Showing papers on "Inverse trigonometric functions published in 2001"

Calculus Early Transcendentals

Abstract: Chapter 0 Before Calculus 0.1 Functions 0.2 New Functions from Old 0.4 Families of Functions 0.5 Inverse Functions Inverse Trigonometric Functions 0.6 Exponential and Logarithmic Functions Chapter 1 Limits and Continuity 1.1 Limits (An Intuitive Approach) 1.2 Computing Limits 1.3 Limits at Infinity End Behavior of a Function 1.4 Limits (Discussed More Rigorously) 1.5 Continuity 1.6 Continuity of Trigonometric, Exponential, and Inverse Functions Chapter 2 The Derivative 2.1 Tangent Lines and Rates of Change 2.2 The Derivative Function 2.3 Introduction to Techniques of Differentiation 2.4 The Product and Quotient Rules 2.5 Derivatives of Trigonometric Functions 2.6 The Chain Rule Chapter 3 Topics in Differentiation 3.1 Implicit Differentiation 3.2 Derivatives of Logarithmic Functions 3.3 Derivatives of Exponential and Inverse Trigonometric Functions 3.4 Related Rates 3.5 Local Linear Approximation Differentials 3.6 L'Hopital's Rule Indeterminate Forms Chapter 4 The Derivative in Graphing and Applications 4.1 Analysis of Functions I: Increase, Decrease, and Concavity 4.2 Analysis of Functions II: Relative Extrema Graphing Polynomials 4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 4.4 Absolute Maxima and Minima 4.5 Applied Maximum and Minimum Problems 4.6 Rectilinear Motion 4.7 Newton's Method 4.8 Rolle's Theorem Mean-Value Theorem Chapter 5 Integration 5.1 An Overview of the Area Problem 5.2 The Indefinite Integral 5.3 Integration by Substitution 5.4 The Definition of Area as a Limit Sigma Notation 5.5 The Definite Integral 5.6 The Fundamental Theorem of Calculus 5.7 Rectilinear Motion Revisited Using Integration 5.8 Average Value of a Function and its Applications 5.9 Evaluating Definite Integrals by Substitution 5.10 Logarithmic and Other Functions Defined by Integrals Chapter 6 Applications of the Definite Integral in Geometry, Science, and Engineering 6.1 Area Between Two Curves 6.2 Volumes by Slicing Disks and Washers 6.3 Volumes by Cylindrical Shells 6.4 Length of a Plane Curve 6.5 Area of a Surface of Revolution 6.6 Work 6.7 Moments, Centers of Gravity, and Centroids 6.8 Fluid Pressure and Force 6.9 Hyperbolic Functions and Hanging Cables Ch 7 Principles of Integral Evaluation 7.1 An Overview of Integration Methods 7.2 Integration by Parts 7.3 Integrating Trigonometric Functions 7.4 Trigonometric Substitutions 7.5 Integrating Rational Functions by Partial Fractions 7.6 Using Computer Algebra Systems and Tables of Integrals 7.7 Numerical Integration Simpson's Rule 7.8 Improper Integrals Ch 8 Mathematical Modeling with Differential Equations 8.1 Modeling with Differential Equations 8,2 Separation of Variables 8.3 Slope Fields Euler's Method 8.4 First-Order Differential Equations and Applications Ch 9 Infinite Series 9.1 Sequences 9.2 Monotone Sequences 9.3 Infinite Series 9.4 Convergence Tests 9.5 The Comparison, Ratio, and Root Tests 9.6 Alternating Series Absolute and Conditional Convergence 9.7 Maclaurin and Taylor Polynomials 9.8 Maclaurin and Taylor Series Power Series 9.9 Convergence of Taylor Series 9.10 Differentiating and Integrating Power Series Modeling with Taylor Series Ch 10 Parametric and Polar Curves Conic Sections 10.1 Parametric Equations Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length, and Area for Polar Curves 10.4 Conic Sections 10.5 Rotation of Axes Second-Degree Equations 10.6 Conic Sections in Polar Coordinates Ch 11 Three-Dimensional Space Vectors 11.1 Rectangular Coordinates in 3-Space Spheres Cylindrical Surfaces 11.2 Vectors 11.3 Dot Product Projections 11.4 Cross Product 11.5 Parametric Equations of Lines 11.6 Planes in 3-Space 11.7 Quadric Surfaces 11.8 Cylindrical and Spherical Coordinates Ch 12 Vector-Valued Functions 12.1 Introduction to Vector-Valued Functions 12.2 Calculus of Vector-Valued Functions 12.3 Change of Parameter Arc Length 12.4 Unit Tangent, Normal, and Binormal Vectors 12.5 Curvature 12.6 Motion Along a Curve 12.7 Kepler's Laws of Planetary Motion Ch 13 Partial Derivatives 13.1 Functions of Two or More Variables 13.2 Limits and Continuity 13.3 Partial Derivatives 13.4 Differentiability, Differentials, and Local Linearity 13.5 The Chain Rule 13.6 Directional Derivatives and Gradients 13.7 Tangent Planes and Normal Vectors 13.8 Maxima and Minima of Functions of Two Variables 13.9 Lagrange Multipliers Ch 14 Multiple Integrals 14.1 Double Integrals 14.2 Double Integrals over Nonrectangular Regions 14.3 Double Integrals in Polar Coordinates 14.4 Surface Area Parametric Surfaces} 14.5 Triple Integrals 14.6 Triple Integrals in Cylindrical and Spherical Coordinates 14.7 Change of Variable in Multiple Integrals Jacobians 14.8 Centers of Gravity Using Multiple Integrals Ch 15 Topics in Vector Calculus 15.1 Vector Fields 15.2 Line Integrals 15.3 Independence of Path Conservative Vector Fields 15.4 Green's Theorem 15.5 Surface Integrals 15.6 Applications of Surface Integrals Flux 15.7 The Divergence Theorem 15.8 Stokes' Theorem Appendix [order of sections TBD] A Graphing Functions Using Calculators and Computer Algebra Systems B Trigonometry Review C Solving Polynomial Equations D Mathematical Models E Selected Proofs Web Appendices F Real Numbers, Intervals, and Inequalities G Absolute Value H Coordinate Planes, Lines, and Linear Functions I Distance, Circles, and Quadratic Functions J Second-Order Linear Homogeneous Differential Equations The Vibrating String K The Discriminant ANSWERS PHOTOCREDITS INDEX

