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Graphs of Elementary and Special Functions Handbook

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Graphs of Elementary and Special Functions Handbook

N. O. Virchenko
Ukrainian National Technical University “Kiev Polytechnic Institute”, Kiev, Ukraine

Ivan I. Lyashko
T. Shevchenko National University, Kiev, Ukraine


This reference book gives essentials on functions and methods of plotting their graphs. Special attention is paid to methods used for plotting graphs of functions that are defined implicitly and in a parametric form, methods for making plots in polar coordinates, and plotting graphs of remarkable curves, special functions, etc.
The book consists of two parts. The first part gives general information about the notion of a function and methods of plotting graphs of functions without the use of derivatives. The second part deals with methods of studying functions and plotting their graphs with the use of calculus. The book contains many examples of functions that are more complicated than commonly considered, graphs of important curves, and properties, and graphs of common special functions (gamma function, integral exponential functions, Fresnel integrals, Bessel functions, orthogonal polynomials, elliptic functions, Mathieu functions, etc.).
The book contains 760 figures and uses material found in existing references, textbooks, articles, and other sources on the subject.

766 pages, © 2001

Table of contents:

Part I. Plots of graphs with elementary methods
Chapter 1. Basic notions of numbers, variables and functions
1.1. Numbers. Variables. Functions
1.1.1. Real numbers
1.1.2. Constants and variables
1.1.3. Concept of a function
1.1.4. Methods of defining a function
1.2. Classification of functions
1.2.1. Composition of functions
1.2.2. Inverse functions
1.2.3. Elementaiy functions
1.2.4. Single-valued and multivalued functions
1.2.5. Bounded and unbounded functions
1.2.6. Monotone functions
1.2.7. Even and odd functions
1.2.8. Periodic functions
1.3. Limit of a function. Continuity of a function
1.3.1. Limit of a number sequence
1.3.2. Limit of a function
1.3.3. Continuity of a function
Chapter 2. Analysis of functions for constructing graphs
2.1. Coordinate systems
2.1.1. Cartesian coordinate system
2.1.2. Polar coordinate system
2.1.3. Transformations of the Cartesian coordinate system
2.2. Analysis of functions in the Cartesian coordinate system
2.2.1. Domain of a function
2.2.2. Range of a function. Graph of a bounded function
2.2.3. Even and odd functions. Properties of graphs
2.2.4. Types of symmetry
2.2.5. Graph of the inverse function
2.2.6. Periodicity of a function
2.2.7. Zeroes and signs of functions
2.2.8. Monotonicity of functions
2.2.9. Convexity of functions
2.2.10. Characteristic points of graph
Chapter 3. Graphs of principal elementary functions
3.1. Power function
3.2. Exponential function
3.3. Logarithmic function
3.4. Trigonometric functions
3.5. Inverse trigonometric functions
Chapter 4. Methods for plotting graphs by using arithmetic transformations. Transformations of graphs in the Cartesian coordinate system
4.1. Arithmetic transformations of graphs
4.1.1. Addition and subtraction of graphs
4.1.2. Multiplication and division of graphs
4.2. Plotting graphs by using simplest transformations
4.2.1. Transformations that do not change the scale
4.2.2. Transformations that change the scale
4.2.3. Plotting graphs of functions the analytic expression of which contains the absolute value sign
Chapter 5. Plotting of graphs of elementary functions
5.1. Plots of compositions of functions
5.1.1. Plotting the graphs of functions у = (f(x))r, where r ∈ Q
