Fundmentl Theorem of Clculus nd Computtions on Some Specil Henstock-Kurzweil Integrls Wei-Chi YANG wyng@rdford.edu Deprtment of Mthemtics nd Sttistics Rdford University Rdford, VA 24142 USA DING, Xiofeng dingxiofeng@hhu.edu.cn College of Computer nd Informtion Engineering Hohi University Nnjing 2198 People s Republic of Chin Abstrct The constructive definition usully begins with function f, then by the process of using Riemnn sums nd limits, we rrive the definition of the integrl of f, f. On the other hnd, descriptive definition strts with primitive F stisfying certin condition(s) such s F = f nd F is bsolutely continuous if f is Lebesgue integrble, nd F is generlized bsolutely continuous if f is Henstock-Kurzweil integrble. For descriptive integrls, the deficiency is tht we need to know primitive F for which F = f nd stisfying some properties. For constructive integrtion, we proposed in [8] using n uneven prtition to get broder fmily functions which includes some improper Riemnn integrls. In this pper, we describe how we cn mke use of the Fundmentl Theorem of Clculus nd the constructive definition to rech description definition for some improper Riemnn integrble functions tht re monotonic or highly oscillting with singulrity on one end. 1 Introduction If f is continuous over closed intervl, [, b], the Riemnn integrl f mkes sense nd in some cses we re ble to find n explicit formul for the expression F (x) = x f(t)dt; in such cse we cll F the primitive of f, we cn check esily tht the condition F (x) = f(x) holds. However, in lmost every cse we re unble to produce simple closed form expression for F (x) by the process of tking limits of sums. We recll tht the constructive definition usully begins with function f, then by the process of using Riemnn sums nd limits, we rrive the definition of the integrl of f, f. On the other hnd, descriptive definition strts with primitive F stisfying certin condition(s) such s F = f. For exmple, F is uniformly continuous over closed intervl if f is Riemnn integrble over the sme intervl. On the other hnd, F is bsolutely continuous if f is Lebesgue integrble nd F is generlized bsolutely continuous if f is Henstock-Kurzweil (HK) integrble. For descriptive integrls, the chllenge is tht we need to know primitive F such tht F = f (nd stisfying some properties). In
this pper, we first describe we cn construct the primitive function from function f tht is Riemnn integrble. Next, we extend this ide to two types of improper Riemnn integrble functions: One is the fmily of monotonic functions with singulrity t one end, which normlly re computed s improper Riemnn integrls, but now they re direct results from Henstock- Kurzweil definition. Second, we will tke cre of functions tht re highly oscilltory with singulrity t one end. In [8], we described how these two types of HK integrble functions cn be computed by introducing n uneven prtition. In this pper, we strt with constructive definition by using uneven prtitions nd rech description definition for these two types of improper Riemnn integrble functions It is known tht Fundmentl Theorem of Clculus (FTC) should be vlid when function F is differentible on (, b) nd we hve F (x)dx = F (b) F (). (1) However, the Lebesgue integrtion requires F to be integrble over [, b]. Essentilly in this pper, we mke use of the computtion methods described in [8] nd show tht there is fmily of computtionl functions tht re HK but not Lebesgue integrble. In summry, the functions we defined by using the Riemnn sum with our computtion scheme(s) llows us to reconstruct the primitive F (x) = x f, nd the primitive F is generlized uniformly continuous in closed intervl. We first recll the following definitions. Definition 1 Let F be defined on set A R. We sy tht F is uniformly continuous on A if for every ε > there exists δ > such tht if x, y A nd x y < δ, then F (x) F (y) < ε. Definition 2 For sequence {F