Integration by parts

Many theorems of integration theory are true for a wide range of definitions of integrals. One such theorem is that giving integration by parts, and we discuss it in this paper.

[1]  P. Bois-Reymond Ueber die allgemeinen Eigenschaften der Klasse von Doppelintegralen, zu welcher das Fouriersche Doppelintegral gehört. , 1868 .

[2]  Ernest William Hobson The Theory of Functions of a Real Variable and the Theory of Fourier's Series , 1907 .

[3]  E. Hobson [On the second mean-value theorem of the integral calculus , 1909 .

[4]  H. Lebesgue Sur l'intégration des fonctions discontinues , 1910 .

[5]  On the Conditions that a Trigonometrical Series Should have the Fourier Form , 1911 .

[6]  W. H. Young On Integration with Respect to a Function of Bounded Variation , 2022 .

[7]  W. H. Young On Multiple Integration by Parts and the Second Theorem of the Mean , 1917 .

[8]  G. Bliss,et al.  Integrals of Lebesgue , 1917 .

[9]  On Non‐Absolutely Convergent, not Necessarily Continuous, Integrals , 1917 .

[10]  W. Sierpinski Sur un problème concernant les ensembles mesurables superficiellement , 1920 .

[11]  E. C. Titchmarsh On Conjugate Functions , 1929 .

[12]  J. C. Burkill The approximately continuous-Perron integral , 1932 .

[13]  The Cesàro‐Perron Integral , 1932 .

[14]  A. Kolmogoroff Grundbegriffe der Wahrscheinlichkeitsrechnung , 1933 .

[15]  The Cesàro-Perron Scale of Integration , 1935 .

[16]  The Perron-Stieltjes integral , 1936 .

[17]  Fractional Orders of Integrability , 1936 .

[18]  T. H. Hildebrandt Definitions of Stieltjes Integrals of the Riemann Type , 1938 .

[19]  On Perron integration , 1942 .

[20]  Convergence factors for generalized integrals , 1945 .

[21]  Differential Properties of Young-Stieltjes Integrals , 1948 .

[22]  Mean-Continuous Integrals , 1949, Canadian Journal of Mathematics.

[23]  The Efficiency of Convergence Factors for Functions of a Continuous Real Variable , 1955 .

[24]  Sur les intégrales (E.R.) et ses applications , 1959 .

[25]  T. H. Hildebrandt On systems of linear differentio-Stieltjes-integral equations , 1959 .

[26]  The Use of Convergence Factors in Ward Integration , 1960 .

[27]  R. Henstock The Equivalence of Generalized Forms of the Ward, Variational, Denjoy‐Stieltjes, and Perron‐Stieltjes Integrals , 1960 .

[28]  R. Henstock Definitions of Riemann Type of the Variational Integrals , 1961 .

[29]  N-Variation and N-Variational Integrals of Set Functions , 1961 .

[30]  Sur une généralisation de l'intégrale (E.R.) et un théorème général de l'intégration par parties , 1962 .

[31]  An integration-by-parts formula , 1963 .

[32]  Ralph Henstock,et al.  Theory of integration , 1966 .

[33]  An elementary proof of integration by parts for the Perron integral , 1967 .

[34]  Ralph Henstock,et al.  A Riemann-Type Integral of Lebesgue Power , 1968, Canadian Journal of Mathematics.

[35]  The N‐Variational Integral and the Schwartz Distributions , 1968 .

[36]  Generalized Integrals of Vector-Valued Functions , 1969 .

[37]  On integration-by-parts for weighted integrals , 1969 .

[38]  A Proof That Henstock's Integral Includes Lebesgue's , 1970 .

[39]  E. J. McShane A Riemann-type integral that includes Lebesgue-Stieltjes, Bochner and stochastic integrals , 1971 .

[40]  R. Henstock Integration in Product Spaces, Including Wiener and Feynman Integration , 1973 .