Showing papers on "Elementary function published in 1995"
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01 Jan 1995TL;DR: In this article, the authors present a glossary of functions and symbols for algebra, geometry and trigonometry, including the elementary functions and the Elementary Functions of Differential Calculus (one variable).
Abstract: 1 Fundamentals. Discrete Mathematics.- 2 Algebra.- 3 Geometry and Trigonometry.- 4 Linear Algebra.- 5 The Elementary Functions.- 6 Differential Calculus (one variable).- 7 Integral Calculus.- 8 Sequences and Series.- 9 Ordinary Differential Equations (ODE).- 10 Multidimensional Calculus.- 11 Vector Analysis.- 12 Orthogonal Series and Special Functions.- 13 Transforms.- 14 Complex Analysis.- 15 Optimization.- 16 Numerical Analysis.- 17 Probability Theory.- 18 Statistics.- 19 Miscellaneous.- Glossary of functions.- Glossary of symbols.
541 citations
TL;DR: In this article, the precision results in terms of generalized, multivariable hypergeometric functions are presented giving explicit series for small and large momenta, and the imaginary parts of these integrals are expressed as complete elliptic integrals.
183 citations
TL;DR: In this paper, an analytical method somewhat analogous to finite wing theory has been developed which enables the flow induced by a linearized propeller actuator disk with variable radial distribution of load to be solved in closed form for the first time.
Abstract: An analytical method somewhat analogous to finite wing theory has been developed which enables the flow induced by a linearized propeller actuator disk with variable radial distribution of load to be solved in closed form for the first time. Analytical solutions are given for various load distributions including the case of an arbitrary polynomial loading. As in finite wing theory, the case of elliptic loading is exceptionally simple and the induced velocities and stream function are simple expressions of elementary functions. Results are also given for a practical propeller load distribution with finite hub. The method can also be used to solve a wide range of analogous electromagnetic problems.
129 citations
TL;DR: In this paper, the scalar two-loop self-energy master diagram is studied in the case of arbitrary masses and analytical results in terms of Lauricella-and Appell-functions are presented for the imaginary part.
118 citations
19 Jul 1995
TL;DR: A general approach decomposing a function into a sum of functions, each with a smaller input site than the original, which can be mapped with essentially the same precision using small ROM tables and adders.
Abstract: We describe a general approach decomposing a function into a sum of functions, each with a smaller input site than the original. Hence we can map such functions with essentially the same precision using small ROM tables and adders. We derive an easy method to compute the worst case error for many elementary functions and an error bound for the rest. Important applications are reciprocals, logarithms, exponentials and others. >
115 citations
TL;DR: In this article, the scalar two-loop self-energy master diagram is studied in the case of arbitrary masses and analytical results in terms of Lauricella-and Appell-functions are presented for the imaginary part.
Abstract: The scalar two-loop self-energy master diagram is studied in the case of arbitrary masses. Analytical results in terms of Lauricella- and Appell-functions are presented for the imaginary part. By using the dispersion relation a one-dimensional integral representation is derived. This representation uses only elementary functions and is thus well suited for a numerical calculation of the master diagram.
99 citations
TL;DR: A new method for the fast evaluation of the elementary functions in single precision based on the evaluation of truncated Taylor series using a difference method, which can calculate the basic elementary functions, namely reciprocal, square root, logarithm, exponential, trig onometric and inverse trigonometric functions, within the latency of two to four floating point multiplies.
Abstract: In this paper we introduce a new method for the fast evaluation of the elementary functions in single precision based on the evaluation of truncated Taylor series using a difference method. We assume the availability of large and fast (at least for read purposes) memory. We call this method the ATA (Add-Table lookup-Add) method. As the name implies, the hardware required for the method are adders (both two/ and multi/operand adders) and fast tables. For IEEE single precision numbers our initial estimates indicate that we can calculate the basic elementary functions, namely reciprocal, square root, logarithm, exponential, trigonometric and inverse trigonometric functions, within the latency of two to four floating point multiplies. >
72 citations
TL;DR: In this article, the Landau-Lifschitz equation for a spin chain with an easy plane is solved by the method of the Darboux transformation matrix in terms of a particular parameter k.
Abstract: The Landau-Lifschitz equation for a spin chain with an easy plane is solved by the method of the Darboux transformation matrix. In terms of a particular parameter k, Jost solutions and Darboux matrices are generated in a recursive manner. The Jost solutions are shown to satisfy the corresponding Lax equations by a suitable choice of the constants involved in the Darboux matrices. A system of linear equations is derived and can yield the expressions for multi-soliton solutions. Asymptotic behaviour in the limits as t to +or- infinity is derived. An expression for the one-soliton solution is given in terms of elementary functions of x and t, as an example.