n : A R}, we sy tht the sequence {F n } converges uniformly on A to F if for every ε > there exists positive integer N such tht F n (x) F (x) < ε for every n N nd for every x A. Definition 3 The sequence of functions {F n : A R} is sid to be uniformly Cuchy if for every ε > there exists n N, F n+k (x) F n (x) < ε for every positive integer n N, every positive integer k, nd for every x A. It is known from the Weierstrss Uniform Convergence Criterion tht sequence {F n } converges uniformly on A to F if nd only if it is uniformly Cuchy. For computtion purpose, checking if sequence is uniformly Cuchy is more convenient thn checking if sequence is uniformly convergence. 1.1 Grphicl Approch to The Riemnn Sums nd Fundmentl Theorem of Clculus We summrize how we my use the grphs of Riemnn sums s described in [9] to provide n intuitive pproch to the First Form of The Fundmentl Theorem of Clculus. Suppose tht f is continuous on n open intervl I. Suppose tht is ny number in I nd tht the function F is defined by F (x) = x f(t)dt
Figure 1: The grph of f for every x I. Then F (x) = f(x) for every number x I. We use the function f(x) = 1 x sin x with x [ 3, 3] for illustrtion. We note tht x f(t)dt does not posses closed form nd note tht f fils to be differentible t severl points, which cn be seen in Figure 1. We use midpoint sums sum to pproximte x f(t)dt. If M f (x, n) is the midpoint pproximtion to x f (t) dt tken over regulr prtition of [, x] into n subdivisions then ( ( ) x 2jx x M f (x, n) = f. n) 2n As we know, if x is ny number then j=1 lim M f(x, n) = x f(t)dt. We shll see how the grph of x f(t)dt looks like for ech number x by mking use of the grphs of M f (x, n), n = 1, 2,.... The grphs of M f (x, 5) nd M f (x, 15) cn be seen in Figure 2(); while the grphs of M f (x, 25) nd M f (x, 35) cn be seen in Figure 2(b), respectively. Since f is continuous everywhere over the intervl [ 3, 3], the sequence M f (x, n) is continuous everywhere for ech n, nd it esy to check tht M f (x, n) converges uniformly to x f(t)dt s n in [ 3, 3]. In other words, we hve lim M f(x, n) = x f(t)dt. (2) Consequently, we expect the grph of x f(t)dt will resemble the one in Figure 2(b). We leve it to reders to verify grphiclly tht ( ) d lim dx M f(x, n) = f(x) (3) Exercise: Use computtionl tool to verify tht lim M f (1, n) = 1 f(t)dt = 1 + cos 1 sin 1.
() (b) Figure 2: The grphs of M f (x, n). (): The grphs of M f (x, 5) nd M f (x, 15); (b): The grphs of M f (x, 25) nd M f (x, 35). 1.2 Extend the FTC to some Henstock Integrls First, we introduce some preliminry nottions nd definitions. Let A = [, b], we sy P = {(A 1, x 1 ),..., (A n, x n )} is prtition of A if A 1,..., A n re non-overlpping subintervls, x i A i, for i = 1, 2,..., n, nd n i=1a i = A. Let δ be positive function defined on A. A prtition P = {(A 1, x 1 ),..., (A n, x n )} is clled δ-fine if A i (x i δ(x i ), x i + δ(x i )), for i = 1, 2,...n. We first give the definition of Henstock- Kurzweil integrtion on one dimension. Definition 4 A rel-vlued function f is sid to be Henstock-Kurzweil integrble (or simply HK-integrble) with vlue I on [, b] if for every ɛ > there is positive function δ on [, b] such tht f(x i ) A i I < ɛ (4) i=1 for ech δ-fine prtition P of A, where A i denotes the length of A i, i = 1, 2,..., n. In such cse, we write fdx or simply f. Alterntively, we cn use the following Definition 5 A rel-vlued function f is sid to be Henstock-Kurzweil integrble (or simply HK-integrble) with vlue I on n intervl A = [, b] if there is sequence of delt functions δ n (x) such tht for every δ n (x)-fine division D n, the Riemnn sums n i=1 f(x i) A i I s n. The Henstock integrl nd our qudrtures cn be stted in higher dimensions s discussed in [7]. However, we describe how we pply the FTC on some HK-integrls in one dimension in this pper. First we use the following definition to introduce the uneven prtitions on n intervl.