5.1.2. Plotting the graph of the function у = 1/f(x) from the graph of у = f(x)
5.1.3. Plotting the graphs of the function у = df(x) where a|a − 1| > 0
5.1.4. Plotting the graphs of the function у = logaf(x), where a|a − 1| > 0
5.1.5. Plotting the graphs of the functions у = sinf(x), у = cosf(x)
5.1.6. Plotting the graphs of functions у = tanf(x), у = cotf(x)
5.1.7. Plotting the graphs of the functions у = arcsinf(x), arccosf(x)
5.1.8. Plotting the graphs of the functions у = arctanf(x), у = arccotf(x)
5.2. Graphs of algebraic functions
5.2.1. Graphs of integer functions
5.2.2. Graphs of fractional rational functions
5.2.3. Graphs of irrational functions
5.3. Graphs of transcendental functions
5.3.1. Hyperbolic functions
5.3.2. Inverse hyperbolic functions
Chapter 6. Plots of parametrically defined curves
6.1. A study of parametrically defined functions
6.2. The method of eliminating the parameter t
6.3. Methods for finding points of intersection of the curve x = φ(t), у = ψt) and the straight line у = kx
Chapter 7. Plots of graphs in polar coordinates
7.1. Functions in polar coordinates
7.2. Plots of graphs in polar coordinates
7.2.1. Plots of the curves ρ = a, ρ = asin φ, ρ = acos φ, ρ = acos (φ − φ0)
7.2.2. Plots of the graphs of curves ρ = asin nφ, ρ = acos nφ (n ∈ N, n > 1)
7.2.3. Plots of the graphs of curves ρ = a(l ± sin φ), ρ = a(1 ± cos φ)
7.2.4. Plots of graphs of curves ρ = a(l ± sin nφ), ρ = a(1 ± cos nφ) (n ∈ N, n > 1)
7.2.5. Plots of the graphs of curves ρ = atan φ, ρ = acot φ
7.2.6. Plots of the graphs of curves ρ = atan nφ, ρ = acot nφ (n ∈ N, n > 1)
7.2.7. Plots of the graphs of curves ρ = a(l ± tan φ), ρ = a(l ± cot φ)
7.2.8. Plots of the graphs of curves ρ = a(l ± tan nφ), ρ = a(1 ± cot nφ) (n ∈ N, n > 1)
7.2.9. Plots of the graphs of curves ρ = b ± asin φ, ρ = b ± acos φ(b≠a)
7.2.10. Plots of the graphs of curves ρ = b ± asin nφ, ρ = b ± acos nφ (n ∈ N, b ≠ a, n > 1)
7.2.ll. Plots of the graphs of curves ρ = b ± a tan nφ, ρ = b ± a cot nφ (b ≠ a, n ∈ N, n > 1)
7.2.12. Plots of the graphs of curves ρ = a sin2nφ, ρ = a cos2nφ(n ∈ N)
7.2.13. Plots of the graphs of curves ρ = asin3nφ, ρ = acos3nφ (n ∈ N)
7.2.14. Plots of the graphs of curves ρ = atan2nφ, ρ = acot2nφ (n ∈ N, n > 1)
7.2.15. Plots of curves ρ = atan3nφ, ρ = acot3nφ (n ∈ N, n > 1)
Plots of curves ρ = asin р/qφ, ρ = acosр/qφ, ρ = atan р/qφ, ρ = acot р/qφ,ρ = a(l ± sin р/qφ), ρ = a(l ± cos р/qφ), ρ = a(l ± tan р/qφ),ρ = a(l ± cot р/qφ), ρ = asinnр/qφ, ρ = acosnр/qφ, ρ = atannр/qφ,ρ = acotnр/qφ (р, q, n ∈ N)
7.2.16. Plots of curves ρ = a/sinknφ, ρ = a/cosknφ (n,k ∈ N)
7.3. Transformations of graphs in polar coordinates
Chapter 8. Plots of implicitly defined functions
8.1. Implicitly defined functions
8.2. Plotting the graphs of implicitly defined functions
8.2.1. Plotting the graphs by transforming an implicitly defined function into an explicit function
8.2.2. Plotting the graphs by using the method of “linear relation”
8.2.3. Plotting of graphs by using the method of “cells”
8.3. Curves defined by second-degree algebraic equation
8.4. Curves defined by a third-degree polynomial
8.4.1. Hyperbolic hyperbola
8.4.2. Defective hyperbola
8.4.3. Parabolic hyperbola
8.4.4. Hyperbolisms of conic sections
8.4.5. Trident (Hyperbola parabolism)
8.4.6. Prolate parabola
8.4.7. Cubic parabola
8.5 Graphs of implicitly defined functions containing the absolute value sign
8.6. Examples of implicitly defined functions for which it is convenient to plot the graphs in polar coordinates
Chapter 9. Plots of more complicated functions
9.1. Functions defined by several analytical expressions
9.2. Function y = signf(x), y = f(signx)
9.3. Plots of functions defined by a recurrence relation
