Ries Via the Generalized Weighted-Averages Method

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Progress In Electromagnetics Research M, Vol. 14, 233–245, 2010

Received 7 October 2010, Accepted 28 October 2010, Scheduled 8 November 2010

Corresponding author: Athanasios G. Polimeridis (athanasios.polymeridis@epﬂ.ch).

234
Polimeridis, Golubović Nićiforović, and Mosig
Progress In Electromagnetics Research M, Vol. 14, 2010
235

∞

=0 =0

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−

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}

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236
Polimeridis, Golubović Nićiforović, and Mosig

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+

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+
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+

+ −1
+2 −1

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( +1)
−1

≥

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( ) −1
( ) 0

−

Progress In Electromagnetics Research M, Vol. 14, 2010
237

(1) 2

0
2 +1

−

−1

−

{
}

{ }

+1 +1
+1

−

0
0

238
Polimeridis, Golubović Nićiforović, and Mosig

+1
+1
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+1

−

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∼
→ ∞

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∼

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0

||
||

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Progress In Electromagnetics Research M, Vol. 14, 2010
239

−

( )
( ) ( )

+1
( )
( +1)

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−

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( )

≈ −

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(0) 0
(1) 0
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(0) 1
(1) 1
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( −2) 2

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· · ·

240
Polimeridis, Golubović Nićiforović, and Mosig

( )

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→

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√
−

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Progress In Electromagnetics Research M, Vol. 14, 2010
241

16 14 12 10 8
16 14 12 10 8

WA _IV(asymptotic)

WA _I(asymptotic) u

ελλ

2
4
6
8
10 12 14 16 18 20

2
4
6
8
10 12 14 16 18 20

(a) Test example (29)
(b) Test example (30)

∞

2
2
=0

∞

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Polimeridis, Golubović Nićiforović, and Mosig

WA_III(asymptotic)
WA _III(asymptotic)

14 12 10 8
14 12 10 8

4
4

2
4
6
8
10 12 14 16 18 20

2
4
6
8
10 12 14 16 18 20

(a) Test example (31)
(b) Test example (32)

16 14 12 10 8
16 14 12 10 8

WA_IV(asymptotic)
WA_IV(asymptotic) vv

4
4

0
0

2
4
6
8
10 12 14 16 18 20

2
4
6
8
10 12 14 16 18 20

(a) Test example (33)
(b) Test example (34)

∞

+1
=0

∞

Progress In Electromagnetics Research M, Vol. 14, 2010
243
244
Polimeridis, Golubović Nićiforović, and Mosig
Progress In Electromagnetics Research M, Vol. 14, 2010
245

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Arxiv:Math/0702243V4

AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS LINAS VEPŠTAS ABSTRACT. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efﬁcient algorithm for computing the Riemann zeta function”[4, 5], to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general values of complex s and a kidney-shaped region of complex z values given by z2/(z 1) < 4. By using the duplication formula and the inversion formula, the − range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler- Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evalua- tions of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable.
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