Definition 6 A mtrix A with positive nk is clled uniformly regulr if the following conditions re stisfied: (i) lim nk = uniformly over k. (ii) n k=1 nk = 1. or We introduce the following qudrtures in 1-D: The closed type qudrture Q 1 nk n(f) = 2 (f(u n,k 1) + f(u nk )) (5) k=1 Q 2 n(f) = 1 2 n1f(u n1 ) + k=2 nk 2 (f(u n,k 1) + f(u nk )) (6) where u nk = + k i=1 ni, nd u n, =. Remrks: 2(b )k (1) By looking the nk = n(n + 1), we hve n1 < n2 <... < nn, n k=1 nk = b for ech n, which is the bsis of our choice of uneven prtitions. (2) Both qudrtures re similr to the trpezoidl rule except we re using uneven prtitions, which re the essence of the HK-integrtion. (3) In the closed type qudrture, if the Eq. (5) contins the singulrity t the end point t x =, or x = b, we set f() = or f(b) = respectively. (4) In Eq. (6), we consider the integrl vlue of f over the first intervl [, u n1 ]. In other words, we ignore the singulrity t x =. We shll see we pply this qudrture for functions tht re monotonic nd hve singulrities ner the end point. 2 Improper Riemnn Integrls 2.1 Monotone functions We summrize the computtion qudrture mentioned from [8] for completeness. Suppose we denote Q n (f) s either closed or open qudrture described bove in the intervl [, b]. We recll the following theorem from [8] without proof, which sys tht there is no improper Riemnn integrls under HK-definition. Theorem 7 If f is Riemnn integrble over [c, b] for ech c (, b] nd improper Riemnn integrble over the intervl [, b] or lim Q 1 n(f) exists. Then f is HK-integrble over [, b], nd we hve ( b ) f = lim Q 1 nk n(f) = lim 2 (f(u n,k 1) + f(u nk )), (7) k=2 where Q 1 n(f) is the qudrture pplied on the intervl [, b]. In prticulr, if function f is monotonic over (, b] nd hs singulrity t x =, Eq. (7) cn be replced by f = lim Q 2 n(f) = lim ( 1 2 n1f(u n1 ) + k=2 nk 2 (f(u n,k 1) + f(u nk )) ). (8)
The following theorem is useful when we use uniformly regulr mtrix nk for computing monotonic nd improper Riemnn but HK-integrble function. Theorem 8 If f is Riemnn integrble over [c, b] for ech c (, b]. If for ech ɛ >, there exists regulr mtrix nk such tht f(u n1 ) n1 < ɛ. Then f is HK integrble over [, b] nd ( ) b f = lim Q 2 1 n(f) = lim 2 nk n1f(u n1 ) + 2 (f(u n,k 1) + f(u nk )). (9) Proof. f is Riemnn integrble on [c, b] for ech c (, b], so f is HK-integrble on [c, b] for ech c (, b]. Let L = f nd Let ε >. By theorem 8 in [8], there exists positive function δ c 1 on (, b] nd P is δ 1 fine prtition in (c, b], then f(p) f < ε. Choose η > so tht c f L < ε for ll c (, + η), define positive function δ on [, b] by c { min{δ1 (b), x } x (c, b] δ(x) = ε min{η, } x =, (1) 1+f(c) Suppose tht P is δ fine prtition of [, b] nd (c, [, c]) cn be subintervl belongs to P where < x < + η. Let P = P {(c, [, c])} nd compute f(p) L f(p ) c f + We see tht f is HK-integrble on [, b] nd ( b f = lim Q 2 1 n(f) = lim 2 n1f(u n1 ) + c k=2 f L + f(c) η < ε + ε + ε = 3ε. k=2 nk 2 (f(u n,k 1) + f(u nk )) ). We next see how we cn reconstruct the primitive F of f through the process of Fundmentl Theorem of Clculus when f is improper Riemnn integrble over n intervl. First we show tht the convergence lim Q n (f)(x) is uniform for ll x [, b], where Q n (f)(x) denotes the function when Q n (f) is pplied on [, x] for ech x (, b]. In other words, we mke use of the Riemnn sum, Q n (f), to reconstruct the primitive F of f. More precisely, we prove the following: Theorem 9 If f is Riemnn integrble over [c, b] for ech c (, b] nd f() =. Given uniformly regulr mtrix nk over [, b], if we write F (x) = lim Q n (f)(x), where Q n (f)(x) denotes the function when Q n (f) is pplied on [, x].then (i) F (x) is continuous over [, b], (ii) f is HK-integrble over [, b], (iii) x f = F (x) nd F (x) = f(x) over [, b]. Proof. First, we note tht Q n (f)() =. For x (, b], the function Q n (f)(x) is continuous, nd we see F (x) = lim Q n (f)(x) = n=1 Q n(f)(x). Since Q n (f)(x) is continuous for ech x (, b],nd the convergence of the infinite series is uniform, F is continuous over [x, b] for ech x (, b]. It remins to show tht F is continuous t x =, x f = F (x) nd F (x) = f(x).