9.4. Plots of functions defined in terms of limits
9.5. Plots of functions y = [f(x)]
9.6. Plots of functions y = f([x])
9.7. Plots of functions y = {f(x)}
9.8. Plots of functions y = f({x})
Part II. Plots of graphs with the use of calculus. Graphs of special functions
Chapter 1. Derivatives and differentials. Their applications to plotting of graphs
1.1. Derivative of a function of one variable. Properties
1.2. Differential of a function of one variable
1.3. Fundamental theorems of calculus
1.4. Study of functions by using derivatives
1.4.1. Condition for constancy
1.4.2. Condition for monotonicity
1.4.3. Maxima and minima of a function
1.4.4. Determining extremums of a function by using the second derivative
1.4.5. Determining extremums of a function by using Taylor’s formula
1.4.6. The maximal and minimal value of a function on an interval
1.4.7. Convexity and concavity of a curve. Inflection points
1.5. Graph plotting with the use of derivatives
1.6. Plotting the graph of f'(x) and f"(x) from the graph of f(x)
1.7. Graphical differentiation
1.8. Graphical integration
1.8.1. The rule of Bemoulli-L’Hospital
1.9. Approximations for solutions of equations
1.9.1. The chord method
1.9.2. The tangent method (Newton’s method)
Chapter 2. Graphs of all types of functions
2.1. Graphs of functions у = f(x) in Cartesian coordinates
2.2. Plots of parametrically defined functions
2.2.1. A study of parametrically defined functions by using derivatives
2.2.2. Graphs of parametrically defined functions
2.3. Graphs of implicitly defined functions
2.4. Graphs of functions in polar coordinates
Chapter 3. Graphs of certain remarkable curves
3.1. Algebraic curves
3.2. Curves of degree two
3.2.1. The circle
3.2.2. The ellipse
3.2.3. The hyperbola
3.2.4. The parabola
3.3. Curves of degree three
3.3.1. Folium of Descartes
3.3.2. Cissoid of Diodes
3.3.3. Strophoid
3.3.4. Ophiurida
3.3.5. Trisectrix of Maclaurin
3.3.6. Cubic of Tschimhausen
3.3.7. Versiera of Agnesi
3.3.8. Semicubic parabola (parabola of Neil)
3.4. Curves of degree four
3.4.1. Conchoid
3.4.2. The limacon of Pascal
3.4.3. Cycloidal curves
3.5. Transcendental curves
3.5.1. Spiral of Archimedes
3.5.2. Algebraic spirals
3.5.3. Catenary
3.5.4. Tractrix
3.5.5. Quadratrix of Dinostratus
3.5.6. Cochleoid
3.5.7. Exponential curves
3.5.8. Cycloid
3.5.9. Curves of Sturm
3.5.10. Involute of a circle
3.5.11. Pursuit curve
3.5.12. Curves of Ribaucour
3.5.13. Cornu’s spiral
3.5.14. Radial curves
Chapter 4. Graphs of special functions
4.1. Factorial function n!. Bernoulli numbers and polynomials. Euler numbers and polynomials. Binomial coefficients. Pochhammer polynomials
4.2. Gamma function and related functions
4.2.1. Gamma function Γ(x)
4.2.2. Beta function B(x, у) (Euler integral of the first kind)
4.2.3. Psi function (digamma function ψ(x))
4.2.4. The function G(x) (Bateman function)
4.2.5. Incomplete gamma functions
4.3. Integral exponential and related functions
4.4. Integral sine and cosine functions
4.5. Fresnel integrals
4.6. Probability integral and related functions
4.7. Bessel functions
4.8. Hypergeometric functions
4.9. Orthogonal polynomials
4.9.1. General properties of orthogonal polynomials
4.9.2. Jacobi polynomials Pn(α,β)(x)
4.9.3. Jacobi functions of the second kind
4.9.4. Gegenbauer polynomials (ultra-spherical polynomials)
4.9.5. Legendre polynomials (spherical polynomials) Pn(x)
4.9.6. Laguerre polynomials Ln(x)
4.9.7. Hermite polynomials Hn(x)
4.9.8. Chebyshev polynomials
4.10. Legendre functions
4.11. Elliptic integrals and elliptic functions
4.12. Mathieu functions
4.13. Zeta numbers and related functions
4.14. Hurwitz function
4.15. Heaviside function. Dirac’s delta function