to Cm , rm ª ` as m ª `.
Let the poles of F Ž z . be ordered in such a way that Cm contains the first mm q 1 of the poles, b 0 s z 0 , b 1 , . . . , bmm, so that mm F mm q1 Ž m s 1, 2, 3, . . . .. Then if condition Ž23. is satisfied, F Ž z . can be represented as F Ž z . s g 0 Ž z . q lim
mª`
Ns
Ý
mm
n
an Ž z y z0 . q
ns0
Ý Gk Ž z, z 0 ; Ns . Ž 34. ks1
Ž Ns s Nm , Nm q 1, . . . . for any finite regular z of F Ž z .. Now, suppose}instead of Ž23. }that for F Ž z . of Theorem 4* and for some integer N s N# G y1 the condition Ž30. is satisfied for any finite regular point z of F Ž z .. Then if N# s y1, F Ž z . can be represented as mm
F Ž z . s lim
Ý gk Ž z . ,
Ž 35 .
mª` ks0
while if N# ) y1, F Ž z . can be represented as N
F Ž z . s g0 Ž z . q
Ý an Ž z y z0 . ns0
n
mm
q lim
Ý Gk Ž z, z 0 ; N .
mª` ks1
Ž 36 .
with N s N# for any finite regular point z of F Ž z .; the a n in Ž36. are the Taylor coefficients of F Ž z . y g 0 Ž z . at z 0 . In particular, if the condition Ž33. is satisfied for some integer N s N# G y1, then Ži. if N# s y1, F Ž z . can be represented both as in Ž35. and as in Ž36. with N s 0, 1, 2, . . . for any finite regular point z of F Ž z ., while
REPRESENTATION AND EXPANSIONS OF F Ž z .
537
Žii. if N# G 0, then F Ž z . can be represented as in Ž36. with N s N#, N# q 1, N# q 2, . . . for any finite regular point z of F Ž z .. EXAMPLE. Consider F Ž z . s tan z with z 0 s 0 and Cm described in w3, p. 164, Examplex. Using w3, Eqs. Ž35., Ž36.x, one finds that Ž23. is satisfied with Nm s 0 in this case, so that we can choose Ns G 0. Let us choose Ns s 2 m y 1. Then from Ž34., and using Ž15. for Gk , we find tan z s lim
mª`
½
m
Ý a n z 2 ny1 ns1
y2 z
2m
z
ž /
m
Ý
pr2
ks1
1
Ž 2 k y 1.
2m
?
1 z y Ž 2 k y 1 . pr2 2
2
5
Ž 37 .
5
Ž 38 .
and Ž29. yields tan z s lim
mª`
½
m
m
a n z 2 ny1 y
Ý ns1
m
y
2z
Ý
z y Ž 2 k y 1 . pr2 2
ks1
m
z
2 ny1
Ý Ý ks1 ns1 Ž 2 k y 1 . pr2
2
2
Ž 2 k y 1 . pr2
in this case; a n are as in w3, Eq. Ž40.x. Both in Ž37. and Ž38., < z < - `, and poles of tan z are deleted. Remark. Equation Ž37. can be obtained from w3, Eq. Ž38.x. ŽRedenote N by m , and then write the sum in k of w3, Eq. Ž38.x as two sums: one from k s 1 to k s m , and another from k s m q 1 to k s `.. Equation Ž38. can be reduced to Ž37.. Remark. Integers Nm G 0 to satisfy condition Ž23. can always be found. Namely, let «m ) 0 Ž m s 1, 2, . . . ., «m ª 0 as m ª `. Choose t # g Ž0, 1. Žt # s 1r2, for example., and for each curve Cm Ž m s 1, 2, . . . . Žof Theorems 4 or 4*. determine Nm as the smallest nonnegative integer such that
Ž hmrrm . Ž t#. N q1 - «m . m
Ž 39 .