(i) We clim tht F is continuous t x = from the right. Let x +, ( ) ( ) ( x ) lim lim Q n(f)(x) = lim lim Q n(f)(x) = lim f = F () =. (11) x + x + x + Therefore F is continuous t x = from the right. (ii) We note tht F (x) = lim Q n (f)(x) = n=1 Q n(f)(x) = x f. (iii) We only need to prove tht F () = f() = : x F F (x) F () () = lim = lim f = f() =. (12) x + x x + x We pply Theorem 8 on the following two exmples, which is to sy tht improper Riemnn integrls re HK-integrble nd we cn use the qudrture mentioned bove to compute their respective integrls. Exmple 1 We define f(x) = 1/ x if if x, nd f() =. We choose nk = 2 then f(u n1 ) n1 =. Thus f is integrble over [, 1] nd it is proved in [8] tht n(n+1) 1 f = lim Q 2 n(f) = 2. Exmple 11 We define f(x) = ex x + 1 if if x, nd f() =.We choose nk = 2k, n(n+1) 2k,then n(n+1) f(u n1 ) n1 = 2 2e n(n+1) (2+n 2 +n)(n(n+1)) n(n+1). (13) Therefore, it follows from Theorem 8 tht f is improper Riemnn integrble but f is HKintegrble over [, 1], nd it cn be shown tht f = lim 1 Q 2 n(f) 1.76. We further note tht if we define F (x) = lim Q n (f)(x), then we re ble to use computtionl tool to simulte the grph of primitive F of f s follows s we did in Section 1.1. We plot y = f(x) nd y = Q 1 (f)(x) over the intervl [ 1, ] in Figures 3() nd 3(b), respectively. 2.2 Highly Oscillting Functions In this section, we discuss those highly oscillting functions with one singulrity ner one end of the intervl. The following exmple shows tht dividing the entire intervl [, b] unevenly by using nk is not sufficient for computing highly oscillting, non-bsolute integrls. Exmple 12 We define f(x) = 1 sin( 1 ) if x, nd f() =. We shll show tht f is x x HK-integrble though not Lebesgue integrble over [, 1] when we prove its two dimensionl extension in the next section. We demonstrte how we pproximte the integrl 1 f. We first note the followings: (1) We cn t pply the uneven prtition nd qudrture over the intervl [, 1] in one step. Insted, we construct sequence {x n } converges to. (2) In other words, we select x i = 5 (i 1) (14)
() (b) Figure 3: The grphs of f nd its primitive F. (): The grph of f; (b): The grph of the primitive F of f. for i = 1, 2,... We pproximte the integrl of f in ech I i = [x i+1, x i ] nd denote the integrl of f over I i by A i when pplying the closed type qudrture Q 2 n(f) = n nk k=1 (f(u 2 n,k 1) + f(u nk )), where u n = x i+1,nd u nk = x i+1 + k i=1 ni. (3) For ech x [x i+1, x i ], we write F i (x) = lim Q i n(f)(x), where Q i n(f)(x) denotes when Q n (f) is pplied on [x i+1, x].then 1 f r F i (x) = i= r A i, (15) 2k for some r. If we use the mtrix nk = n(n + 1) nd the closed type qudrture in ech I i = [x i+1, x i ] for i = 1, 2,...6,nd with the help of Mtlb nd the closed type Q 2 n(f), we hve shown in [8] tht 1 f is pproximtely equl to 6.2473274145915e 1. Remrk: We demonstrte the convergence of F (for the function in bove Exmple) being uniform in the intervl of [5 3, 5 2 ] grphiclly in http://mthndtech.org/yng/henstock_ integrl/henstock21_feb1.html, but the convergence is not uniform in [, 1].In other words, F is countble union of uniformly continuous function. We cn use computble tool to demonstrte tht the grphiclly tht i=1 y = Q 2 n(f)(x) is getting closer to y = F 2 (x) in [5 3, 5 2 ], nd (16) y = Q 3 n(f)(x) is getting closer to y = F 3 (x) in [5 4, 5 3 ], (17) lthough it is evidently tht the second grphicl representtion will consume much more computtion time. We demonstrte the grphs of y = F 2 (x) nd y = F 3 (x) re shown respectively in Figures 4() nd 4(b). In view of Exmple 12, we hve the following definition. Definition 13 If function F is sid to be generlized uniformly continuous (UCG) on set X if X is countble union of X i, or X = i=1 X i, such tht F is uniformly continuous on ech X i.
