GRAPHS of ELEMENTARY and SPECIAL FUNCTIONS

GRAPHS of ELEMENTARY and SPECIAL FUNCTIONS HANDBOOK N.O. Virchenko I.I. Lyashko BEGELL HOUSE, INC. PUBLISHERS New York

Part I. Plots of graphs with elementary methods Chapter 1. Basic notions of numbers, variables and functions 1.1. Numbers. Variables. Functions 3 1.1.1. Real numbers 3 1.1.2. Constants and variables 7 1.1.3. Concept of a function 7 1.1.4. Methods of defining a function 8 1.2. Classification of functions 13 1.2.1. Composition of functions 13 1.2.2. Inverse functions 14 1.2.3. Elementary functions 15 1.2.4. Single-valued and multivalued functions 17 1.2.5. Bounded and unbounded functions 17 1.2.6. Monotone functions 18 1.2.7. Even and odd functions 19 1.2.8. Periodic functions 20 1.3. Limit of a function. Continuity of a function 24 1.3.1. Limit of a number sequence 24 1.3.2. Limit of a function 24 1.3.3. Continuity of a function 28 Chapter 2. Analysis of functions for constructing graphs 2.1. Coordinate systems 31 2.1.1. Cartesian coordinate system 31 2.1.2. Polar coordinate system 32" 2.1.3. Transformations of the Cartesian coordinate system 34 2.2., Analysis of functions in the Cartesian coordinate system 37 2.2.1. Domain of a function 37 2.2.2. Range of a function. Graph of a bounded function 38 2.2.3. Even and odd functions. Properties of graphs 38 2.2.4. Types of symmetry "? -,._ 39 2.2.5. Graph of the inverse function 41 2:2.6. Periodicity of a function.42 2.2.7. Zeroes and signs of functions 45 2.2.8. Monotonicity of functions 46 2.2.9. Convexity of functions 47 2.2.10. Characteristic points of graph 50 III

Chapter 3. Graphs of principal elementary functions 3.1. Power function 57 3.2. Exponential function 75 3.3. Logarithmic function 75 3.4. Trigonometric functions 76 3.5. Inverse trigonometric functions 80 Chapter 4. Methods for plotting graphs by using arithmetic transformations. Transformations of graphs in the Cartesian coordinate system 4.1. Arithmetic transformations of graphs 85 4.1.1. Addition and subtraction of graphs 85 4.1.2. Multiplication and division of graphs 93 4.2. Plotting graphs by using simplest transformations 107 4.2.1. Transformations that do not change the scale 107 4.2.2. Transformations that change the scale 109 4.2.3. Plotting graphs of functions the analytic expression of which contains the absolute value sign 117 Chapter 5. Plotting of graphs of elementary functions 5.1. Plots of compositions of functions 144 5.1.1. Plotting the graphs of functions y = (/[x)) r, where r e Q. 148 5.1.2. Plotting the graph of the function >> = I/fix) from the graph of y =/x).. 150 5.1.3.Plottingtliegraphsofthefunction - v = a^x),wherea a-l >0 151 5.1.4. Plottingthe graphs of the function^ = \og a j{x), where a\a- l >0.... 152 5.1.5. Plotting the graphs of the functions y = sin/(x),y = cosffx) 155 5.1.6. Plotting the graphs of functions > = tan - /(x),y = cot^x) 171 5.1.7. Plotting the graphs of the functions y = sacsiaj[x),mccosj{x) 180 5.1.8. Plottingthe graphs of the functions >> = arctanxx),>> = arccot./(jf)... 182 5.2. Graphs of algebraic functions 191 5.2.1. Graphs of integer functions 191 5.2.2. Graphs of fractional rational functions 207 5.2.3. Graphs of irrational functions 214 5.3. Graphs of transcendental functions 228 5.3.1. Hyperbolic functions > 228 5.3.2. Inverse hyperbolic functions 235 Chapter 6. Plots of parametrically defined curves 6.1. A study of parametrically defined functions 241 6.2. The method of eliminating the parameter/ 244 6.3. Methods for finding points of intersection of the curve x = <p(t), y = \\i(t) and the straight line y = kx 244 IV

