VERSIONS OF THE FUNDAMENTAL THEOREM OF CALCULUS

Transcription

1 VERSIONS OF THE FUNDAMENTAL THEOREM OF CALCULUS A tlk by Gunther Hormnn in the DIANA Seminr Fkultt fur Mthemtik, Universitt Wien corrected nd updted version, Mrch 16, FTC before 1800 Cmbridge : lecture courses by Isc Brrow on mthemtics, optics, nd geometry; bsic ide tht \nding tngents is inverse to qudrture (clcultion of res)" is mentioned, but without cler nottion, notions or lgorithms; Isc Newton ttends s student, nd is explicitly mentioned by Brrow (in 1670) to hve mde corrections nd dditions to the course mteril; in Newton's mnuscript from 1666 we nd the rst explicit sttement of the form da dx = f(x), where A = re underneth the grph of f. Leibniz 1686 (in Hnnover): convinced of the importnce of pproprite symbolic nottion nd notions; introduces integrl sign (to remind of the notion of sum) nd the term `function'; sttes FTC in the form d f(x) dx = f(x). dx 1

2 Tsk of integrtion: for Leibniz: nd n (explicit) ntiderivtive (or primitive); for Newton: nd (power) series expnsion of the integrnd, interchnge sum nd integrl. (For more informtion nd references see [Vol88, HW96, Sti02, Wu08, Wu09].) Bsic nottion nd conventions We x, b R with < b. We will work exclusively on the bounded closed intervl [, b] nd consider functions f: [, b] R (everywhere dened nd with nite rel vlue t ech point). A prtition of [, b] is nite collection of contiguous subintervls [x k 1, x k ] (k = 1,..., N) with = x 0 < x 1 < < x N = b. The points x 0,..., x N re the division points of the prtition. A tgged prtition of [, b] is set P = {(c 1, [x 0, x 1 ]),..., (c N, [x N 1, x N ])}, where c k [x k 1, x k ] (k = 1,..., N) nd x 0,..., x N re the division points of prtition. We cll c k the tg of the intervl [x k 1, x k ]. If P is tgged prtition, we cll µ(p) := mx{x k x k 1 k = 1,..., N} the mesh size of P (which clerly depends only on the underlying prtition). 1. FTC ccording to Cuchy (1823) 1.1. DEF A Cuchy prtition of [, b] is tgged prtition P with tgs c k = x k 1 for ll k = 1,..., N (tgs re the left endpoints of the subintervls). The Cuchy sum of f corresponding to P is the rel number C P N f x := f(x k 1 )(x k x k 1 ). Cuchy used the concept of Cuchy sequences nd uniform continuity to prove the following result (cf. [Bur07, Theorem 2.2.1]), which mrks the rst ppernce of systemtic integrtion theory (i.e. bstrct denition of n integrl nd existence proof for whole clss of functions in contrst to trditionl clcultions of mny specic res by vriety of ingenious d-hoc tricks). 2

3 1.2. THM If f: [, b] R is continuous then there exists unique rel number A with the following property: ε > 0 δ > 0 such tht, for ny Cuchy prtition P with mesh size µ(p) < δ, C f x A < ε. P The unique number A is clled the Cuchy integrl of f nd we write C f(x) dx = A. The following two results present the two spects of the fundmentl theorem of clculus seprtely, nmely tht of \recovering function by integrtion of its derivtive" nd \recovering function by dierentition of its integrl". The proofs of these sttements re nowdys stndrd in introductory nlysis courses on functions of rel vrible (nevertheless, detils cn lso be found in [Bur07, Sections 2.3 nd 2.4]) THM (FTC - Prt I) If F: [, b] R is continuously dierentible, then x [, b] : C F (t) dt = F(x) F() THM (FTC - Prt II) If f: [, b] R is continuous, nd we dene F: [, b] R by F(x) := C f(t) dt (x [, b]), then F is (continuously) dierentible on [, b] nd x [, b] : F (x) = f(x). Serching for conditions on functions tht would gurntee convergence of the Fourier series Dirichlet ws led to the question of n extension of Cuchy integrtion to wider clss of functions (round 1829). He observed tht for mny exmples of discontinuous functions the Cuchy sums still converge, while for the, nowdys so-clled, Dirichlet function (the chrcteristic function of Q [0, 1]) he filed to decide whether it ws possible to pply n \integrtion process" to it. Around he hd discussions bout this question with the student Riemnn. A few yers lter Riemnn set out to nd notion of integrl tht would hndle exmples like the Dirichlet function. Although Riemnn filed to chieve this prticulr gol he developed signicnt extension to Cuchy's theory of integrtion. 3

