Index

Abelian group, 379addition formulae for hyperbolic functions,

56agreement between functions, 35Aitken’s δ2 process, 143algebraic properties of a group, 380angle

between a line and a plane, 253between two lines, 240between two planes, 255

area formula in polar coordinates, 105area of a triangle using vector product, 283argument of a complex number, 294associative, 372associative axiom, of a group, 376asymptote(s), 77auxiliary equation, 268axioms, of a group, 376

binary operation, 370associative, 372closed, 370combination table for, 373commutative, 371

binomial expansion, 42, 348bounds, 121

for sums of series, 125for the value of an integral, 121

cancellation law for a group, 378cartesian coordinates

converting to and from polar coordinates,95

converting to and from polar equations, 102cartesian equations of a line, 240cartesian equations of a plane, 250closed binary operation, 370closure axiom, of a group, 376cobweb diagram, 138combination table for a binary operation, 373commutative, 371

complement rulesfor inverse trigonometric functions, 15

complementary function, 265alternative form, 362for first order equation with constant

coefficients, 267for general second order equation, 271for second order equation with constant

coefficients, 268, 270, 360complex number(s), 294

and vectors, 305applied to trigonometric series, 349argument, 294binomial expansion of, 348exponential form of, 302exponential function of, 303exponential series, 349geometric representation of multiplication,

307, 311infinite geometric series, 347modulus, 294modulus–argument form, 295multiplication and division, 298nth roots of, 325, 331polar form, 295power series, 347powers of, 320real and imaginary parts, 336roots of unity, 325spiral enlargement, 307square roots of, 300

conchoids, 114conventions for polar coordinates, 94, 111coplanar lines, 286corollaries of Lagrange’s theorem, 401cosec, 12coset, 402cosine, series for, 42cot, 12cross product of vectors, 279cyclic group, 394

© Cambridge University Press www.cambridge.org

Cambridge University Press0521548993 - Further Pure 2 and 3Hugh Neill and Douglas QuadlingIndexMore information

Index 457

de Moivre’s theorem, 321applications to trigonometry, 336, 345, 349hyperbolic analogies, 351

derivative ofcomplex exponential function, 360cos−1x, 7cosec−1x, 16hyperbolic functions, 59, 63integral, 49inverse hyperbolic functions, 67sec−1x, 13sin−1x, 7tan−1x, 4

differential equations, 217, 262auxiliary equation, 268complementary function, 265, 267, 360first order, 217first order linear, standard form of, 218general solution, 265linear, 218, 264particular integral, 265, 272, 363second order, 262solution curve, 220solution using integrating factor, 219, 222solution using the product rule, 217transforming by substitution, 225

dihedral groupof the square (D4), 389of the triangle (D3), 388

directrix, of parabola, 113distance

between two lines, 289of a point from a line, 243, 287of a point from a plane, 252

divided out form, 27application to graphs of rational functions,

78domain, symmetrical, 53

elementidentity, 375inverse, 375of a group, 376of a set, 370order of, 393

equating coefficients method for partialfractions, 20

equiangular spiral, 229, 324error, 133

in iterative method, 137in Newton–Raphson method, 158in quadratic convergence, 143

even function, 53exponential

form of complex number, 302function of complex number, 303, 359function, derivative of, 360series, 42, 349

first order differential equation, 217focus, of parabola, 109four-group (V ), 414function

even, 53hyperbolic, 54odd, 53

generator(s), of a group, 394, 412graph(s)

in polar coordinates, 96of complementary inverse trigonometric

functions, 15of cosh−1x, 65of hyperbolic functions, 55, 62of inverse trigonometric functions, 3, 5, 6,

12of rational functions, 76of solution curves, 220

graph(s) of the form y2 = f(x), 197, 200gradient when y = 0, 199properties of gradient, 198stationary values, 198

group(s), 376abelian, 379algebraic properties of, 380axioms of, 376cancellation law, 378cyclic, 394dihedral, 388four, 414

© Cambridge University Press www.cambridge.org

Cambridge University Press0521548993 - Further Pure 2 and 3Hugh Neill and Douglas QuadlingIndexMore information

458 Index

group(s) (cont.)generated by an element, 394, 399generators, 394, 412index notation, 391integers modulo n under addition,

383integers ( = 0) modulo p under

multiplication, 385isomorphism of, 407, 409Lagrange’s theorem, 401Latin square, 378multiplicative notation, 391of infinite order, 376of order 4, 413of order 6, 414of symmetries, 387order of, 376quaternion (Q4), 403subgroup(s), 396subgroup(s) of a finite, 398

hyperbolic functions, 54, 61, 351addition formulae, 56and rectangular hyperbola, 71and trigonometric functions, 55basic identity, 56differentiation and integration, 59,

63differentiation of inverse, 67inverse, 65logarithmic forms of inverse, 66Maclaurin expansions of, 60

identityaxiom, of a group, 376element, 375uniqueness of identity element, 380

image, of element of group, 408improper fractions, 26, 31index notation for a group, 391infinite geometric series, 348infinite order

of a group, 376of an element of a group, 393

initial line, 93

integersmodulo n under addition, 383modulo p under multiplication, 385

integralsestimating with sums of series, 120relating to inverse hyperbolic functions,