Chapter 7. Plots of graphs in polar coordinates 7.1. Functions in polar coordinates 259 7.2. Plots of graphs in polar coordinates 263 7.2.1. Plots of the curves p = a, p = asin cp, p = acos 9, p = acos (cp cp 0 )... 264 7.2.2. Plots of the graphs of curves p = asin mp, p = acos mp (n e N, n > 1).. 267 7.2.3. Plots of the graphs of curves p = a(l ± sin cp), p = a(l ± cos 9) 272 7.2.4. Plots of graphs of curves p = a(l ± sin mp), p = a(l ± cos mp) (n e N, n > 1) 274 7.2.5. Plots of the graphs of curves p = atancp,p = acotcp 281 7.2.6. Plots of the graphs of curves p = otan mp, p = acot mp (n e N, n > 1).. 283 7.2.7. Plots of the graphs of curves p = a(l ± tan 9), p = a(l ± cot 9) 284 7.2.8. Plots of the graphs of curves p = a(l ± tan mp), p = a(l ± cot mp) (n e N, n > 1). 284 7.2.9.Plotsofthegraphsofcurvesp = 6±asin9,p = b±acosq>(b*a)... 288 7.2.10. Plots of the graphs of curves p = b ± asin nq>, p = b± acos mp (/7eN,6*a,w>l) 288 7.2.11.Plots of the graphs of curves p = b ±atanmp, p = b ±acotmp (6*a,neN,wkl) 288 1 1 7.2.12. Plots of the graphs of curves p = a sin /19, p = acos mp(/j e N)... 306 7.2.13. Plots of the graphs of curves p = a sin 3 «9, p = a cos 3 n9 (/? e N).... 306 7.2.14. Plots of the graphs of curves p = a tan 2 «9, p = a cot 2 «9 («e N, n > 1). 306 7.2.15. Plots of curves p = otan 3 mp, p = aco^mp (n e N, n> 1) 306 Plots of curves p = a sin^ 9, p = a cos^ 9, p = atan^ 9, p = a cot^ 9, p ( ^ 9 ), p = a(l ±cos^9), p = a( a p = a(l ± cot ^ 9), p = a sin" ^ 9, p = a cos" 9, p = a tan" ^ 9, p = OCot" 9(p,a,«eN) 311 7.2.16. Plots of curves p = a/sin* mp, p = a/cos fc 779 («,& e N) 311 7.3. Transformations of graphs in polar coordinates 338 Chapter 8. Plots of implicitly defined functions 8.1. Implicitly defined functions 348 8.2. Plotting the graphs of implicitly defined functions 350 8.2.1. Plotting the graphs by transforming an implicitly defined function into an explicit function 350 8.2.2. Plotting the graphs by using the method of "linear relation" 351 8.2.3. Plotting of graphs by using the method of "cells" 353 8.3. Curves defined by second-degree algebraic equation 363

8.4. Curves defined by a third-degree polynomial 373 8.4.1. Hyperbolic hyperbola 373 8.4.2. Defective hyperbola 383 8.4.3. Parabolic hyperbola 391 8.4.4. Hyperbolisms of conic sections 394 8.4.5. Trident (Hyperbola parabolism) 400 8.4.6. Prolate parabola 400 8.4.7. Cubic parabola 403 8.5 Graphs of implicitly defined functions containing the absolute value sign... 403 8.6. Examples of implicitly defined functions for which it is convenient to plot the graphs in polar coordinates 412 Chapter 9. Plots of more complicated functions 9.1. Functions defined by several analytical expressions 414 9.2. Function^ = sign/joe),.*'=./lsign*) 414 9.3. Plots of functions defined by a recurrence relation 420 9.4. Plots of functions defined in terms of limits 424 9.5. Plots of functions y=lf(x)] 425 9.6. Plots of functions y=/(m) 430 9.7. Plots of functions.y = {/(*)} 430 9.8 Plots of functions y =/({*}) 442 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 VI 1.1. Derivativeof afunctionof one variable. Properties 447 1.2. Differential of a function of one variable 449 1.3. Fundamental theorems of calculus 450 1.4. Study of functions by using derivatives 454 1.4.1. Condition for constancy 454 1.4.2. Condition for monotonicity 454 1.4.3. Maxima and minima of a function 455 1.4.4. Determining extremums of a function by using the second derivative.. 457 1.4.5. Determining extremums of a function by using Taylor's formula... 457 1.4.6. The maximal and minimal value of a function on an interval 458 1.4.7. Convexity and concavity of a curve. Inflection points 458 1.5. Graph plotting with the use of derivatives 460 1.6. Plotting the graph of/' (x) and/" (x) from the graph off(x) 460 1.7. Graphical differentiation 462 1.8. Graphical integration 464 1.8.1. The rule of Bernoulli-L'Hospital 466 1.9. Approximations for solutions of equations 468