4 2. FTC with the Riemnn integrl (1854) 2.1. DEF Let P = {(c 1, [x 0, x 1 ]),..., (c N, [x N 1, x N ])} be tgged prtition of [, b]. The Riemnn sum of f corresponding to P is given by R P N f x = f(c k )(x k x k 1 ). A (bounded 1 ) function f: [, b] R is sid to be Riemnn integrble if there exists rel number A with the following property: ε > 0 δ > 0 such tht, for ny tgged prtition P with mesh size µ(p) < δ, R f x A < ε. P In this cse A is unique nd is clled the Riemnn integrl of f; we write R f(x) dx = A REM The following re stndrd introductory text book results: (i) If f is continuous on [, b], then it is Riemnn integrble. By n esy exercise we then obtin tht C b f = R b f. (ii) Monotone functions on [, b] re R-integrble. (iii) One could dene Cuchy integrbility similrly s in Denition 2.1, but with Cuchy prtitions replcing rbitrry tgged prtitions. As reported in [Tl09] it ws shown by Gillespie in 1915 tht Cuchy nd Riemnn integrbility re, in fct, equivlent. Riemnn knew tht countble number of discontinuities does not prevent R-integrbility. But here we will brek the strict chronologicl ow nd lredy stte Lebesgue's criterion for Riemnn integrbility from 1902, which llows for shrper sttements of the FTC for the R-integrl. (Elementry proofs of the following three theorems, i.e. on n introductory course level only with the dditionl notion of Lebesgue null 2 subsets of [, b], cn be found in [Bur07, Theorems 3.6.1, 3.7.1, nd 3.7.2].) 1 It cn be shown tht boundedness of f is implied by the stted condition on the Riemnn sums ([KS04, Proposition 2.11]). 2 N [, b] is null set, if ε > 0 we cn nd countble collection of intervls I 1, I 2,... which cover N, i.e. N I k, nd stisfy length(i k) < ε. A property is sid to hold lmost everywhere on [, b], if there is null set N [, b] such tht it holds for ll x [, b] \ N. 4

5 2.3. THM Let f: [, b] R be bounded. Then f is Riemnn integrble i f is continuous lmost everywhere THM (FTC - Prt I) If F: [, b] R is dierentible, nd F is bounded nd continuous lmost everywhere on [, b] (equivlently, F is R-integrble), then x [, b] : R F (t) dt = F(x) F() THM (FTC - Prt II) If f: [, b] R is R-integrble, nd we dene F: [, b] R by F(x) := R f(t) dt (x [, b]), then F is Lipschitz continuous on [, b]. At points x [, b] of continuity of f the function F is dierentible nd F (x) = f(x) holds. Thus, we obtin F = f lmost everywhere on [, b]. Some, by now folklore, deciencies of the Riemnn integrl concern non-uniformly converging sequences of functions, in prticulr the inbility to integrte certin limit functions of convergent Fourier series. But it ws n exmple by Volterr (1881) of (n everywhere) dierentible function with bounded, but non-r-integrble, derivtive which illustrted gp in connecting FTC I with II (detils of the exmple re in Appendix A) 3. Lebesgue set out to nd notion of integrl tht would be ble to hndle every bounded derivtive. He founded fr-reching new pproch to integrtion theory nd esily dels with problem Riemnn hd to leve unsolved hlf century go: integrting the Dirichlet function. 3. FTC with the Lebesgue integrl (1904) Bsed on the concept of Lebesgue mesure λ nd L-mesurble functions (on R) there exist nowdys few lterntive pproches to dene Lebesgue integrbility for functions f: [, b] R (besides the originl Lebesgue ide, e.g., vi simple functions nd monotone limits or Young's with Lebesgue sums for mesurble prtitions). We recll Lebesgue's originl pproch from 1902: It strts with bounded mesurble f such tht α f(x) β, prtitions the rnge by α = y 0 < < y M = β, which 3 An exmple of dierentible function on [0, 1] with unbounded derivtive is f(x) = x 2 sin(π/x 2 ) (0 < x 1), f(0) = 0. However, here 1 0 f exists s n improper Riemnn integrl nd FTC I holds. In view of the Henstock-Kurzweil theory (see 4-5) this is no ccident. 5