68relating to inverse trigonometric functions,

9integration

by parts, 170of hyperbolic functions, 59using a hyperbolic substitution, 177using complex numbers, 363using reduction formulae, 186using the substitution t = tan 1

2 x, 173using trigonometric identities, 167using trigonometric substitution, 168, 173,

177intersection

of a line and a plane, 247, 251of two lines, 239of two planes, 256, 284

inverse (functions)complement rules, 15cos, 6cos, derivative of, 7cosec, 13cosec, derivative of, 16cot, 13hyperbolic functions, 65hyperbolic functions, derivative of, 67hyperbolic functions, logarithmic forms,

66reciprocal rules, 13sec, 13sec, derivative of, 13sin, 5sin, derivative of, 7tan, 3tan, derivative of, 4

inverse (groups)axiom, 376element, 375uniqueness of inverse element, 380

© Cambridge University Press www.cambridge.org

Cambridge University Press0521548993 - Further Pure 2 and 3Hugh Neill and Douglas QuadlingIndexMore information

Index 459

isomorphismof cyclic groups, 411of groups, 407, 409

Kepler’s equation, 151

Lagrange’s theorem, 401proof of, 402

Latin square, 378Limacon, 113line

cartesian equations of, 240coplanar, 286foot of perpendicular from a point, 243shortest distance between two, 288vector equation of, 238

linear differential equation, 218, 264logarithmic function, series for, 42logarithmic forms of

inverse hyperbolic functions, 66lower bound, 121

Maclaurin expansion (series), 39, 40, 301for composite functions, 43for hyperbolic functions, 60for standard functions, 42general term, 37interval of validity, 41polynomial, 36

mathematical structure, 369modular arithmetic

and addition, 382and multiplication, 383

modulus of a complex number, 294modulus–argument form of a complex

number, 295multiple angle formulae, 336multiplication and division

of complex numbers, 298

Newton–Raphson method, 153as an iteration, 159convergence of, 162error in, 158graphical representation, 155

normalequation of a plane, 250to a plane, 249

nth roots of a complex number, 331nth roots of unity, 325, 328

applications in algebra, 326applications in geometry, 327

odd function, 53one–one function, 408order

of an element of a group, 393of a group, 376

paraboladirectrix, 113focus, 109polar equation for, 108

parallel vectors, 247partial fractions, 19

equating coefficients method, 20for improper fractions, 28substituting and equating coefficients

method, 21substitution and algebraic method, 20

particular integral, 265, 272, 363plane, 246

cartesian equation of, 249foot of perpendicular from a point, 253normal equation of, 250through three given points, 247, 254, 284vector equation of, 247

polar coordinates, 93area formula, 105conventions for, 94, 111converting to cartesian coordinates, 95converting to cartesian equations, 102graphs, 96graphs, symmetry of, 98graphs, tangent at the origin to, 101

polar form of a complex number, 295pole, 93polynomial equations, 339power series, 347powers of complex numbers, 320

© Cambridge University Press www.cambridge.org

Cambridge University Press0521548993 - Further Pure 2 and 3Hugh Neill and Douglas QuadlingIndexMore information

460 Index

prism, 257proper subgroup, 396

quadratic convergence, 143, 162quaternion group (Q4), 403

radius vector, 93maximum and minimum values of, 100

range, of function, 408rational functions, graphs, 76

an algebraic technique, 87asymptote(s), 77linear denominators, 76quadratic denominators, 81, 85summary of methods for sketching, 80

real and imaginary parts of a complexnumber, 336

reciprocal rules for inversetrigonometric functions, 13

rectangular hyperbola and hyperbolicfunctions, 71

reduction formula, 186using algebraic or trigonometric identities,

191relation between successive errors, 137, 143,

158Riccati equations, 229right coset, 402right-handed rule, 278

sandwich inequality, 121sec, 12second order differential equation, 262set notation, 369sheaf, 258shortest distance between two lines, 289similar triangles in an Argand diagram, 311sin, series for, 42spiral

equiangular, 229, 324of Archimedes, 111

spiral enlargement, 307square roots of complex number(s), 300staircase diagram, 138

strophoid, 114, 115subgroup, 396

generated by an element, 399of a finite group, 398proper, 396testing whether a subset is a subgroup,

397trivial, 396

substitution and algebraic methodfor partial fractions, 20

substitution in differential equations, 225substitution in integrals

hyperbolic, 177trigonometric, 168, 173, 177

substituting and equating coefficientsmethod for partial fractions, 20

successive errors, relation between, 137, 143,158

sums of seriesfinding bounds for integrals, 121finding bounds using integrals, 125using complex numbers, 347

symmetrical domain, 53symmetry of polar graphs, 98symmetry(ies), 387

a group of, 388

tangent at the origin to polar graphs, 101Taylor expansion, 136trigonometric series, 349trigonometry, multiple angles, 336, 345trivial subgroup, 396

upper bound, 121

vectorequation of a line, 238equation of plane, 247

vector product, 279area of a triangle using, 283components, 282of basic unit vectors, 281rules for manipulating, 280

vectors and complex numbers, 305

© Cambridge University Press www.cambridge.org

Cambridge University Press0521548993 - Further Pure 2 and 3Hugh Neill and Douglas QuadlingIndexMore information