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A Sequence of Series for The Lambert Function
, 1997
"... We give a uniform treatment of several series expansions for the Lambert W function, leading to an infinite family of new series. We also discuss standardization, complex branches, a family of arbitraryorder iterative methods for computation of W , and give a theorem showing how to correctly solve ..."
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Cited by 20 (4 self)
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We give a uniform treatment of several series expansions for the Lambert W function, leading to an infinite family of new series. We also discuss standardization, complex branches, a family of arbitraryorder iterative methods for computation of W , and give a theorem showing how to correctly solve another simple and frequently occurring nonlinear equation in terms of W and the unwinding number. 1 Introduction Investigations of the properties of the Lambert W function are good examples of nontrivial interactions between computer algebra, mathematics, and applications. To begin with, the standardization of the name W by computer algebra (see section 1.2 below) has had several effects. First, this standardization has exposed a great variety of applications; second, it has uncovered a significant history, hitherto unnoticed because the lack of a standard name meant that most researchers were unaware of previous work; and, third, it has now stimulated current interest in this remarkable ...
Emerging Tools for Experimental Mathematics
 American Mathematical Monthly
, 1999
"... This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to ..."
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Cited by 17 (9 self)
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This paper is not a tutorial on how lattice basis reduction algorithms such as LLL or PSLQ actually work; rather, we discuss some of the ways these tools can be used to generate conjectures, and for that, a detailed understanding of the underlying algorithms is not necessary. We do hope, however, to convey some appreciation of their power. We begin with some warmup examples, using the Inverse Symbolic Calculator (ISC); http:// www.cecm.sfu.ca/ MRG/ INTERFACES.html. The basic idea is simple: given the first few decimal digits of some real number, we want the ISC to guess a formula for what it `really' is. For example, if we input K 1 = 3:14626436994198, and click on simple lookup
Reasoning About the Elementary Functions of Complex Analysis
, 2001
"... There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make "howlers" or not to simplify enough. In this paper we outline the "unwinding number" approach to such problems, and show how it can be used to prevent errors and to syst ..."
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Cited by 17 (10 self)
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There are many problems with the simplification of elementary functions, particularly over the complex plane. Systems tend to make "howlers" or not to simplify enough. In this paper we outline the "unwinding number" approach to such problems, and show how it can be used to prevent errors and to systematise such simplification, even though we have not yet reduced the simplifiation process to a complete algorithm. The unsolved problems are probably more amenable to the techniques of artificial intelligence and theorem proving than the original problem of complexvariable analysis.
Some Applications of the Lambert W Function to Physics
"... : Two standard physics problems are solved in terms of the Lambert W function, in order to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien's displacement law and the fringing fie ..."
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Cited by 4 (0 self)
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: Two standard physics problems are solved in terms of the Lambert W function, in order to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien's displacement law and the fringing fields of a capacitor, the latter problem being representative of some problems solved using conformal transformations. The physical content of the solutions remains unchanged, but they gain a new elegance and convenience. 1. Introduction Many physicists have experienced, during their education, the surprise of seeing a known mathematical function appear in a new physical context. An example in elementary physics is one that arises when students are first taught about simple harmonic motion. We hope that some readers can remember their amazement on learning that the motion of objects bobbing on springs, or moving in circles, can be described using trigonometric functions. At the time, they m...
Graphing Elementary Riemann Surfaces
 SIGSAM Bulletin
, 1998
"... This paper discusses one of the prettiest pieces of elementary mathematics or computer algebra, that we have ever had the pleasure to learn. The tricks that we discuss here are certainly \wellknown" (that is, in the literature) , but we didn't know them until recently, and none of our immediate col ..."
