INDEX 389

equivalence class, 50equivalence relation, 49Euclidean algorithm, 22Euler’s constant, 229even, 293exponential function, 165

properties, 166exponential growth, 241exponential sine and cosine functions, 162extreme value theorem, 108

F sigma, 330Fatou’s lemma, 339Fibonacci sequence, 58field axioms, 10finite intersection property, 74first order linear differential euqations, 223Fourier cosine formula, 295Fourier cosine transform, 297Fourier series, 279

pointwise convergence, 281uniform convergence, 293

Fourier sine transform, 297Fourier transform, 294Fresnel integrals, 248

contour integration, 275Fubini’s theorem., 216function, 46

even, 143odd, 143uniformly continuous, 114

function of many variablescontinuous, 335

fundamental theorem of algebra, 183, 184fundamental theorem of arithmetic, 23fundamental theorem of calculus, 208, 209

Lebesgue, 354future value of an annuity, 24

G delta, 330G delta set, 81gamma function, 322

existence and convergence, 240properties, 240

gauge function, 303generalized Riemann integral

Cauchy criterion, 304geometric series, 86

geometric series, 86greatest common divisor, 22

characterization, 22greatest lower bound, 27Gronwall’s inequality, 230

Hardy Littlewood maximal functionestimate, 346

Hausdorffmaximal principle, 50

Heine Borel, 67Heine Borel theorem, 72Henstock, 303Henstock’s lemma, 308Holder condition, 127Holder continuous, 376Holder functions, 127Holder’s inequality, 221, 375holomorphic, 258

identitiescosine and sine, 162

improper integral, 222infinite series

raised to a power, 186infinite sums

properties, 86integer combinations of irrational numbers,

39integral

continuous function, 191continuous functions, 155decreasing function, 201, 336finding them, 211p variation integrator, 382piecewise continuous, 192uniform convergence, 210

integralsmean value theorem, 215, 239second mean value theorem, 239, 250

integration by parts, 208, 220integration by parts formula, 205interest

compounded continuously, 176interior point, 68intermediate value theorem, 108, 109intersection of dense open sets, 81intervals, 13