Two-dimensional phase unwrapping by quad-tree decomposition.

Abstract: One problem to be tackled when interferometric phase-shifting techniques are used is the method in which the phase can be reconstructed. Because an inverse trigonometric function appears in the formulation, the final data are not the phase, but the phase modulo 2pi. A new phase-unwrapping algorithm based on a two-step procedure is presented. In the first step, the digital image to be analyzed is divided into continuous patches by a quad-tree-like recursive procedure; in the second step, the same level patches are joined together by an error-norm-minimizing approach to obtain larger, almost continuous ones. The basic idea of the procedure is to simplify the problem by factoring the complete image into square, variable-size, homogeneous areas (i.e., regions with no internal phase jump) so that only interfaces need to be dealt with. By hierarchically recombining the so-obtained subimages, an unwrapped phase field can be obtained. After a complete description of the algorithm, some examples of its use on synthesized digital images are illustrated. As the algorithm can be used with and without quality masks and the error-minimizing step can use different norms, a full class of unwrapping algorithms can be implemented by this approach.

Non-Uniform Sampling in Higher Dimensions: From Trigonometric Polynomials to Bandlimited Functions

Abstract: The chapter discusses the relation between finite-dimensional models of non-uniform sampling as they are used for numerical algorithms and the infinite-dimensional theory of non-uniform sampling of bandlimited functions. It is shown that interpolation and approximation by trigonometric polynomials provides a correct finite-dimensional discretization of the sampling problem for bandlimited functions. The results hold in arbitrary dimension and in particular they validate some recent fast reconstruction methods in dimension two.

Volume of the Intersection of Three Spheres

Abstract: The volume of the intersection of three spheres is represented as a continuous piecewise analytic combination of algebraic and inverse trigonometric functions of the radii and the distances between the centers of the spheres.

Monogenic multiperiodic functions in Clifford analysis

Abstract: In this paper we deal with generalizations of the classical Eisenstein series in the frame-work of Clifford analysis. They provide monogenic generalizations of the classical tangent, cotangent, secant and cosecant function and the elliptic functions to higher dimensions. We study properties of them and characterize them by certain duplication formulas and their principal parts.

Joint cosine transforming and quantizing device, and joint inverse quantizing and inverse cosine transforming device

Abstract: Each of the discrete cosine transforming (DCT) unit of a joint cosine transforming and quantizing device, and the inverse discrete cosine transforming (IDCT) unit of a joint inverse quantizing and inverse cosine transforming device includes a multiplication operation unit that is implemented in a look-up table device having a look-up table and an output multiplexer. The look-up table has a number of data fields, each of which contains products associated with a respective one of a corresponding number of coefficients used in multiplication-involving stages of a DCT or IDCT fast algorithm. The look-up table further has a number of outputs corresponding respectively to the data fields, and can be addressed in order to output the products, that correspond to the address data and that are stored in the data fields, at the outputs of the look-up table. The output multiplexer has data inputs connected to the outputs of the look-up table, and is operable so as to select one of the data inputs thereof and provide data at the selected one of the data inputs thereof to a data output of the output multiplexer.

Method and apparatus for reproducing timing, and a demodulating apparatus that uses the method and apparatus for reproducing timing

Abstract: A base band signal is sampled, based on an asynchronous sampling clocking, to output two sampled data series. A transmission side complex symbol frequency generator generates a complex transmission symbol frequency component combining an envelope value and an amplitude change amount value, which are each calculated from the two sampled data series. The transmission side complex symbol frequency component is provided to a correlation value calculator that outputs a correlation data series correlated to the complex symbol frequency and the symbol frequency generated by a cosine wave generator. The correlation data series is converted a timing error by an inverse tangent calculator. By the above configuration, a timing reproducing unit, which includes the above components, determines the timing error at a sampling speed of twice the symbol rate.