6 gives complicted mesurble \prtition" of [, b] by setting E k = f 1 ([y k 1, y k [) (k = 1,..., M), nd considers the lower nd upper sum M y k 1 λ(e k ) M y k λ(e k ). He shows tht the supremum, sy A, of the lower sums over ll possible prtitions of the rnge [α, β] is nite nd equls the corresponding inmum of the upper sums. In this sense, f is Lebesgue integrble (on [, b]) nd we dene L f(x) dx := A. For mesurble unbounded, but non-negtive, function f we consider sequence of trunctions f k (x) = f(x) if 0 f(x) k, nd f k (x) = k if f(x) > k. If the monotone sequence of rel numbers (L b f k) k N is bounded, then f is L-integrble nd L f(x) dx := lim k L f k (x) dx For the generl cse of mesurble, possibly unbounded, function f: [, b] R we consider the representtion s dierence of two mesurble non-negtive functions in the form f = f + f, where f + (x) := mx(f(x), 0), f (x) := min(f(x), 0). Then f is L-integrble i both f + nd f re L-integrble; in this cse we set L f(x) dx := L f + (x) dx L f (x) dx We collect few bsic properties (proofs re stndrd mteril of mesure theory courses, see lso [Bur07, Gor94, KS04]; prt (i) is n immedite from Lebesgue's denition.) Proposition (i) Let f: [, b] R be bounded. mesurble. Then f is Lebesgue integrble i f is Lebesgue (ii) Let f: [, b] R be L-mesurble. Then f is L-integrble i f is L-integrble. (iii) Every R-integrble function f is L-integrble nd R f(x) dx = L f(x) dx. We note tht the derivtive of dierentible function F is L-mesurble, since it is obtined s the pointwise limit of sequence of mesurble functions by F (x) = lim n n(f(x+ 1/n) F(x)). This observtion in combintion with the stndrd convergence theorems of Lebesgue's integrtion theory, yields direct proof of the following FTC vrint (cf. [Bur07, Theorem 6.4.2]). 6

7 3.2. THM (FTC - Prt I) If F: [, b] R is dierentible nd F is bounded, then x [, b] : L F (t) dt = F(x) F(). Michel Grosser hs pointed out to me the following sttement of FTC I from clssic textbook (cf. [Nt54, pge 270]), which is stronger thn tht in Theorem 3.2: If F is dierentible, nd F is L-integrble, then L x F (t) dt = F(x) F() holds for ll x [, b]. For further improvement of the FTC we recll the following notion due to Vitli (1904). A function F: [, b] R is bsolutely continuous if ε > 0 δ > 0 such tht the following holds: for ll nite disjoint collections of subintervls ] k, b k [ [, b] (k = 1..., n) with n (b k k ) < δ we hve n F(b k) F( k ) < ε Note: dierentible with bounded derivtive Lipschitz continuous bsolutely continuous continuous; bsolutely continuous bounded vrition; none of these implictions holds in the reversed direction! [Exercise: give proofs nd nd counter exmples.] Wrning: dierentible bsolutely continuous. 4 However, s consequence of Lebesgue's Dierentition Theorem for monotone functions (1904) together with Jordn's representtion of functions of bounded vrition s the dierence of two monotone (incresing) functions (1894), we deduce the following result. (Agin, [Bur07, Gor94, KS04] my serve s references for this nd the results following below.) 3.4. Lemm Absolutely continuous functions re dierentible lmost everywhere Convention: If F is dierentible lmost everywhere, how do we dene derivtive of F s function on ll of [, b]? At points x [, b] of dierentibility we clerly tke F (x). If N [, b] denotes the Lebesgue null set of points, where F is not dierentible, we prescribe for ech y N the vlue F (y) := 0. In the sequel we will lwys refer to the derivtive function in this sense. Since L-integrbility nd the vlues of L-integrls re not ected by chnges on subsets of mesure zero, the subsequent sttements do not depend on the precise choice of function vlues on the exception set N THM (FTC - Prt I'; 1904) If F: [, b] R is bsolutely continuous, then F is L-integrble nd x [, b] : L F (t) dt = F(x) F(). 4 E.g. consider the function in footnote 3. 7

