- Email: [email protected]
A short proof of Darboux's lemma
App!. Math. Lett. Vol. 2, No. 4, pp. iii-iv, 1989 Pnnted in Great Britain. All rights reserved
A Short
Proof
08939659/89 $3.00 + 0.00 Copyright@ 1989 Pergamon Press plc
of Darboux’s
DONALD E. KNUTH and
Lemma
HERBERT S. WILF
Computer Science Department, Stanford University Mathematics Department, University of Pennsylvania Abstract. Elementary asymptotic methods suffice to establish asymptotic series for the coeficients of analytic functions with algebraic singularities.
Darboux’s lemma [l] g ives the complete asymptotic expansion of the coefficients of power series that represent functions whose singularity nearest to the origin is algebraic. Most proofs of this result use contour integration. We present here a short and more elementary derivation. Our proof is an outgrowth of the method used in an example on p. 71 of [2]. Essentially, we are observing here that that method is perfectly general. The symbol ‘[z”] . . . ’ will mean ‘the coefficient of z” in the power series . . . .’ We consider here only the case where there is just one singularity on the circle of convergence, and w.1.o.g. we can suppose that the singularity is at I = 1. THEOREM (DARBOUX). Let V(Z) be analytic in some disk 1~1< 1 + q, and suppose that in a neighborhood of z = 1 it has the expansion V(Z) = c vj(l - z)j. Then for every ,B and every integer m 2 0 we have
[Z”]{(l
- Z)‘V(.Z)} = [%“I { FVj(l-
Z)‘+j} + 0(n-m-p-2)
j=O m
= CC j=O
n-p-j-1 9
n
+ 0(n-m-P--2), )
asn+co. We begin
the proof with three lemmas.
LEMMA 1. Let {a,} and {b,} b e t wo sequences that satisfy b, = O(P) (0 c 0 < 1). Then n c akb,,_k = O(nSr).
(a) a, = O(n-7)
and (b)
kc0
PROOF:
We have first (the C’s are not all the same)
5 max(C, Cn-r)(C0”‘2)
5 Co’”
(0 < e’ < 1).
Further,
Supported in part by National Science Foundation grant CCR-8610181 and by Office of NavaI Reseerch contracts N00014-87-K-0502 and NOO014-85-H-0320. Typeset by A,#-TjQC
D.E. KNUTH, H.S. WILF
iv
Next, an easy exercise in Stirling’s formula yields LEMMA 2. If /3 is fixed, then as n + 00 we have
LEMMA 3. Let u(%) = (1 - %)Tv(%), w h ere V(Z) is analytic
(q>
in some disk 1.~1< 1 + q
0). Then [z”] u(z) = 0(n-7-l).
PROOF: Apply Lemma 1 with a, = [%“I (1 - %)’ and b, = [P] V(Z). Since v is analytic The result follows by Lemma 2. in a disk /%I < 1 + 77we have b, = O(P).
Now, to prove the theorem, we have (1 - %)&I(%) - &(I
- %)p+j = c
j=O
Vj(l - %)p+j
j>m = (1 - %)P+m+G(%),
where the regions of analyticity of 6 and II are the same. The result now follows from Lemma 3. REFERENCES 1. G. Darboux, Mtmoire &endue
mr I’appsoximation des fonctiona de trbs grands nombres, et I)‘ELP une chase des dCveloppementa en a&-ie, J. Math. Pures Appl. 4 (1878), 5-56, 377-416.
2. Daniel H. Greene and Donald E. Knuth, “Mathematics for the Analysis of Algorithms,” Birkhiiuser Boston, 1981.
Computer Science Depwtment, Stanford Universit.y, Stanford, CA 94305, U.S.A. Mathematics Department, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.
A Short
Proof
08939659/89 $3.00 + 0.00 Copyright@ 1989 Pergamon Press plc
of Darboux’s
DONALD E. KNUTH and
Lemma
HERBERT S. WILF
Computer Science Department, Stanford University Mathematics Department, University of Pennsylvania Abstract. Elementary asymptotic methods suffice to establish asymptotic series for the coeficients of analytic functions with algebraic singularities.
Darboux’s lemma [l] g ives the complete asymptotic expansion of the coefficients of power series that represent functions whose singularity nearest to the origin is algebraic. Most proofs of this result use contour integration. We present here a short and more elementary derivation. Our proof is an outgrowth of the method used in an example on p. 71 of [2]. Essentially, we are observing here that that method is perfectly general. The symbol ‘[z”] . . . ’ will mean ‘the coefficient of z” in the power series . . . .’ We consider here only the case where there is just one singularity on the circle of convergence, and w.1.o.g. we can suppose that the singularity is at I = 1. THEOREM (DARBOUX). Let V(Z) be analytic in some disk 1~1< 1 + q, and suppose that in a neighborhood of z = 1 it has the expansion V(Z) = c vj(l - z)j. Then for every ,B and every integer m 2 0 we have
[Z”]{(l
- Z)‘V(.Z)} = [%“I { FVj(l-
Z)‘+j} + 0(n-m-p-2)
j=O m
= CC j=O
n-p-j-1 9
n
+ 0(n-m-P--2), )
asn+co. We begin
the proof with three lemmas.
LEMMA 1. Let {a,} and {b,} b e t wo sequences that satisfy b, = O(P) (0 c 0 < 1). Then n c akb,,_k = O(nSr).
(a) a, = O(n-7)
and (b)
kc0
PROOF:
We have first (the C’s are not all the same)
5 max(C, Cn-r)(C0”‘2)
5 Co’”
(0 < e’ < 1).
Further,
Supported in part by National Science Foundation grant CCR-8610181 and by Office of NavaI Reseerch contracts N00014-87-K-0502 and NOO014-85-H-0320. Typeset by A,#-TjQC
D.E. KNUTH, H.S. WILF
iv
Next, an easy exercise in Stirling’s formula yields LEMMA 2. If /3 is fixed, then as n + 00 we have
LEMMA 3. Let u(%) = (1 - %)Tv(%), w h ere V(Z) is analytic
(q>
in some disk 1.~1< 1 + q
0). Then [z”] u(z) = 0(n-7-l).
PROOF: Apply Lemma 1 with a, = [%“I (1 - %)’ and b, = [P] V(Z). Since v is analytic The result follows by Lemma 2. in a disk /%I < 1 + 77we have b, = O(P).
Now, to prove the theorem, we have (1 - %)&I(%) - &(I
- %)p+j = c
j=O
Vj(l - %)p+j
j>m = (1 - %)P+m+G(%),
where the regions of analyticity of 6 and II are the same. The result now follows from Lemma 3. REFERENCES 1. G. Darboux, Mtmoire &endue
mr I’appsoximation des fonctiona de trbs grands nombres, et I)‘ELP une chase des dCveloppementa en a&-ie, J. Math. Pures Appl. 4 (1878), 5-56, 377-416.
2. Daniel H. Greene and Donald E. Knuth, “Mathematics for the Analysis of Algorithms,” Birkhiiuser Boston, 1981.
Computer Science Depwtment, Stanford Universit.y, Stanford, CA 94305, U.S.A. Mathematics Department, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.