BASIC EXPONENTIAL FUNCTIONS ON A q-QUADRATIC GRID

Abstract: A review of basic exponential functions, basic trigonometric functions, and basic Fourier series on a q-quadratic grid is given.

Method and apparatus for symbol timing recovery for phase modulated signals

Abstract: Two A/D converters (3a, 3b) sample received base band signal Sb by asynchronous sampling clock at double speed of symbol rate, and based on the sampled data series I i , Q i . A transmission complex symbol frequency generator (5) generates data series E i , D i of transmission complex symbol frequency component. A correlation value calculator (10) outputs correlation data series SM i as the correlation value of cosine wave data series C i , and data series E i , D i of symbol frequency generated by a cosine wave generator (9), based on the asynchronous sampling clock CK. An inverse tangent calculator (11) outputs timing error τ based on the correlation data series SM i .

Inverse tangent arithmetic circuit in digital signal processing

Abstract: PROBLEM TO BE SOLVED: To provide an inverse tangent arithmetic circuit capable of reducing the arithmetic processing amount and decreasing the capacity of the ROM table. SOLUTION: A phase range reduction part 31 obtains the magnitudes |X|, |Y| of input signals X, Y, shifts bits of the magnitudes |X|, |Y| which is grater so as to obtain a maximum value within an input dynamic range of an inverse tangent function ROM 32 and shifts bits of the magnitudes |X|, |Y| which is smaller by the same amount of bit shift as that of the greater magnitude so as to normalize the result. Thus, the arithmetic amount of the phase range reduction part 31 is reduced and the capacity of an arithmetic table of the inverse tangent function ROM 32 is decreased. COPYRIGHT: (C)2003,JPO

A characterization of Fourier series of Stepanov almost-periodic functions

Abstract: We show by example that the classical characterization of the Fourier series of periodic functions in Lp, 1

Series over the product of bessel and trigonometric functions

Abstract: This paper is a contribution to the mathematics of Bessel series which is used extensively in engineering. The sums of series (1) and (2) are found by considering series involving the product of two trigonometric functions. The obtained sums are in terms of Riemann zeta and related functions.

Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions

Abstract: The sum of the series Sα = Sα s, a, b, f(y), g(x) = ∞ X n=1 (s)n−1f (an− b)y g (an− b)x (an− b)α involving the product of two trigonometric functions is obtained using the sum of the series ∞ X n=1 (s)(n−1)f((an− b)x) (an− b)α = cπ 2Γ(α)f( 2 ) xα−1 + ∞ X i=0 (−1) F (α− 2i− δ) (2i + δ)! x whose terms involve one trigonometric function. The first series is represented as series in terms of the Riemann zeta and related functions, which has a closed form in certain cases. Some applications of these results to the summation of series containing Bessel functions are given. The obtained results also include as special cases formulas in some known books. We further show how to make use of these results to obtain closed form solutions of some boundary value problems in mathematical physics.

Discovering integrals with geometry

Abstract: In this paper, students and teachers are provided with problems that lead them in finding the integrals of logarithmic and inverse trigonometric functions early in the calculus sequence by using the Fundamental Theorem of Calculus and the concept of area, and without the use of integration by parts. The methods link geometric and symbolic representations, and allow students to visually interpret these concepts.

Electronic circuit for generation of inverse trigonometric signal, for use in phase-locked loops and phase detectors

Abstract: The electronic circuit for the generation of an output analogue signal corresponding to at least one mathematical operation on the basis of at least two input analogue signals, comprises a set of sub-circuits which can deliver a signal corresponding to the square-root of product of input signals, that is of the form 2 square root (I1I2), and a circuit for delivering a signal corresponding to the raising to the power of 2 of input signal, that is of the form K.Vin . The electronic circuit comprises a generator with an input receiving an analogue signal x(t), and an output delivering a signal f(x), which is given analytically. The electronic circuit comprises a differential pair, and also means for division, multiplication and summation. A phase detector circuit comprises a generator of analogue signal of inverse trigonometric type. A frequency synthesizer with a phase-locked loop (PLL), utilized in radio-telephones, comprises a circuit for the generation of a signal as a function of phase-shift, which incorporates specified circuits. The method for the generation of analogue signal as a function of phase-shift between two analogue signals is by use of specified circuit to obtain the analogue signal f(x) given analytically and corresponding to inverse sine and cosine, and also another analytical form corresponding to inverse tangent and cotangent. The maximum error with respect to exact values of inverse sine is 0.22%. An application of electronic circuit is for measuring the phase-shift of impedance and the in-circuit test of application of electronic circuit is for measuring the phase-shift of impedance and the in-circuit test of PCB. The equipment for the generation of output signal corresponding to the application of one of specified mathematical operations comprises an electronic circuit of proposed type. The phase detector specified mathematical operations comprises an electronic circuit of proposed type. The phase detector circuit is used for a direct evaluation of phase error without filter, and comprises a tracker-holder.