Thus, F Ž z . Žof Theorem 4 or 4*. can always be represented as in Ž24., Ž29., Ž34. Žcondition Ž33. or Ž30. does not need to be satisfied for that..1 1
That is to say that any meromorphic Žin the entire finite z-plane. function F Ž z . can always be represented in the form Ž24., Ž29., or Ž34. wsince it is evident that the system of curves Cm4 Žof Theorems 4 or 4*. can always be constructed Žfor any meromorphic function F Ž z ...
538
DAVID S. TSELNIK
In conclusion of this section, we shall give an example showing how slowly convergent partial fraction expansions obtained by Cauchy can be. EXAMPLE. Consider for tan z its partial fraction expansion w3, Eq. Ž37.x, which for z s x is tan x s Ž 8 xrp 2 . S Ž x . ,
Ž 40 .
where SŽ x . s
`
Ý f Ž t , x . < tsk ,
2
f Ž t , x . s Ž 2 t y 1 . y Ž 2 xrp .
2 y1
. Ž 41 .
ks1
For x g Ž0, pr2. we use the Maclaurin]Cauchy estimate for the remainder of the series S after its K th Ž K s 1, 2, 3, . . . . term D SŽ K, x . w1, pp. 281]284x to find yF Ž t s K q 1, x . F D S Ž K , x . F yF Ž t s K , x . ,
Ž 42 .
where F Ž t , x . s Ž pr8 x . ln Ž 1 y 2 xrp Ž 2 t y 1 . . r Ž 1 q 2 xrp Ž 2 t y 1 .
.4. Ž 43 .
For the corresponding error in the value of tan x, calculated by Ž40. with 1 F k F K in Ž41., denoted D tanŽ x, K ., we find for K large D tan Ž x s pr4, K . ( 1r2p K .
Ž 44 .
Using Ž40., Ž41., and a calculator with the eight-digit-of-mantissa display, to obtain the mantissa of tanŽpr4. on the display as 1, we need to take into account such number of terms Ž K . of the series Ž41., that the corresponding D tanŽ x s pr4, K . would be less than 5 ? 10y8 . For that we must have K ( 10 7rp ( 3.18 ? 10 6 . In other words, using Cauchy’s partial fraction expansion w3, Eq. Ž37.x for tan z, we need to take into account more than three million Ž!. terms of that expansion to get tanŽpr4., in the display of our calculator, as 1. This is an enormously slow convergence, of course. For comparison, it takes five terms Ž N s 5. of the sum in n and one term of the sum in k of w3, Eq. Ž39.x to get tanŽpr4. s 1 in the display of our calculator!
6. OBTAINING ESTIMATES FOR REMAINDERS OF TAYLOR SERIES Expansions w3, Eq. Ž18.x are good instruments for obtaining estimates for remainders of Taylor series.
REPRESENTATION AND EXPANSIONS OF F Ž z .
539
EXAMPLE. Let us use w3, Eq. Ž39.x to obtain an estimate for the remainder of the Taylor series `
Ý Ž y1. ny 1 a n z 2 ny1 Ž < z < - pr2.
tanh z s
Ž 45 .
ns1
on the interval 0 - z s x - pr2 from below. For a n of Ž45. we have
a n s 2 2 n Ž 2 2 n y 1 . Bnr Ž 2 n . !
Ž n s 1, 2, 3, . . . .
Ž 46 .
w4, Eq. Ž44.x, where the Bn are the Bernoulli numbers, B1 s 1r6, B2 s 1r30, B3 s 1r42, . . . . From w3, Eq. Ž39.x, for the remainder of Ž45. after its Nth term Ž N s 1, 2, 3, . . . . we get rN Ž x . s Ž y1 .
N
2N
x
`
2x
ž / pr2
x 2 q Ž pr2 .
2
Ý ks1
1
Ž 2 k y 1.