15 citations
08 Jun 1995
TL;DR: This APL*PLUS III implementation loops through one nested reference array and takes sub-arrays from another for a practical solution to this problem that can make tremendous demands on time and space.
Abstract: Automatic differentiation, manipulating numerical vectors of coefficients, is the efficient way to compute multivariable Taylor series. This does not require symbolic differentiation or numerical approximation but uses exact formulas applied to numerical arrays. Arrays of Taylor series coefficients of any elementary function can be built-up, as the array for each component (combination or function) is a combination of its argument arrays. The functions TIMES and EXP display the algorithmic ideas that enable all of the other standard functions. We study the interesting recursive formulas for these combinations, the resulting algorithms, and the implementation in APL. To handle all coefficients in n variables up to order m, the arrays are hyper-pyramid data structures, considered conceptually as n-dimensional but implemented as one-dimensional arrays. Unlike previous work, this implementation does not require huge arrays for binomial coefficients and indirect referencing. This APL*PLUS III implementation loops through one nested reference array and takes sub-arrays from another for a practical solution to this problem that can make tremendous demands on time and space.
13 citations
TL;DR: In this paper, Knoebel [K] showed that En = a? is convergent if and only if e~ < a < e/. This result has been rediscovered by many authors.
Abstract: (1.1) En = a? . We will call {an} a sequence of exponents and the sequence {En} an infinite exponential. As in the study of sums and products one would like to develop tests of convergence of an infinite exponential. Euler [E] was the first to give such a test. He showed that in the special case a\\ = α2 = a^ = = α, En is convergent if and only if e~ < a < e/. This result has been rediscovered by many authors. An extensive bibliography of papers containing this and related results may be found in the survey paper by Knoebel [K]. In the general case of non-constant exponents the best known results are due to Barrow [B]. He showed (although some of his arguments are rather sketchy) that {En} is convergent for e~ e < 0\"n < e , n > ΠQ. He also considered the cases an > e 1//e and aTM < e~. In the first case, writing an — e 1//e + en, with en > 0, he showed that {En} is convergent if
11 citations
TL;DR: This paper is the description and the representation of shape functions when the element has trianoular shape and has been done by using two algorithmic schemes: Neville-Aitken and De Casteljau.
TL;DR: In this paper, the authors use group theory to reduce the equations of motion of the CP 1 model in (2+1) dimensions to sets of two coupled ordinary differential equations, and decouple and solve many of these equations in terms of elementary functions, elliptic functions and Painlev{\'e} transcendents.
Abstract: We use the methods of group theory to reduce the equations of motion of the $CP^{1}$ model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary functions, elliptic functions and Painlev{\'e} transcendents. Some of the reduced equations do not have the Painlev{\'e} property thus indicating that the model is not integrable, while it still posesses many properties of integrable systems (such as stable ``numerical'' solitons).
TL;DR: In this article, several integral representations for the axially symmetric Helmholtz Green's function are considered, and it is demonstrated that contour deformation schemes lead to rapidly convergent integrals, particularly in cases where the integrands of the standard forms are highly oscillatory.
01 Jan 1995
TL;DR: In this article, the authors define an integral of the form ∫ R(x, √P(x))dx, in which R is a rational function of its arguments and P(x) is a third-or fourth-degree polynomial with distinct zeros.
Abstract: Publisher Summary
This chapter describes the elliptic integrals and functions. An elliptic integral is an integral of the form ∫ R(x, √P(x))dx, in which R is a rational function of its arguments and P(x) is a third- or fourth-degree polynomial with distinct zeros. Every elliptic integral can be reduced to a sum of integrals expressible in terms of algebraic, trigonometric, inverse trigonometric, logarithmic, and exponential functions (the elementary functions), together with one or more of the three special types of integral. The integrals are said to be expressed in the Legendre normal form. The number k is called the modulus of the elliptic integral, and the number k' = √1- k2 is called the complementary modulus of the elliptic integral. It is usual to set k = sin α, to call a the modular angle, and m = k2= sin2 α the parameter. The number n is called the characteristic parameter of the elliptic integral of the third kind.
01 Jan 1995
TL;DR: In this paper, the authors apply Lagrangian distributions to modelling claim frequency data in an insurance portfolio and use Maple to overcome the difficulty of expressing the generating functions of these distributions in terms of elementary functions.
Abstract: Applications of Lagrangian distributions to modelling claim frequency data in an insurance portfolio is a relatively new concept. The major difficulty is that the generating functions of these distributions cannot be expressed in terms of elementary functions. Also deriving moments and/or cumulants is somewhat tedious. This article illustrates how to use Maple effectively to overcome these difficulties.