8 3.7. REM If F is nondecresing on [, b] then F is L-integrble nd x [, b] : L F (t) dt F(x) F() THM (FTC - Prt II; 1904) If f: [, b] R is L-integrble, nd we dene F: [, b] R by F(x) := L f(t) dt (x [, b]), then F is bsolutely continuous on [, b] nd F = f lmost everywhere on [, b] REM (i) Observe tht FTC I' nd II do not leve gp between them in the sense tht one my now conclude: f is L-integrble F bsolutely continuous: F = f lmost everywhere. (ii) Improperly R-integrble non-negtive functions re L-integrble nd the vlues of the integrls gree. (Cn be deduced from [KS04, Remrks on bottom of pge 167 combined with Theorem 4.79 (on pge 189)]; direct proof for the cse of n unbounded domin cn be found in [Els05, Kpitel IV, Stz 6.3]) This is no longer true, if f chnges sign: f, where f is s in footnote 3, yields n exmple of n improperly R-integrble function which is not L-integrble (on [0, 1]). (This is n esy ddition to [Bur07, Exercises c nd d].) (iii) One of the key technologies now vilble to nlysis nd probbility theory through Lebesgue-sytle integrtion theory is the group of fmous convergence theorems (Levi 1906: monotone convergence; Ftou's lemm 1906; Lebesgue's dominted convergence 1910). A prominent consequence of these is the completeness of the spces L p ([, b]). The fct tht (everywhere) dierentible functions with non-l-integrble derivtive exist rised new question concerning the FTC: Cn we extend Lebesgue's theory to n integrtion process which gurntees for ny dierentible function F tht F is integrble nd tht x F (t) dt = F(x) F() holds? 8

9 4. Integrtion à l Denjoy (1912), Perron (1914), nd Kurzweil (1957) - Henstock (1961) The good news re: For functions f: [, b] R the three integrtion theories mentioned in the title of this Section re ll equivlent. (Detiled proofs re discussed in [Gor94].) The issue of n FTC vlid for ll dierentible functions F is ddressed successfully. (So is lso the comptibility with Lebesgue integrtion.) 4.1. The three dierent pproches nd their equivlence: Denjoy 1912: f is D-integrble if F ACG ([, b]) [= spce of functions tht re generlized bsolutely continuous in the restricted sense ] such tht F = f lmost everywhere; then D b f(x) dx := F(b) F(). The techniclly demnding tsk is the denition of ACG ([, b]) (which is lredy simpliction due to Lusin ; [Gor94, Chpters 4 nd 7]) Perron 1914: bsed on the notion of upper nd lower derivtives (dened by tking only lim sup nd lim inf of the dierence quotients) nd the concept of mjor nd minor functions for f; e.g. U: [, b] R continuous is clled mjor function for f, if U() = 0 nd the lower derivtives stisfy DU(x) f(x) for ll x. If inf {U(b) U mjor function of f} = sup {u(b) u minor function of f}, then the common vlue is dened to be P b f(x) dx. (Cf. [Gor94, Chpter 8] or [KS04, Section 4.2].) Kurzweil 1957, Henstock 1961: Kurzweil studies ODEs (with non-lipschitz righthnd side;cf. [Kur57]) in integrted form x(t) = t 0 f(x(τ), τ) dτ nd investigtes convergence upon pproximting f by sequence f k, which in turn leds to n pproximtion of the integrl; he introduces Riemnn sums with n dditionl guge condition, mixes this with Perron's constructions nd clls this the generlized Perron integrl, but mentions tht it could lso be clled the generlized Riemnn integrl, nd furthermore clims tht his new integrl is, in fct, equivlent to Perron integrtion... A systemtic investigtion nd clriction comes with Henstock from 1961 onwrds. We will describe the denition in detil below nd cll it the HK-integrl (cf. [Gor94, Chpter 9], [KS04, Chpter 4], or [Bur07, Chpter 8]). 9