2N
n k Ž x . , Ž 47 .
where 2
n k Ž x . s x 2 q Ž pr2 . r x 2 q Ž 2 k y 1 . pr2
2
4
Ž k s 1, 2, 3, . . . . , Ž 48 .
and then we find < rN Ž x . < )
x
2N
ž / pr2
`
2x x 2 q Ž pr2 .
2
Ý ms0
1
Ž 2 m q 1.
2 Nq2
Ž 49 .
Ž0 - x - pr2, N s 1, 2, 3, . . . .. Using that `
Ý Ž 2 m q 1. y2 Ny2 s Ž 1 y 2y2 Ny2 . z Ž 2 N q 2.
Ž 50 .
ms0
Ž N s 0, 1, 2, . . . ., where z is the Riemann zeta-function w12, Sect. 13.3x, and also that
z Ž 2 N q 2 . s Ž 2p .
2 Nq2
BNq 1r2 Ž 2 N q 2 . !
Ž N s 0, 1, 2, . . . . Ž 51 .
w4, Eq. Ž65.x, we find < rN Ž x . < )
a Nq1 x 2 Nq1 1 q xr Ž pr2 .
2
Ž 52 .
for x g Ž0, pr2., N s 1, 2, 3, . . . . This is the estimate for the absolute value of the remainder of the Taylor series Ž45. after its Nth term on the interval x g Ž0, pr2. from below, expressed in terms of the absolute value of the first neglected Žthe Ž N q 1.st. term of that series.
540
DAVID S. TSELNIK
Now, on the circle of convergence of Ž45., < z < s pr2, tanh z has two poles, z s "p ir2, both simple. Thus, Ž45. will diverge at z s x s pr2 Žsee Special Case in Section 4 above.. Therefore, one cannot calculate tanhŽpr2. as N
lim
Ý Ž y1. ny 1 a n Ž pr2. 2 ny1 .
Ž 53 .
Nª` ns1
From w3, Eq. Ž39.x it follows, however, that tanhŽpr2. can be calculated as Žfor example. tanh
p 2
2M
s lim
2
Ý Ž y1. ny 1 a n Ž pr2. 2 ny1 q p
Ž 54 .
Mª` ns1
an interesting fact. Finally, for < x < g Žpr2, `., when Ž45. diverges, w3, Eq. Ž39.x is still valid, of course, and for N large we find the following approximation for the segment of Ž45. N
Ý Ž y1. ny 1 a n x 2 ny1 ( tanh
N
x y Ž y1 . 2 x
ns1
x
2N
ž / pr2
1 x q Ž pr2 . 2
2
Ž 55 . also an interesting fact.
7. FOUR COMMENTS ON MITTAG-LEFFLER’S THEOREM In this section, we give concrete expressions for series which Mittag-Leffler uses in the course of proof of his theorem on partial fraction expansions. These expressions are given in the comments to follow. But first we need to state the theorem itself. MITTAG-LEFFLER’S THEOREM w2, Vol. 2, pp. 299]301x.
Let
b 0 s z0 , b1 , . . . , bk , . . .
Ž 56 .
be a sequence of distinct complex numbers such that < b 1 y z 0 < F < b 2 y z 0 < F ??? F < b k y z 0 < F ???
Ž 57 .
and < bk y z0 < ª `
as k ª `,
Ž 58 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
541
and let g 0 Ž z ., g 1Ž z ., . . . , g k Ž z ., . . . be a sequence of rational functions of the form Ž1.. Then there exists a meromorphic function F Ž z . whose poles coincide with the points Ž56. and whose principal part at the pole b k equals g k Ž z . for each k s 0, 1, 2, . . . . ŽThe special case when g 0 Ž z . s 0 is included in the theorem.. Remark. We used notations of this paper in writing out this theorem. Moreover, in Mittag-Leffler’s theorem as it is given in w2x, z 0 s 0. We have chosen to set b 0 s z 0 Žnot zero necessarily . for more generality and to be able to use expressions for Gk Ž z, z 0 ; N . of w3, 4x below. Mittag-Leffler starts the proof of the theorem from the Taylor series w2, Vol. 2, p. 300x n
g k Ž z . s aŽ0k . q a1Ž k . Ž z y z 0 . q ??? qaŽnk . Ž z y z 0 . q ???
