Summability methods hFrom Wikipedia, the free encyclopediaContents1 Hadamard regularization 11.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Hlder summation 32.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4iChapter 1Hadamard regularizationIn mathematics, Hadamard regularization (also called Hadamard nite part or Hadamards partie nie) is amethod of regularizing divergent integrals by dropping some divergent terms and keeping the nite part, introducedby Hadamard (1923, book III, chapter I, 1932). Riesz (1938, 1949) showed that this can be interpreted as taking themeromorphic continuation of a convergent integral.If the Cauchy principal value integralCbaf(t)t x dt (for a < x < b)exists, then it may be dierentiated with respect to x to obtain the Hadamard nite part integral as follows:ddx(Cbaf(t)t x dt)= Hbaf(t)(t x)2dt (for a < x < b).Note that the symbols C and H are used here to denote Cauchy principal value and Hadamard nite-part integralsrespectively.The Hadamard nite part integral above (for a < x < b) may also be given by the following equivalent denitions:Hbaf(t)(t x)2dt =lim0+{xaf(t)(t x)2dt +bx+f(t)(t x)2dt 2f(x)},Hbaf(t)(t x)2dt =lim0+{ba(t x)2f(t)((t x)2+ 2)2dt f(x)2f(x)2(1b x 1a x)}.The denitions above may be derived by assuming that the function f (t) is dierentiable innitely many times at t =x for a < x < b, that is, by assuming that f (t) can be represented by its Taylor series about t = x. For details, see Ang(2013). (Note that the term f (x)/2(1/b x 1/a x) in the second equivalent denition above is missing in Ang(2013) but this is corrected in the errata sheet of the book.)Integral equations containing Hadamard nite part integrals (with f (t) unknown) are termed hypersingular integralequations. Hypersingular integral equations arise in the formulation of many problems in mechanics, such as infracture analysis.1.1 ReferencesAng, Whye-Teong (2013), Hypersingular Integral Equations in Fracture Analysis, Oxford: Woodhead Publish-ing, pp. 1924, ISBN 978-0-85709-479-7.12 CHAPTER 1. HADAMARD REGULARIZATIONBlanchet, Luc; Faye, Guillaume (2000), Hadamard regularization, Journal of Mathematical Physics 41 (11):76757714, doi:10.1063/1.1308506, ISSN 0022-2488, MR 1788597, Zbl 0986.46024.Hadamard, Jacques (1923), Lectures on Cauchys problemin linear partial dierential equations, Dover Phoenixeditions, Dover Publications, New York, p. 316, ISBN 978-0-486-49549-1, JFM 49.0725.04, MR 0051411,Zbl 0049.34805.Hadamard, J. (1932), Le problme de Cauchy et les quations aux drives partielles linaires hyperboliques (inFrench), Paris: Hermann & Cie., p. 542, Zbl 0006.20501.Riesz, Marcel (1938), Intgrales de Riemann-Liouville et potentiels., Acta Litt. Ac Sient. Univ. Hung.Francisco-Josephinae, Sec. Sci. Math. (Szeged) (in French) 9 (11): 142, JFM 64.0476.03, Zbl 0018.40704.Riesz, Marcel (1938), Rectication au travail Intgrales de Riemann-Liouville et potentiels"", Acta Litt. AcSient. Univ. Hung. Francisco-Josephinae,Sec. Sci. Math. (Szeged) (in French)9 (22): 116118, JFM65.1272.03, Zbl 0020.36402.Riesz, Marcel (1949), L'intgrale de Riemann-Liouville et le problme de Cauchy, Acta Mathematica 81:1223, doi:10.1007/BF02395016, ISSN 0001-5962, MR 0030102, Zbl 0033.27601Chapter 2Hlder summationIn mathematics, Hlder summation is a method for summing divergent series introduced by Hlder (1882).2.1 DenitionGiven a seriesa1 + a2 + ,deneH0n= a1 + a2 + + anHk+1n=Hk1+ + HknnIf the limitlimnHknexists for some k, this is called the Hlder sum, or the (H,k) sum, of the series.2.2 ReferencesHlder, O. (1882), Grenzwerthe von Reihen an der Konvergenzgrenze, Math. Ann. 20: 535549, doi:10.1007/bf01540142Hazewinkel, Michiel, ed. (2001), Hlder summation methods, Encyclopedia of Mathematics, Springer, ISBN978-1-55608-010-434 CHAPTER 2. HLDER SUMMATION2.3 Text and image sources, contributors, and licenses2.3.1 Text HadamardregularizationSource: https://en.wikipedia.org/wiki/Hadamard_regularization?oldid=641973717Contributors: MichaelHardy, Andreas Kaufmann, R.e.b., Alaibot, Headbomb, Cardamon, Daniele.tampieri, Robertang, Sednivo~enwiki, RjwilmsiBot, Glay-hours and Roberthong Hlder summation Source: https://en.wikipedia.org/wiki/H%C3%B6lder_summation?oldid=649088968 Contributors: Michael Hardy,Rjwilmsi, R.e.b., Yobot, AnomieBOT, K9re11 and Ethically Yours2.3.2 Images File:Lebesgue_Icon.svgSource: https://upload.wikimedia.org/wikipedia/commons/c/c9/Lebesgue_Icon.svgLicense: Public domainContributors: w:Image:Lebesgue_Icon.svg Original artist: w:User:James pic2.3.3 Content license Creative Commons Attribution-Share Alike 3.0