10 History of equivlence proofs: D = P [Hke 1921], P = D [Aleksndrov 1924, Loomn 1925], P HK [Kurzweil nd Henstock 's]. (Direct proofs of D HK seem very hrd nd hve not been chieved until series of rticles during the lte 1980's.) 4.2. DEF A guge on [, b] is positive function δ: [, b] R, i.e. δ(t) > 0 for ll t [, b]. A tgged prtition P = {(c 1, [x 0, x 1 ]),..., (c N, [x N 1, x N ])} of [, b] is sid to be δ-ne if we hve for k = 1,..., N: c k δ(c k ) < x k 1 c k x k < c k + δ(c k ) REM (i) Observe tht for ny δ-ne tgged prtition P we hve, in prticulr, tht 0 x k x k 1 < 2δ(c k ) (k = 1,..., N). The (vrying!) size of δ controls the \loclly llowed" mximl length of subintervls in the prtition. (ii) If δ 1 nd δ 2 re guges on [, b] such tht δ 1 δ 2, then ny δ 1 -ne tgged prtition is lso δ 2 -ne. (iii) If δ > 0 is constnt, then ny tgged prtition with mesh size µ(p) < δ is δ-ne. More generlly, this is still true in the cse 0 < µ(p) inf t b δ(t). (iv) Due to lemm by Cousin (1895) δ-ne tgged prtitions lwys exist. (Exercise: proof by contrdiction constructing sequence of nested intervls.) 4.4. DEF A function f: [, b] R is sid to be HK-integrble if there exists rel number A with the following property: ε > 0 guge δ on [, b] such tht, for ny δ-ne tgged prtition P, we hve R f x A < ε. P In this cse A is unique nd is clled the HK-integrl of f; we write HK f(x) dx = A REM (Bsic properties in comprison with other notions of integrl) (i) If f is improperly R-integrble on [, b] then f is HK-integrble nd the vlues of the integrls re equl. (Follows from [KS04, Theorem 4.46].) (ii) HK-integrble L-mesurble ([KS04, Corollry 4.86] or [Bur07, Theorem 8.8.1,3]). 10

11 (iii) If f is L-integrble then f is lso HK-integrble nd the vlues of the integrls gree. ([KS04, Theorem 4.46] or [Bur07, Theorem 8.7.1].) (iv) Let f 0 nd L-mesurble: f is HK-integrble i f is L-integrble. In this cse the vlues of the integrls gree. ([KS04, Theorem 4.79].) (v) Wrning: f HK-integrble f HK-integrble (use exmples from improper R-integrl). (vi) Let HK([, b]) denote the set of clsses of HK-integrble functions modulo equlity (Lebesgue) lmost everywhere. We obtin normed vector spce with the Alexiewicz norm f := sup x b HK f(y) dy ( = F L, where F(x)=HK f(y) dy). Note tht L 1 ([, b]) HK([, b]) continuously, since for f L 1 ([, b]) we hve f = HK x f(y) dy L x f(y) dy = f L 1. However, (HK([, b]),. ) is not complete ([KS04, Exmple 4.106]). (Although HK-nlogues of the Lebesgue-type convergence theorems do exist, these do not overcome deciencies relted to (v) bove: we cnnot use HK f s norm.) 5. FTC with the HK integrl We wnt to present t lest the one proof which shows why the dditionl guge condition for Riemnn sums in the denition of the HK-integrl ensures integrbility of ny derivtive. As preprtion we stte simple consequence of dierentibility, lso known s strddle lemm Lemm If f: [, b] is dierentible t y [, b], then the following holds: ε > 0 δ(y) > 0 such tht we hve f(v) f(u) f (y)(v u) ε(v u) whenever u, v [, b] nd y δ(y) < u y v < y + δ(y). Proof: Let ε > 0. Dierentibility of f t y provides δ(y) > 0 so tht for x [, b] with x y < δ(y) we hve f(x) f(y) f (y)(x y) ε x y. 11