() (b) Figure 4: The grphs of y = F 2 (x) nd y = F 3 (x). (): The grph of F 2 (x); (b): The grph of F 3 (x). Theorem 14 Let {x r } +, nd A r = lim Q n (f) in [x r+1, x r ], where x = b. If r= A r converges, then f is HK-integrble over [, b] nd f = A r. (18) Furthermore, for ech x [x r+1, x r ], if we write F r (x) = lim Q r n(f)(x), where Q r n(f)(x) denotes when Q n (f) is pplied on [x r, x].then (i) F r (x) is uniformly continuous (UC) on [x r+1, x r ], (ii) F r (x r ) = A r,(iii) f = r= F r(x r ) = r= A r. In other words, if we write F (x) = r= F r(x), then F is UCG on [, b]. Proof. The proof of the first prt of this Theorem is done in [8]. As A r = lim Q n (f) in [x r+1, x r ], we hve tht f is HK-integrble over [x r+1, x r ]. So for ech x [x r+1, x r ], lim Q r n(f)(x) exists nd x r+1 f = lim x Q r n(f)(x) = F r (x). So the primitive F r (x) of f is continuous in [x r+1, x r ], nd F r (x) is UC there from the Theorem 5.48 in [7]. xr In ddition, x r+1 f = lim Q r n(f)(x r ) = F r (x r ) = A r, nd f = r= F r(x) = r= A r. Theorem 15 Assume the conditions of the preceding Theorem re met, nd we write F (x) = r= F r(x), then F (x) = f(x) for ll x in [, b] nd F is generlized uniformly continuous on [, b]. Proof. For ech x (, b], there exists N nd x [x N+1, x N ]. Then F (x) cn be rewritten s: r= F (x) x [x 1, x ] F 1 (x) + A x [x 2, x 1 ] F (x) =. F N (x) + N 1 r= A r x [x N+1, x N ]
In [x r+1, x r ], r =, 1,.., N, s every F r (x), is continuous nd is the primitive of f, so (F r (x)) = f(x) nd F r (x) is UC. So for ech x (, b], F (x) = f(x). F r (x) is UC in [x r+1, x r ], let r then we hve F () = f(). So F (x) = f(x) for ll x in [, b] nd F is generlized uniformly continuous on [, b]. We conclude the discussion of FTC by reminding reders tht theoreticlly, we hve more generl theorem regrding the HK-integrble functions below. In other words, we cn llow more singulrities in n intervl nd the theorem cn be extended into higher dimensions, which is so clled the Divergence Theorem, see [5]. Theorem 16 If F = f on [, b], then f is HK-integrble over [, b] nd f = F (b) F (). (19) Proof. This follows from the definition of HK integrtion, see ([5], Theorem 6.1.2, pge 13). 3 Conclusion In this pper, we described how we cn replce the even prtitions with n uneven prtition in Riemnn sum definition to rech HK-definition. This llows us to hndle some improper Riemnn integrls. We lso discussed the properties for these specil types of primitive functions, which re generlized uniformly continuous functions, which re nturl extensions from the primitive functions of Riemnn integrl functions. The pproches here re intuitive without the knowledge of Lebesgue mesure nd integrtion theories, which we think re more ccessible to undergrdute students. Since the primitive function for highly oscilltory with singulrity t one end point nd non-bsolute HK-integrble function is generlized uniformly continuous, our future works will involve the use of different uneven prtitions in ech subintervls. References [1] Dvis nd Rbinowitz (1983), Method of Numericl Integrtion, 2nd ed., Acdemic Press, New York, ISBN978-1226362 [2] P. Y. Lee (1989), Lnzhou Lectures on Henstock Integrtion, World Scientific, Singpore, ISBN 9971-5-892-3. [3] P. Y. Lee & R. Výborný (2), Integrl: An Esy Approch fter Kurzweil nd Henstock, Cmbridge University Press, Cmbridge UK, ISBN 521 77968 5. [4] R. K. Miller (1971), On Ignoring the Singulrity in Numericl Qudrture, Mthemtics of Computtion, Volume 25, Number 115, July. [5] W.F. Pfeffer (1993), The Riemnn Approch to Integrtion: Locl Geometric Theory, Cmbridge University Press, ISBN -521-4435-1.
[6] P. Rbinowitz nd I. H. Slon (1984), Product Integrtion in the Presence of Singulrity, Sim J. Numer. Anl. Vol 21, No. 1, Februry. [7] B. S. Thomson, J. B. Bruckner nd A. M. Bruckner (2), Elementry Rel Anlysis, 1st edition, Prentice Hll, ISBN -13-1975-6. [8] W.-C. Yng, P.-Y. Lee nd X. Ding (29), Numericl Integrtion On Some Specil Henstock-Kurzweil Integrls, Electronic Journl of Mthemtics nd Technology (ejmt), ISSN 1933-2823, Issue 3, Vol. 3. [9] W.-C. Yng nd J. Lewin, Exploring Mthemics with Scientific Notebook, pge 91, Springer Verlg, 1998. ISBN 981-383-88-3.