INDEX 411

epigraph, 120equicontinuous, 126equivalence class, 50equivalence relation, 49Euclidean algorithm, 22Euler’s constant, 233even, 303exponential function, 169

properties, 170exponential growth, 246exponential sine and cosine functions, 166extreme value theorem, 112

F sigma, 339Fatou’s lemma, 350Fibonacci sequence, 58field axioms, 10finite intersection property, 74first mean value theorem integrals, 219first order linear differential euqations, 227Fourier cosine formula, 305Fourier cosine transform, 307Fourier series, 289

pointwise convergence, 292uniform convergence, 303

Fourier sine transform, 307Fourier transform, 304Fresnel integrals, 253

contour integration, 283Fubini’s theorem., 220function, 46

even, 147odd, 147uniformly continuous, 118

function of many variablescontinuous, 345

fundamental theorem of algebra, 187, 188fundamental theorem of arithmetic, 23fundamental theorem of calculus, 212, 213

Lebesgue, 365future value of an annuity, 24

G delta, 339G delta set, 82gamma function

existence and convergence, 245properties, 245

gauge function, 313

generalized Riemann integralCauchy criterion, 314

geometric series, 88geometric series, 88greatest common divisor, 22

characterization, 22greatest lower bound, 27Gronwall’s inequality, 234

Hardy Littlewood maximal functionestimate, 357

Hausdorffmaximal principle, 50

Heine Borel, 68Heine Borel theorem, 72Henstock, 313Henstock’s lemma, 318Holder condition, 132Holder continuous, 371Holder functions, 132Holder’s inequality, 225, 369holomorphic, 267

identitiescosine and sine, 166

improper integral, 226infinite series

raised to a power, 190infinite sums

properties, 88integer combinations of irrational numbers,

39integral

continuous function, 195continuous functions, 159decreasing function, 205, 346finding them, 215mean value theorem, 243p variation integrator, 378piecewise continuous, 196uniform convergence, 214

integration by parts, 212, 224integration by parts formula, 209interest

compounded continuously, 180interior point, 68intermediate value theorem, 112, 113intersection of dense open sets, 81