A constructive method for approximating trigonometric functions and their integrals
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A constructive method for approximating trigonometric functions and their integrals CHEN Xiao-diao
WANG Long-quan
WANG Yi-gang∗
Abstract. This paper presents an interpolation-based method (IBM) for approximating some trigonometric functions or their integrals as well. It provides two-sided bounds for each function, which also achieves much better approximation effects than those of prevailing methods. In principle, the IBM can be applied for bounding more bounded smooth functions and their integrals as well, and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.
§1
Introduction
In many applications such as operations research, computer science, mathematics, physical sciences and engineering [1,9,22], computing the integrals of some bounded functions in an interval, is needed such as the trigonometric functions sin(x) x f1 (x) = and f2 (x) = 3 + cos(x), x ∈ [0, π/2], x sin(x) whose integrals can not be explicitly expressed and must be numerically solved. Many authors consider estimating the bounds of the integrals, by estimating the bounds of the given bounded functions, which leads to many researches on the corresponding inequalities [2-6, 8, 10-20, 23-25], including some unbounded functions f (x) = ( sin(x) )2 + tan(x) , 3 x x sin(x) tan(x) + , x ∈ [0, π/2]. f4 (x) = 2 · x x sin(x) tan(x) f5 (x) = + , x x Several famous inequalities with x ∈ (0, π/2) are summarized as follows [3] Received: 2017-07-28. Revised: 2019-11-01. MR Subject Classification: 26D05 , 26D07, 42A10, 42A15. Keywords: Pad´ e approximant, trigonometric function, constructive method, interpolation-based method, two-sided bounds. Digital Object Identifier(DOI): https://doi.org/10.1007/s11766-020-3562-z. Supported by the National Natural Science Foundation of China (61672009,61502130). ∗ Corresponding author.
294
Appl. Math. J. Chinese Univ.
Vol. 35, No. 3
1 + cos(x) 2 + cos(x) < L1 (x) < f1 (x) < U1 (x) < , 2 3 f2 (x) > L2 (x), f3 (x) > L3 (x) > 2, f4 (x) > L4 (x) > 3,
(1) (2) (3) (4)
f5 (x) > L5 (x) > 2, (5) 60 − 7x2 11x4 − 360x2 + 2520 x4 + 8x2 + 96 where L1 (x) = , U1 (x) = , L2 (x) = , 60 + 3x2 60x2 + 2520 2x2 + 24 6 4 2 4 2 −303x + 5550x − 32400x + 108000 81x − 945x + 2700 L3 (x) = , L4 (x) = , −54x6 − 2025x4 − 16200x2 + 54000 −18x4 − 315x2 + 900 4 2 39x − 480x + 1800 L5 (x) = . −18x4 − 315x2 + 900 In principle, there are two key issues for an inequality. One is to find the bounds, and the other is to prove it [28]. Malesevic and his coauthors provided several methods for proving some inequalities of some special functions [15,27]. This paper presents an interpolation-based method (IBM) for constructing the bounds of some famous trigonometric functions, including fi (x), i = 1, 2, · · · , 5. We take the inequalities (1 ∼ 5) for example, and provide two-sided bounding functions which can achieve much better approximation effects. In principle, the IBM can be applied in approximating any smooth bounded function within some bounded interval, a
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