Ž 59 .
Ž k s 1, 2, 3, . . . .. Our Comment. Written in concrete form, series Ž59. is gk Ž z . s
gk
`
Ý Ý ns0
ss1
Žk.
ž
Ays nqsy1 s n Ž z0 y bk .
/
ž
n
z y z0
bk y z0
/
Ž 60 .
Ž k s 1, 2, 3, . . . . w4, Eq. Ž25.x. Then Mittag-Leffler introduces a sequence « k 4 of positive numbers such that `
Ý «k - `
Ž 61 .
ks1
w2, Vol. 2, Eq. Ž10.35.x, and he chooses integers n1 , n 2 , n 3 , . . . such that on the disk D k : < z y z 0 < - 12 < b k y z 0 <, < g k Ž z . q pk Ž z . < - « k
Ž k s 1, 2, 3, . . . . ,
Ž 62 .
where pk Ž z . s yaŽ0k . y a1Ž k . Ž z y z 0 . y ??? yaŽnkk. Ž z y z 0 .
nk
Ž 63 .
Žsee w2, Vol. 2, Eqs. Ž10.36., Ž10.37.x.. Our Comment. From Ž59., Ž60., Ž63., and w4, Eq. Ž26.x, we find that g k Ž z . q pk Ž z . s Gk Ž z, z 0 ; n k .
Ž 64 .
542
DAVID S. TSELNIK
Ž k s 1, 2, 3, . . . .. Thus, Mittag-Leffler’s condition Ž62. can be written as < Gk Ž z, z 0 ; n k . < - « k
Ž 65 .
Ž k s 1, 2, 3, . . . . for all z g D k ; concrete expressions for Gk are given in w3, 4x Žreplace N of these expressions by n k .. With « k already chosen, Ž65. can be used to find the corresponding n k . For example, if the pole b k is simple, then using Ž15. one finds from Ž65. that it is sufficient to have
Ž 12 .
nk
k. < < < AŽy1 r bk y z0 < F « k
Ž 66 .
for Ž65. to be satisfied on D k . Which leads to n k G y Ž ln 2 .
y1
k. < ln Ž « k < b k y z 0
Ž 67 .
in the case in question. Finally, Mittag-Leffler finds F Ž z . in the form FŽ z. s
`
Ý
g k Ž z . q pk Ž z . ,
Ž 68.
ks0
where p 0 Ž z . is an arbitrary polynomial w2, Vol. 2, p. 300x. Our Comment. Expression Ž68. for F Ž z . in its concrete form is F Ž z . s g 0 Ž z . q p0 Ž z . q
`
Ý Gk Ž z, z 0 ; n k . ,
Ž 69 .
ks1
where the Gk are as described above. Also, according to the corollary to Mittag-Leffler’s Theorem w2, Vol. 2, p. 301x, a meromorphic function F Ž z . with poles Ž56. and the corresponding principal parts g k Ž z . can be represented by expansion FŽ z. s vŽ z. q
`
Ý
g k Ž z . q pk Ž z . ,
Ž 70.
ks0
where v Ž z . is an entire function and p 0 Ž z . is an arbitrary polynomial. Our Comment. In a more concrete form, Ž70. is Ž p 0 is omitted in Ž71.. F Ž z . s v Ž z . q g0 Ž z . q
`
Ý Gk Ž z, z 0 ; n k . . ks1
Ž 71 .
REPRESENTATION AND EXPANSIONS OF F Ž z .
