LEADER 06777nam 22005772 450 001 9910791746803321 005 20151002020706.0 010 $a1-61444-209-6 035 $a(CKB)2560000000081401 035 $a(SSID)ssj0000577625 035 $a(PQKBManifestationID)11347937 035 $a(PQKBTitleCode)TC0000577625 035 $a(PQKBWorkID)10576756 035 $a(PQKB)10272527 035 $a(UkCbUP)CR9781614442097 035 $a(MiAaPQ)EBC3330374 035 $a(Au-PeEL)EBL3330374 035 $a(CaPaEBR)ebr10728523 035 $a(OCoLC)929120469 035 $a(RPAM)14779275 035 $a(EXLCZ)992560000000081401 100 $a20120305d2007|||| uy| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 12$aA garden of integrals /$fFrank Burk$b[electronic resource] 210 1$aWashington :$cMathematical Association of America,$d2007. 215 $a1 online resource (xiv, 281 pages) $cdigital, PDF file(s) 225 0 $aDolciani Mathematical Expositions, $vv. 31 225 0$aDolciani mathematical expositions ;$vno. 31 300 $aTitle from publisher's bibliographic system (viewed on 02 Oct 2015). 311 $a0-88385-356-6 311 $a0-88385-337-X 320 $aIncludes bibliographical references and index. 327 $tForeword --$gAn historical overview --$g1.1.$tRearrangements --$g1.2.$tThe lune of Hippocrates --$g1.3.$tExdoxus and the method of exhaustion --$g1.4.$tArchimedes' method --$t1.5.$tGottfried Leibniz and Isaac Newton --$g1.6.$tAugustin-Louis Cauchy --$g1.7.$tBernhard Riemann --$g1.8.$tThomas Stieltjes --$g1.9.$tHenri Lebesgue --$g1.10.$tThe Lebesgue-Stieltjes integral --$g1.11.$tRalph Henstock and Jaroslav Kurzweil --$g1.12.$tNorbert Wiener --$g1.13.$tRichard Feynman --$g1.14.$tReferences --$g2.$tThe Cauchy integral --$g2.1.$tExploring integration --$g2.2.$tCauchy's integral --$g2.3.$tRecovering functions by integration --$g2.4.$tRecovering functions by differentiation --$g2.5.$tA convergence theorem --$g2.6.$tJoseph Fourier --$g2.7.$tP.G. Lejeune Dirichlet --$g2.8.$tPatrick Billingsley's example --$g2.9.$tSummary --$g2.10.$tReferences --$g3.$tThe Riemann integral --$g3.1.$tRiemann's integral --$g3.2.$tCriteria for Riemann integrability --$g3.3.$tCauchy and Darboux criteria for Riemann integrability --$g3.4.$tWeakening continuity --$g3.5.$tMonotonic functions are Riemann integrable --$g3.6.$tLebesgue's criteria --$g3.7.$tEvaluating a? la Riemann --$g3.8.$tSequences of Riemann integrable functions --$g3.9.$tThe Cantor set --$g3.10.$tA nowhere dense set of positive measure --$g3.11.$tCantor functions --$g3.12.$tVolterra's example --$g3.13.$tLengths of graphs and the Cantor function --$g3.14.$tSummary --$g3.15.$tReferences. 327 $g4.$tRiemann-Stieltjes integral --$g4.1.$tGeneralizing the Riemann integral--$g4.2.$tDiscontinuities --$g4.3.$tExistence of Riemann-Stieltjes integrals --$g4.4.$tMonotonicity of [null] --$g4.5.$tEuler's summation formula --$g4.6.$tUniform convergence and R-S integration --$g4.7.$tReferences --$g5.$tLebesgue measure --$g5.1.$tLebesgue's idea --$g5.2.$tMeasurable sets --$g5.3.$tLebesgue measurable sets and Carathe?odory --$g5.4.$tSigma algebras --$g5.5.$tBorel sets --$g5.6.$tApproximating measurable sets --$g5.7.$tMeasurable functions --$g5.8.$tMore measureable functions --$g5.9.$tWhat does monotonicity tell us? --$g5.10.$tLebesgue's differentiation theorem --$g5.11.$tReferences --$g6.$tThe Lebesgue-Stieltjes integral --$g6.1.$tIntroduction --$g6.2.$tIntegrability : Riemann ensures Lebesgue --$g6.3.$tConvergence theorems --$g6.4.$tFundamental theorems for the Lebesgue integral --$g6.5.$tSpaces --$g6.6.$tLē[-pi, pi] and Fourier series --$g6.7.$tLebesgue measure in the plane and Fubini's theorem --$g6.8.$tSummary--$tReferences --$g7.$tThe Lebesgue-Stieltjes integral --$g7.1.$tL-S measures and monotone increasing functions --$g7.2.$tCarathe?odory's measurability criterion --$g7.3.$tAvoiding complacency --$g7.4.$tL-S measures and nonnegative Lebesgue integrable functions --$g7.5.$tL-S measures and random variables --$g7.6.$tThe Lebesgue-Stieltjes integral --$g7.7.$tA fundamental theorem for L-S integrals --$g7.8.$tReferences. 327 $g8.$tThe Henstock-Kurzweil integral --$g8.1.$tThe generalized Riemann integral --$g8.2.$tGauges and [infinity]-fine partitions --$g8.3.$tH-K integrable functions --$g8.4.$tThe Cauchy criterion for H-K integrability --$g8.5.$tHenstock's lemma --$g8.6.$tConvergence theorems for the H-K integral --$g8.7.$tSome properties of the H-K integral --$g8.8.$tThe second fundamental theorem --$g8.9.$tSummary--$g8.10.$tReferences --$g9.$tThe Wiener integral --$g9.1.$tBrownian motion --$g9.2.$tConstruction of the Wiener measure --$g9.3.$tWiener's theorem --$g9.4.$tMeasurable functionals --$g9.5.$tThe Wiener integral --$g9.6.$tFunctionals dependent on a finite number of t values --$g9.7.$tKac's theorem --$g9.8.$tReferences --$g10.$tFeynman integral --$g10.1.$tIntroduction --$g10.2.$tSumming probability amplitudes --$g10.3.$tA simple example --$g10.4.$tThe Fourier transform --$g10.5.$tThe convolution product --$g10.6.$tThe Schwartz space --$g10.7.$tSolving Schro?dinger problem A --$g10.8.$tAn abstract Cauchy problem --$g10.9.$tSolving in the Schwartz space --$g10.10.$tSolving Schro?dinger problem B --$g10.11.$tReferences --$tIndex --$tAbout the author. 330 $aThe derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there are a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reason for their existence and their uses are given. Historical information is plentiful. Advanced undergraduate mathematics majors, graduate students, and faculty members are the audience for the book. Even experienced faculty members are unlikely to be aware of all of the integrals in the Garden of Integrals and the book provides an opportunity to see them and appreciate the richness of the idea of integral. Professor Burke's clear and well-motivated exposition makes this book a joy to read. 606 $aIntegrals 615 0$aIntegrals. 676 $a515/.43 700 $aBurk$b Frank$0253701 702 $aScully$b Terence$f1935- 801 0$bUkCbUP 801 1$bUkCbUP 906 $aBOOK 912 $a9910791746803321 996 $aA garden of integrals$93694343 997 $aUNINA