A two points Taylor's formula for the generalised Riemann integral

• Autorzy: Dragomir S. S. , Thompson H. B.
• Języki publikacji: EN

Abstrakty

EN

A two points Taylor's formula for the generalised Riemann integral and various bounds for the remainder are established. Moreover, particular instances of interest are given.

Słowa kluczowe

EN

generalised Riemann integral; Kurzweil integral; Perron integral; Taylor's formula; integral inequalities

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2010
• Tom: Vol. 43, nr 4
• Strony: 827--840
• Opis fizyczny: Bibliogr. 21 poz.

Twórcy

autor

Dragomir S. S.

autor

Thompson H. B.

• School of Engineering & Science Victoria University Po Box 14428 Melbourne, Victoria, Australia, 8001,

Bibliografia

• [1] A. Aleksandrov, Über die Äquivalenz des Perronschen und des Denjoyschen Integralbegriffes, Math. Z. 20 (1924), 213–222.
• [2] J. C. Burkill, A First Course in Mathematical Analysis, Cambridge University Press, Cambridge, 1967.
• [3] A. Denjoy, Une extension de l’intégrale de M. Lebesgue, C. R. Acad. Sci. Paris 154 (1912), 895–862.
• [4] H. Hake, Über de la Vallée Poussins Ober-und Unterfunktionen, Math. Ann. 83 (1921), 119–142.
• [5] R. Henstock, Definitions of Riemann type of variational integrals, Proc. London Math. Soc. 1 (1961), 402–418.
• [6] R. Henstock, A Riemann-type integral of Lebesgue power, Canadian J. Math. 20 (1968), 79–87.
• [7] J. Kurzweil, Generalised ordinary differential equations and continuous dependenceon a parameter, Czechoslovak Math. J. 82 (1957), 418–449.
• [8] J. Kurzweil, On Fubini theorem for general Perron integral, Czechoslovak Math. J. 98 (1973), 286–297.
• [9] H. Looman, Über die Perronsche Integral Definition, Math. Ann. 93 (1935), 153–156.
• [10] N. N. Luzin, Sur les propriétés de l’intégrale de M. Denjoy, C. R. Acad. Sci. Paris 155 (1912), 1475–1478.
• [11] J. Mărík, Foundation of the theory of an integral in Euclidean space (translated into English by L. I. Trudzik, Dept. of Math., Univ. of Melbourne, Parkville, Victoria 3052, Australia) Časopis Pěst. Mat. 77 (1952), 125–144.
• [12] J. Mawhin, L’introduction á l’Analyse, CABAY, Louvain-La-Neuve, 1984.
• [13] E. J. McShane, A Riemann-type Integral that Includes Lebesgue–Stieltjes, Bochner and Stochastic Integrals, Vol. 88, Memoirs of Amer. Math. Soc., 1969.
• [14] R. M. McLeod, The Generalized Riemann Integral , Carus Mathematical Monographs, Mathematical Association of America, 1980.
• [15] O. Perron, Über den Integralbegriff, Sitzber, Heidelberg Akad.Wiss. Abt. A 16 (1914), 1–16.
• [16] M. Spivak, Calculus, W. A. Benjamin, New York, 1967.
• [17] K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Belmont, California, 1981.
• [18] C. Swartz, B. S. Thomson, The teaching of mathematics; More on the fundamental theorem of calculus, Amer. Math. Monthly 95 (1988), 644–641.
• [19] H. B. Thompson, Taylor’s theorem with the integral remainder under very weak differentiability assumptions, Austral. Math. Soc. Gazette 12 (1985), 1–6.
• [20] R. Výborný, Kurzweil’s integral and arclength, Austral. Math. Soc. Gazette 8 (1981), 19–22.
• [21] H. B. Thompson, Taylor’s theorem using the generalised Riemann integral, Amer. Math. Monthly 96 (4) (1989), 346–350.