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Cited by 3 (2 self)
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This paper discusses one of the prettiest pieces of elementary mathematics or computer algebra, that we have ever had the pleasure to learn. The tricks that we discuss here are certainly \wellknown" (that is, in the literature) , but we didn't know them until recently, and none of our immediate colleagues knew them either. Therefore we believe that it is useful to publicize them further. We hope that you nd these ideas as pleasant and useful as we do. We show how to use a computer algebra system (or even a purely numerical graphing package) to graph the Riemann surfaces of various elementary functions. We rst noticed the technique in Cleve Moler's Matlab programs cplxroot, cplxgrid, and cplxmap, which are part of the Matlab 5.1 Demo package (in plots of complex functions). The command type cplxroot in Matlab
A short survey of automated reasoning
"... Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so f ..."
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Abstract. This paper surveys the field of automated reasoning, giving some historical background and outlining a few of the main current research themes. We particularly emphasize the points of contact and the contrasts with computer algebra. We finish with a discussion of the main applications so far. 1 Historical introduction The idea of reducing reasoning to mechanical calculation is an old dream [75]. Hobbes [55] made explicit the analogy in the slogan ‘Reason [...] is nothing but Reckoning’. This parallel was developed by Leibniz, who envisaged a ‘characteristica universalis’ (universal language) and a ‘calculus ratiocinator ’ (calculus of reasoning). His idea was that disputes of all kinds, not merely mathematical ones, could be settled if the parties translated their dispute into the characteristica and then simply calculated. Leibniz even made some steps towards realizing this lofty goal, but his work was largely forgotten. The characteristica universalis The dream of a truly universal language in Leibniz’s sense remains unrealized and probably unrealizable. But over the last few centuries a language that is at least adequate for
Not seeing the roots for the branches: multivalued functions in computer algebra
 SIGSAM Bulletin vol
"... We discuss the multiple definitions of multivalued functions and their suitability for computer algebra systems. We focus the discussion by taking one specific problem and considering how it is solved using different definitions. Our example problem is the classical one of calculating the roots of a ..."
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We discuss the multiple definitions of multivalued functions and their suitability for computer algebra systems. We focus the discussion by taking one specific problem and considering how it is solved using different definitions. Our example problem is the classical one of calculating the roots of a cubic polynomial from the Cardano formulae, which contain fractional powers. We show that some definitions of these functions result in formulae that are correct only in the sense that they give candidates for solutions; these candidates must then be tested. Formulae that are based on singlevalued functions, in contrast, are efficient and direct. 1
The Difficulties of Definite Integration
"... Indefinite integration is the inverse operation to differentiation, and, before we can understand what we mean by indefinite integration, we need to understand what we mean by differentiation. 1.1 What is differentiation? 1. An analytic operation: f ′ (x0) = limx→x0 f(x)−f(x0) x−x0 2. An algebraic ..."
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Indefinite integration is the inverse operation to differentiation, and, before we can understand what we mean by indefinite integration, we need to understand what we mean by differentiation. 1.1 What is differentiation? 1. An analytic operation: f ′ (x0) = limx→x0 f(x)−f(x0) x−x0 2. An algebraic operation, satisfying (a + b) ′ = a ′ + b ′ , (ab) ′ = a ′ b + b ′ a, x ′ = 1. We note the different ways in which the fact that we mean “differentiation with respect to x ” is expressed in the two formulations. 1.1.1 Two interpretations of atan Analytically, atan(x) = y: y = tan(x) and −π/2 < y < π/2. Algebraically, atan(f) ′ ′ f = 1+f 2. Therefore only defined “up to a constant”. Analytically, c is a constant iff c(x1) = c(x2)∀x1, x2, and, in this view, the Heaviside function is not a constant. Algebraically, c is a constant iff c ′ = 0, and, in this view, the Heaviside function is a constant. ∗ The author is grateful to many people for their discussions of definite integration, notably Andrews Adams, Jacques Carrette, Tony Hearn and Daniel Lichtblau. 1 1.1.2 Comparing the two approaches • The success of computer algebra is that one can model the first by the second. • The problem of computer algebra is that the model, particularly when it comes to inverse differentiation and the handling of “up to a constant”, is not perfect. The first deals with functions (say R ↦ → R), the second with algebraic expressions.