12 Let u nd v be s in the sttement, then f(v) f(u) f (y)(v u) ( ) = f(v) f(y) f (y)(v y) + ( ) f(y) f(u) f (y)(y u) f(v) f(y) f (y)(v y) + (f(y) f(u) f (y)(y u) ε(v y) + ε(y u) = ε(v u) THM (FTC - Prt I) If F: [, b] R is dierentible, then F is HK-integrble nd x [, b] : HK F (t) dt = F(x) F(). Proof: Let ε > 0. For ech t [, b] choose δ(t) > 0 s in the strddle lemm. This de- nes guge δ on [, b]. Let x [, b] be rbitrry. If P = {(c 1, [t 0, t 1 ]),..., (c N, [t N 1, t N ])} is n rbitrry tgged δ-ne prtition of [, x], then R F t ( F(x) F() ) N = F (c k )(t k t k 1 ) ( F(x) F() ). P Since F(x) F() = N (F(t k) F(t k 1 )) we my bring ll terms into one sum nd proceed by N ( F (c k )(t k t k 1 ) F(t k ) + F(t k 1 ) ) N F(t k ) F(t k 1 ) F (c k )(t k t k 1 ) }{{} [strddle lemm!] ε(t k t k 1 ) N ε (t k t k 1 ) = ε(x ) ε(b ). Hence we hve shown HK-integrbility of F (on [, b], in fct on every [, x] with x b) nd tht HK x F (t) dt = F(x) F() holds THM (FTC - Prt I') If F: [, b] R is continuous nd dierentible nerly everywhere (i.e. except on countble subset), then F is HK-integrble nd x [, b] : HK F (t) dt = F(x) F(). (For proof see [KS04, Theorem 4.24] or [Bur07, Theorem 8.7.3].) 5.4. THM (FTC - Prt II) If f: [, b] R is HK-integrble, nd we dene F: [, b] R by F(x) := HK f(t) dt (x [, b]), 12

13 then F is continuous nd dierentible lmost everywhere on [, b] nd F = f lmost everywhere on [, b]. (For proof see [KS04, Theorem 4.83] or [Bur07, Theorem 8.8.1].) 6. FTC with the distributionl Denjoy integrl (2008), non-stndrd functions, Colombeu generlized functions,... This Section title is simply ment to suggest possible topic for mster thesis nd I will not elborte on it (for now). Let me just briey report on the bsic concept of very recent pper by Tlvil in (2008; [Tl08]): He shows tht the completion of HK([, b]) with respect to the Alexiewicz norm yields the following subspce of the distributions on the line A C ([, b]) := {f D (], b[) F C([, b]) : F() = 0 nd F = f (in D )}. Furthermore, A C ([, b]) is seprble, isomorphic (s Bnch spce) to (C([, b]), L ), nd hs L 1 ([, b]) s dense subspce. If f A C ([, b]) then the integrl is dened by x f := F(x) (note tht F() = 0) nd we clerly get (chep) FTC F = F(x) F() x [, b]. Bibliogrphy [Bur07] F. E. Burk. A grden of integrls, volume 31 of The Dolcini Mthemticl Expositions. Mthemticl Assocition of Americ, Wshington, DC, [Els05] J. Elstrodt. M- und Integrtionstheorie. Springer-Verlg, Berlin, Auge. [Gor94] R. A. Gordon. The integrls of Lebesgue, Denjoy, Perron, nd Henstock, volume 4 of Grdute Studies in Mthemtics. Americn Mthemticl Society, Providence, RI, [HW96] E. Hirer nd G. Wnner. Anlysis by its history. Springer-Verlg, Berlin,

14 [KS04] [Kur57] D. S. Kurtz nd C. W. Swrtz. Theories of integrtion, volume 9 of Series in Rel Anlysis. World Scientic Publishing Co. Inc., River Edge, NJ, The integrls of Riemnn, Lebesgue, Henstock-Kurzweil, nd Mcshne. Jroslv Kurzweil. Generlized ordinry dierentil equtions nd continuous dependence on prmeter. Czechoslovk Mth. J., 7 (82):418{449, [Nt54] I. P. Ntnson. Theorie der Funktionen einer rellen Vernderlichen. Akdemie-Verlg, Berlin, [Sti02] J. Stillwell. Mthemtics nd its history. Springer-Verlg, New York, 2nd edition, [Tl08] E. Tlvil. The distributionl Denjoy integrl. Rel Anl. Exchnge, 33(1):51{ 82, (rxiv: mth/ v2). [Tl09] E. Tlvil. Review of ' grden of integrls, by f. e. burk'. Amer. Mth. Monthly, 118:90{94, [Vol88] K. Volkert. Geschichte der Anlysis. BI Wissenschftsverlg, Mnnheim, [Wu08] H. Wuing Jhre Mthemtik, von den Anfngen bis Leibniz und Newton. Springer-Verlg, Berlin Heidelberg, [Wu09] H. Wuing Jhre Mthemtik, von Euler bis zur Gegenwrt. Springer- Verlg, Berlin Heidelberg,