International Mathematical Forum, 2, 2007, no. 46, 2269 - 2276
An Accurate Estimate of Mathieu’s Series Vito Lampret University of Ljubljana Faculty of Civil and Geodetic Engineering Department of Mathematics and Physics Jamova 2, Ljubljana 1000, Slovenia 386
[email protected] We dedicate this article to Leonard Euler, the father of the Euler-Maclaurin formula, on the occasion of three hundreds anniversary of his birthday, April 15, 1707. Abstract Using Hermite’s, i.e. the Euler-Maclaurin summation ∑ formula of or2k der four, new approximations to Mathieu’s series S(x) ≡ ∞ k=1 (k2 +x2 )2 are obtained, which are more accurate than the approximations presented recently in the literature.
Mathematics Subject Classification: 26D15, 33E20, 33F05, 40A05, 40A25, 65B10, 65B15, 65D20 Keywords: accuracy, approximation, bound, estimate, inequality, Euler, Maclaurin, Mathieu, relative error, series
1
Introduction
In 1890 Mathieu [11] defined S(x) , now called Mathieu’s series, as S(x) ≡
∞ ∑ k=1
(k 2
2k . + x2 )2
(1)
Since that time Mathieu’s series has been investigated intensively many times, see for example [4], [5], [15], [16], [14], and [17]. Alzer [2] found that, for every real x, 1 1 < S(x) < 2 . (2) 2 x + 1/(2ζ(3)) x + 1/6
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V. Lampret
Moreover, recently, F. Qi et al. [13, Th. 4.2, p. 2550], have estimated Mathieu’s series as A(x) ≤ S(x) ≤ B(x), (3) for x > 0, where ( ) 4 (1 + x2 ) e−π/x + e−π/(2x) − 4x2 − 1 A(x) ≡ (e−π/x − 1) (1 + x2 ) (1 + 4x2 ) ( ) (1 + 4x2 ) e−π/x − e−π/(2x) − 4 (x2 + 1) B(x) ≡ . (e−π/x − 1) (1 + x2 ) (1 + 4x2 ) Although the estimate (2) is asymptotically sharp, it fails for x ≈ 0, since the upper bound in (2) approximates S(x) very roughly. However, the lower bound in (2) is considerably closer to S(x), as is evident from Figure 5. Unfortunately, for larger values of x the estimate (3) is worse than (2). This( is illustrated ) in Figure 1,( where)we plotted the graphs of functions 1 2 x 7→ 1/ x + 2ζ(3) and x 7→ 1/ x2 + 16 (dashed lines), together with the graphs of the bounds x 7→ A(x) and x 7→ B(x) (solid lines). In addition, the lower bound A(x) becomes negative, consequently useless for larger values of x. This is illustrated in Figure 2. In this note we wish to find an estimate of S(x) applicable on the entire R. 6 5 0.3
4 0.2
3 2
0.1
1 0.5
1.0
1.5
2.0
(
4
)
6
8
2
1 2ζ(3)
(
10
) Figure 1: The graphs of functions x 7→ 1/ x + and x 7→ 1/ x2 + 61 (dashed lines), together with the graphs of the bounds x 7→ A(x) and x 7→ B(x) (solid lines). 2
A summation formula
According to [10, Theorem, p. 320] or1 [9, Theorem 2, p. 119] ∑ we have at our disposal ∫a theorem comparing the convergence of a series ∞ k=1 f (k) and an ∞ integral 1 f (x) dx. 1
Additional literature: [1], [3], [6], [7], [8].
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An accurate estimate of Mathieu’s series 1 20
0.8
40
60
80
100
-0.005
0.6
-0.01
0.4
-0.015 -0.02
0.2
-0.025
1
2
3
4
5
6
Figure 2: The graph of lower bound A(x). ∫∞ Theorem. If f ∈ C 4 [1, ∞), 1 f (4) (x) dx converges, and the finite limits λ0 := lim f (n) and λ1 := lim f ′ (n) exist, then: n∈N
n∈N
n→∞
n→∞
(a) The series converges.
∑∞
(b) If the series ∞ ∑
f (k) =
k=1 f (k)
∑∞
k=1 f (k)
m−1 ∑
n
f (k) + lim
f (x) dx +
n∈N
m
n→∞
for every integer m ≥ 1, where 1 |ρ(m)| ≤ 384
3
⇐⇒
The sequence n 7→
∫n 1
f (x) dx
converges, then λ0 = 0 and ∫
k=1
k=1
converges.
∫
∞
f (m) λ1 f ′ (m) + − + ρ(m), (4) 2 12 12
(4) f (x) dx.
(5)
m
An approximation
Mathieu’s series originates from the function fx (t) ≡
2t , (t2 + x2 )2
x ∈ R being a parameter,
(6)
having the derivatives, 3t2 − x2 , (t2 + x2 )3 3t5 − 10t3 x2 + 3tx4 fx(4) (t) ≡ 240 , (t2 + x2 )6 fx′ (t) ≡ −2
(7) (8)
and the integral, ∫
∞
m
[
1 fx (t) dt = − 2 t + x2
]∞ = t=m
m2
1 . + x2
(9)
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V. Lampret
To use the quoted summation theorem we have to estimate the error term ρ(m, x). Thanks to (5) and (8), we estimate, ∫ ∞ (4) 1 fx (t) dt |ρ(m, x)| ≤ 384 m ∫ 240 ∞ 3t5 + 10t3 x2 + 3tx4 ≤ dt 384 m (t2 + x2 )6 5m4 + 15m2 x2 + 6x4 = δ(m, x) := (10) 16 (m2 + x2 )5 1 < ∆(m, x) := , (11) 2 2 (m + x2 )3 valid for every integer m ≥ 1 and any real x. Therefore, using the formula (4), and relations (6)–(7), we find the expression S(x) = σ(m, x) + ρ(m, x),
(12)
true for every real x and any integer m ≥ 1, where σ(m, x) ≡
m−1 ∑ k=1
2k 1 m 3m2 − x2 + + + . (k 2 + x2 )2 m2 + x2 (m2 + x2 )2 6 (m2 + x2 )3
(13)
We have, for m ≥ 1, m 3m2 − x2 6m3 + 3m2 + (6m − 1)x2 + = (m2 + x2 )2 6 (m2 + x2 )3 6 (m2 + x2 )3 2m3 + m2 6m3 + 3m2 = > 0. ≥ 6 (m2 + x2 )3 2 (m2 + x2 )3 Therefore, the relative error, r(m, x) :=
S(x) − σ(m, x) , σ(m, x)
(14)
of the approximation S(x) ≈ σ(m, x) can be estimated, using the relations (11)–(13), as ρ(m, x) |r(m, x)| = σ(m, x)
