文档介绍：To appear: Math. Comp. 2001
LANDEN TRANSFORMATIONS AND THE INTEGRATION OF
RATIONAL FUNCTIONS
E BOROS AND VICTOR H. MOLL
Abstract. We present a rational version of the classical Landen transfor-
mation for elliptic integrals. This is employed to obtain explicit closed-form
expressions for a large class of integrals of even rational functions and to de-
velop an algorithm for numerical integration of these functions.
1. Introduction
We consider the space of even rational functions of degree 2p
p−1 p
P (z)
( 2(p−1−k) 2(p−k))
E2p := R(z) = P (z) := bkz and Q(z) := akz
Q(z) X X
k=0 k=0
with positive real coefficients ak, bk ∈ R+ normalized by the condition a0 = ap = 1,
the space
∞
E∞:= E
[ 2p
p=1
of normalized even rational functions, and the set of 2p − 1 parameters
P2p := {a1, · · · , ap−1; b0, · · · , bp−1}.
We describe an algorithm to determine, as a function of the parameter set P2p, a
closed-form expression of the integral
∞
() I := Z R(z) dz
0
for a large class of functions R ∈ E∞. The function R is called symmetric if its
denominator Q satisfies Q(1/z) = z−2pQ(z). This is equivalent to its coefficients
being palindromic, . aj = ap−j for 1 ≤ j ≤ p.
The class of symmetric functions plays a crucial role in this algorithm. Define
s { ∈}
E2p := R E2p den(R) is symmetric
(where den(R) denotes the denominator of R), the class of rational functions with
symmetric denominators of degree 2p, and
∞
Es := Es .
∞[ 2p
p=1
Date: April 25, 2001.
1991 Mathematics Subject Classification. Primary 33.
Key words and phrases. Rational functions, Landen transformation, Integrals.
The second author was supported in part by NSF Grant DMS-0070567.
1
2 E BOROS AND VICTOR H. MOLL
For m ∈ N define
m { ∈ 1/(m+1) }
E2p := R E∞ (den(R)) is even, symmetric of degree 2p
m,s m ∩ s ∈ m,s
and E2p := E2p E2p, so a function R E2p can be written in the form
P (z)
R(z) = ,
Qm+1(z)
where P (