543
Remark. For the case b 0 s z 0 s 0 Žconsidered in w2, pp. 299]301x. one should use Gk Ž z, 0; n k . instead of Gk Ž z, z 0 ; n k .. The concrete expressions for g k Ž z . Ž60., for g k Ž z . q pk Ž z . Ž64., and for F Ž z . Ž69., Ž71. given above Žwith concrete expressions for Gk Ž z, z 0 ; n k . referred to. will be of help if one wants to use Mittag-Leffler’s theorem to actually construct expansions Ž68. or Ž70. in any concrete cases.
8. CONCLUDING REMARKS This paper originated from the paper ‘‘A Note on Representation and Series Expansions of F Ž z . in Complex Domains’’ Žits contents reflected Žto a very large extent. by abstracts w8]10x. the present author submitted to J. Math. Anal. Appl. in July 1995. With some material deleted and new material added, that paper evolved into the paper entitled ‘‘On Representation and Series Expansions of F Ž z .’’ submitted to J. Math. Anal. Appl. in May 1996. The present paper is a compression Žto approximately 60% of its original length. of the latter manuscript. Some related Žto the material of w3, 4x, and of this paper. developments are described in the present author’s works w5, 7x Žfor linear operators, for Fredholm integral equations, in particular. and in w6x Žfor Jacobian elliptic functions.. On January 8, 1997, the present author made a presentation entitled ‘‘On Representation of F Ž z . in multiply-connected domains Žin the complex z-plane.’’ w11x at the 103rd Annual Meeting of the AMS in San Diego, California. It was expedient, from the author’s point of view, to include the essence of the Theorems 4 and 4*, and of the Remark with Eq. Ž39. of the present paper with that presentation. And they were included, with indication that these results are from a different paper by the author. From the Žnumbered. equations of the present paper, Eqs. Ž22. ] Ž24., Ž34., and Ž39. were presented Žwith pertinent explanations. in San Diego.
REFERENCES 1. G. M. Fikhtengol’ts, ‘‘A Course of the Differential and Integral Calculuses,’’ 6th ed., Vol. 2, Nauka, Moscow, 1966. wIn Russianx 2. A. I. Markushevich, ‘‘Theory of Functions of a Complex Variable,’’ Chelsea, New York, 1977. 3. D. S. Tselnik, Representation and series expansions for meromorphic functions, Complex Variables 25 Ž1994., 159]171. 4. D. S. Tselnik, On series expansions for meromorphic functions, J. Math. Anal. Appl. 193 Ž1995., 522]542.
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5. D. S. Tselnik, A bound for the remainder of the Hilbert-Schmidt series and other results on representation of solutions to the functional equation of the second kind with a self-adjoint compact operator as an infinite series, Comput. Math. Appl. 29 Ž1995., 61]68. 6. D. S. Tselnik, On trigonometric series expansions of twelve Jacobian elliptic functions, SIAM J. Math. Anal., 28 Ž1997., 715]730. 7. D. S. Tselnik, A simple bound for the remainder of the Neumann series in the case of a self-adjoint compact operator, Appl. Math. Lett. 7 Ž1994., 71]74. 8. D. S. Tselnik, Four theorems on representation and series expansions of F Ž z . in complex domains, Part 1, Abstracts Amer. Math. Soc. 17 Ž1996., 258. 9. D. S. Tselnik, Four theorems on representation and series expansions of F Ž z . in complex domains, Part 2, Abstracts Amer. Math. Soc. 17 Ž1996., 258]259. 10. D. S. Tselnik, Expressions for functions in Mittag-Leffler’s theorem, Abstracts Amer. Math. Soc. 17 Ž1996., 449. 11. D. S. Tselnik, ‘‘On representation of F Ž z . in multiply-connected domains Žin the complex z-plane.,’’ Abstracts Amer. Math. Soc., 18 Ž1997., 66. 12. E. T. Whittaker and G. N. Watson, ‘‘A Course of Modern Analysis,’’ 4th ed., Cambridge Univ. Press, Cambridge, UK, 1927.