INDEX 413

decreasing sequences of sets, 334increasing sequences of sets, 334

Merten’s theorem, 102, 189modular group, 281monic

polynomial, 35monotone convergence theorem, 349monotone decreasing, 76monotone function

differentiable, 360monotone increasing, 76Morera’s theorem, 265multinomial expansion, 31

nested closed setsshrinking diameters, 81

nested interval lemma, 65nonremovable discontinuity, 105norms, 372nowhere differentiable function, 143

odd, 303open mapping theorem

analytic functions, 281open set, 68

connected components, 116order, 13ordered

partial, 50totally ordered, 50

orientationan interval, 257

oriented interval, 194oriented simple curve, 258outer measure, 356

p series, 93p variation, 371

approximation, 374convergence of approximations, 376

partial fractionsexpansion, 40theory, 40

partial order, 50partial summation formula, 95partially ordered set, 50partition

oriented, 258perfect sets, 118

periodic function, 289permutations, 18permutations and combinations, 30pi systems, 335piecewise continuous, 196, 294piecewise smooth

curve, 258piecewise smooth curve, 257pointwise convergence

sequence, 120series, 123

pole, 272polynomial, 34

addition, 34degree, 34division, 34equality, 34leading term, 34monic, 34, 35multiplication, 34root, 34zero, 34

polynomialsgreatest common divisor, 35relatively prime, 35

power series, 161exponential sine and cosine, 166multiplication, 183of a quotient, 192raising to a power, 190

power set, 43, 52present value of an annuity, 25preservation of compactness, 112prime number, 22

less than a given number, 103product rule, 143

quadratic formula, 31quotient rule, 145

Raabe’s test, 157Rademacher’s theorem, 364radius of convergence, 162raising a number to a power, 171range, 55ratio test, 96rational function, 57rational numbers, 9