1.9.1. The chord method 468 1.9.2. The tangent method (Newton's method) 469 Chapter 2. Graphs of all types of functions 2.1. Graphs of functions y=f(x) in Cartesian coordinates 472 2.2. Plots of parametrically defined functions 489 2.2.1. A study of parametrically defined functions by using derivatives... 489 2.2.2. Graphs of parametrically defined functions 516 2.3. Graphs of implicitly defined functions 520 2.4. Graphs of functions in polar coordinates 527 Chapter 3. Graphs of certain remarkable curves 3.1. Algebraic curves 531 3.2. Curves of degree two 533 3.2.1. The circle 533 3.2.2. The ellipse 536 3.2.3. The hyperbola 538 3.2.4. The parabola > 541 3.3. Curves of degree three 543 3.3.1. Folium of Descartes 544 3.3.2. Cissoid of Diodes 547 3.3.3. Strophoid 550 3.3.4. Ophiurida ; 552 3.3.5. Trisectrix of Maclaurin 553 3.3.6. Cubic of Tschirnhausen 553 3.3.7. Versieraof Agnesi 555 3.3.8. Semicubic parabola (parabola of Neil)... 556 3.4. Curves of degree four 556 3.4.1. Conchoid 558 3.4.2. The limacon of Pascal 562 3.4.3. Cycloidal curves 563 3.5. Transcendental curves 596 3.5.1. Spiral of Archimedes 596 3.5.2. Algebraic spirals 598 3.5.3. Catenary : 604 3.5.4. Tractrix >-. 606 3.5.5. QuadratrixofDinostratus 608 3.5.6. Cochleoid 610 3.5.7. Exponential curves 610 3.5.8. Cycloid 612 3.5.9. Curves of Sturm 615 3.5.10. Involute of a circle 616 3.5.11. Pursuit curve 618 3.5.12. Curves of Ribaucour 629 VII

3.5.13. Cornu's spiral 620 3.5.14. Radial curves 622 Chapter 4. Graphs of special functions 4.1. Factorial function n\. Bernoulli numbers and polynomials. Euler numbers and polynomials. Binomial coefficients. Pochhammer polynomials 624 4.2. Gamma function and related functions 635 4.2.1. Gamma function T(x) 635 4.2.2. Beta function B(x,y) (Euler integral of the first kind) 637 4.2.3. Psi function (digamma function v /(x)) 638 4.2.4. The function G(x) (Bateman function) 642 4.2.5. Incomplete gamma functions 642 4.3. Integral exponential and related functions 645 4.4. Integral sine and cosine functions 650 4.5. Fresnel integrals 655 4.6. Probability integral and related functions 658 4.7. Bessel functions 663 4.8. Hypergeometric functions 692 4.9. Orthogonal polynomials 703 4.9.1. General properties of orthogonal polynomials 703 4.9.2. Jacobi polynomials/^a>p) (x) 704 4.9.3. Jacobi functions of the second kind 708 4.9.4. Gegenbauer polynomials (ultra-spherical polynomials) 709 4.9.5. Legendre polynomials (spherical polynomials) P n (x) 711 4.9.6. Laguerre polynomials L n (x) 714 4.9.7. Hermite polynomials H n (x) ' 718 4.9.8. Chebyshev polynomials 722 4.10. Legendre functions 726 4.11. Elliptic integrals and elliptic functions 731 4.12. Mathieu functions 744 4.13. Zeta numbers and related functions 750 4.14. Hurwitz function 753 4.15. Heaviside function. Dirac's delta function 756 Bibliography 760 Index 761 VIII