scispace - formally typeset
Search or ask a question
Topic

Inverse trigonometric functions

About: Inverse trigonometric functions is a research topic. Over the lifetime, 854 publications have been published within this topic receiving 11141 citations. The topic is also known as: arcus function & antitrigonometric function.


Papers
More filters
Proceedings ArticleDOI
30 May 1999
TL;DR: A new hardware approach to high-speed computation of nonlinear functions with all of the functions needed can be regularized into a single efficient algorithm, and highly reduced cycle implementations can be achieved.
Abstract: Several advanced DSP algorithms, arising in applications such as wireless communications, computer graphics, computerized tomography, and speech compression, require extensive use of nonlinear functions. We discuss a new hardware approach to high-speed computation of nonlinear functions. With this approach all of the functions needed can be regularized into a single efficient algorithm. Further, highly reduced cycle implementations can be achieved. Specifically, for real arguments, a new result can be produced every cycle-in a pipelined mode. The underlying principle which has made the combined goals of high-speed and multi-functionality possible is significance-based polynomial interpolation of very small ROM tables. Considered are the following seven functions: arctangent, cosine, logarithm, reciprocal, reciprocal-square-root, sine, and square-root. Also presented is a theoretical development for error prediction, a tool for the selection of architectural parameters. Finally, the paper presents a novel technique, named here as 'microshaping', for avoiding overflows, thereby eliminating exception handling.

23 citations

Journal ArticleDOI
TL;DR: In this article, the power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived and applied to trigonometric functions.
Abstract: Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c k , converge rapidly to zero as k??. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc?(at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.

22 citations

Journal ArticleDOI
TL;DR: In this article, the degree of convergence of a sequence of linear operators connected with the Fourier series of a function of class Lp (p > 1) to that function and some inverse results in relating the convergence to the classes of functions are discussed.
Abstract: The paper is concerned with the determination of the degree of convergence of a sequence of linear operators connected with the Fourier series of a function of class Lp (p > 1) to that function and some inverse results in relating the convergence to the classes of functions. In certain cases one can obtain the saturation results too. In all cases Lp norm is used. 1980 Mathematics subject classification (Amer. Math. Soc): 41 A 40. 1 Let f(x) be a periodic, Lebesgue integrable function with period 2m. Let the Fourier series for/(x) be given by OC 00 (1) 2#o + 2 (a*coskx + bksin kx) = 2 Ak{x). A r = l k = 0 Let Sn(f\ x) be the nlh. partial sum of the series (1). The conjugate series of the series (1) is 00 00 2 Bn(x) = 2 (bkcoskx — aksin kx). n=\ k=\ The conjugate function/of/, is given by (2) f{x) = (2^)"' f {f(x + t) -fix t)}cot±dt the integral being interpreted as a Cauchy integral. It is known that/exists almost everywhere whenever/is integrable. Copyright Australian Mathematical Society 1983 143 use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870002317X Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 03 Feb 2019 at 14:42:39, subject to the Cambridge Core terms of 144 R. N. Mohapatra and D. C. Russell [2 ] The space Lp\ — m,-n\ when p — oo will be replaced by the space c2w of all continuous functions defined over [ — 77,77-]. Throughout the paper, norms will be taken with respect to the variable JC and || • II ̂ will denote the usual Lp norm for 1 = 00. F o r / G Lp[ — ir, 77] (1 1) will be as usual (see [5], page 612; also see [18], pages 42, 45). The class Lip(o, p) with/; — 00 will be taken as Lip a. Two functions / and g are said to be equivalent if f(x) — g(x) almost everywhere. Let {cn}, {dn} be two non-zero sequences with cn, dn > 0. Suppose Cn = 2^= 0 ck a n d Dn = 2 n k = 0 dk. L e t R n = c o d n + c x d n _ x + ••• + c n d 0 (n = 0 , 1 , . . . ) . Given/, let us associate with it the operator tn{ f) defined by (5) tn(f;x) = (Rny i ic^kdkSk(x). k=\ It should be remarked that tn(f; x) is the (N, c, d) transform of {Sk(f; x)} (see [2]). We shall write tn(f; x) = Nn(f; x) or Nn(f; x) according as dn = 1 for all n or cn = 1 for all n. If there exists a positive non-increasing function §(n) and a normed linear space K of functions such that (6) II f(x) — tn(f; x)\\ = o((n)) =>/is a constant a.e., (7) \\f(x)-tn(f;x)\\ = O(4>(n))~fGK, and (8) f&K~\\f(x)-tn(f;x)\\ then we say that the operator tn{f) or the corresponding method (N,c,d) is saturated with order («) and class K. Ever since the definition of saturation of summability methods was given by Favard [3] many authors have studied the saturation property of operators which are obtained as transforms of the nth partial sum of the Fourier series by use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870002317X Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 03 Feb 2019 at 14:42:39, subject to the Cambridge Core terms of 13] Approximation of functions 145 summability methods. Sunouchi and Watari [15], [16] have obtained the saturation order and class for Cesaro, Abel and the Riesz method (R, n^, 1) (f = 1,2,...). Mohapatra and Sahney [11] have obtained results on saturation for a general class of summability methods in the supremum norm. Sunouchi [14] has studied the local saturation properties of the convolution operator (also see [13], [17]). Concerning the saturation property of the Norlund method, Goel, Holland, Nasim and Sahney [4] have proved the following theorem: THEOREM A ([4], compare [9]). Letf G c2w and Cn > 0 (all n). Then the following hold: (9) Wf-Nn{f)\\K =o{~\ ^ f is a constant a.e.

22 citations

Journal ArticleDOI
TL;DR: In this paper, the authors extended the T-spline approach to trigonometric generalized B-splines, a particularly relevant case of non-polynomial splines.

22 citations


Network Information
Related Topics (5)
Differential equation
88K papers, 2M citations
81% related
Matrix (mathematics)
105.5K papers, 1.9M citations
80% related
Bounded function
77.2K papers, 1.3M citations
79% related
Boundary value problem
145.3K papers, 2.7M citations
78% related
Nonlinear system
208.1K papers, 4M citations
77% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202335
202298
202134
202027
201918
201814