Necessary and Sufficient Conditions for Differentiating Under the Integral Sign

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1 Necessry nd Sufficient Conditions for Differentiting Under the Integrl Sign Erik Tlvil 1. INTRODUCTION. When we hve n integrl tht depends on prmeter, sy F(x f (x, y dy, it is often importnt to know when F is differentible nd when F (x f 1(x, y dy. A sufficient condition for differentiting under the integrl sign is tht f 1(x, y dy converges uniformly; see [6, p.60].whenwehve bsolute convergence, the condition f 1 (x, y g(y with g(y dy < suffices (Weierstrss M-test nd Lebesgue Dominted Convergence. If we use the Henstock integrl, then it is not difficult to give necessry nd sufficient conditions for differentiting under the integrl sign. The conditions depend on being ble to integrte every derivtive. If g :[, b] R is continuous on [, b] nd differentible on (, b it is not lwys the cse tht g is Riemnn or Lebesgue integrble over [, b]. However, the Henstock integrl integrtes ll derivtives nd thus leds to the most complete version of the Fundmentl Theorem of Clculus. The Henstock integrl s definition in terms of Riemnn sums is only slightly more complicted thn for the Riemnn integrl (simpler thn the improper Riemnn integrl, yet it includes the Riemnn, improper Riemnn, nd Lebesgue integrls s specil cses. Using the very strong version of the Fundmentl Theorem we cn formulte necessry nd sufficient conditions for differentiting under the integrl sign.. AN INTRODUCTION TO THE HENSTOCK INTEGRAL. Here we ly out the fcts bout Henstock integrtion tht we need. There re now quite number of works tht del with this integrl; two good ones to strt with re [1]nd[3]. Let f :[, ] (,. Aguge is mpping γ from [, ] to the open intervls in [, ]. Byopen intervl we men (, b, [, b, (, ], or [, ] for ll < b (the two-point compctifiction of the rel line. The defining property of the guge is tht for ll x [, ], γ(x is n open intervl contining x. Atgged prtition of [, ] is finite set of pirs P {(z i, I i } N, where ech I i is nondegenerte closed intervl in [, ] nd z i I i. The points z i [, ] re clled tgs nd need not be distinct. The intervls {I i } N form prtition:fori j, I i I j is empty or singleton nd N I i [, ].We sy P is γ -fine if I i γ(z i for ll 1 i N.Let I denote the length of n intervl with I 0for n unbounded intervl. Then, f is Henstock integrble, nd we write f A, if there is rel number A such tht for ll ɛ>0there is guge function γ such tht if P {(z i, I i } N is ny γ -fine tgged prtition of [, ] then N f (z i I i A <ɛ. Note tht N is not fixed nd the prtitions cn hve ny finite number of terms. We cn integrte over n intervl [, b] [, ] by multiplying the integrnd with the chrcteristic function χ [,b]. 5 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

2 The more drmticlly function chnges ner point z, the smller γ(z becomes. With the Riemnn integrl the intervls re mde uniformly smll. Here they re loclly smll. A function is Riemnn integrble on finite intervl if nd only if the guge cn be tken to ssign intervls of constnt length. It is not too surprising tht the Henstock integrl includes the Riemnn integrl. Wht is not so obvious is tht the Lebesgue integrl is lso included. And, the Henstock integrl cn integrte functions tht re neither Riemnn nor Lebesgue integrble. An exmple is the function f g where g(x x sin(1/x 3 for x 0ndg(0 0; the origin is the only point of nonbsolute summbility. See [5, p. 18] for function tht is Henstock integrble but whose points of nonbsolute summbility hve positive mesure. A key feture of the Henstock integrl is tht it is nonbsolute: n integrble function need not hve n integrble bsolute vlue. The convention I 0 for n unbounded intervl performs essentilly the sme trunction tht is done with improper Riemnn integrls nd the Cuchy extension of Lebesgue integrls. A consequence of this is tht there re no improper Henstock integrls. This fct is cptured in the following theorem, which is proved for finite intervls in [3]. Theorem 1. Let f be rel-vlued function on [, b] [, ]. Then fexists nd equls A R if nd only if f is integrble on ech subintervl [, x] [, b] nd f exists nd equls A. lim x b x Lebesgue integrls cn be chrcterised by the fct tht the indefinite integrl F(x x f is bsolutely continuous. A similr chrcteristion is possible with the Henstock integrl. We need three definitions. Let F :[, b] R. WesyF is bsolutely continuous (AConsetE [, b] if for ech ɛ>0thereissomeδ>0such tht N F(x i F(y i <ɛfor ll finite sets of disjoint open intervls {(x i, y i } N with endpoints in E nd N (y i x i <δ. We sy tht F is bsolutely continuous in the restricted sense (AC if insted we hve N sup x,y [x i,y i ] F(x F(y <ɛ under the sme conditions s with AC. And, F is sid to be generlised bsolutely continuous in the restricted sense (ACG iff is continuous nd E is the countble union of sets on ech of which F is AC. Two useful properties re tht mong continuous functions, the ACG functions re properly contined in the clss of functions tht re differentible lmost everywhere nd they properly contin the clss of functions tht re differentible nerly everywhere (differentible except perhps on countble set. A function f is Henstock integrble if nd only if there is n ACG function F with F f lmost everywhere. In this cse F(x F( x f.forn unbounded intervl such s [0, ], continuity of f t is obtined by demnding tht lim x F(x exists. We hve given n exmple tht shows tht not ll derivtives re Lebesgue integrble. However, ll derivtives re Henstock integrble. This leds to very strong version of the Fundmentl Theorem of Clculus. A proof cn be pieced together from results in [3]. Theorem. (Fundmentl Theorem of Clculus I Let f :[, b] R. Then f exists nd F(x x f for ll x [, b] if nd only if F is ACG on [, b], F( 0, nd F f lmost everywhere on (, b. If d x f exists nd f is continuous t x (, b then f f (x. June/July 001] 55 dx

3 II Let F :[, b] R. ThenFisACG if nd only if F exists lmost everywhere on (, b, F is Henstock integrble on [, b], nd x F F(x F( for ll x [, b]. Here is useful sufficient condition for integrbility of the derivtive: Corollry 3. Let F :[, b] R be continuous on [, b] nd differentible nerly everywhere on (, b. ThenF is Henstock integrble on [, b] nd x F F(x F( for ll x [, b]. The improvement over the Riemnn nd Lebesgue cses is tht we need not ssume the integrbility of F. Integrtion nd differentition re now inverse opertions. To mke this explicit, let A be the vector spce of Henstock integrble functions on [, b] [, ], identified lmost everywhere. Let B be the vector spce of ACG functions vnishing t.let e the integrl opertor defined by [ f ](x x f for f A.LetD be the differentil opertor defined by D[ f ](x f (x for f B.The Fundmentl Theorem then sys tht D I A nd D I B. 3. DIFFERENTIATION UNDER THE INTEGRAL SIGN. Theorem. Let f :[α, β] [, b] R. Suppose tht f (, y is ACG on [α, β] for lmost ll y (, b. ThenF: f (, y dy is ACG on [α, β] nd F (x f 1(x, y dy for lmost ll x (α, β if nd only if xs y f 1 (x, y dydx y xs f 1 (x, y dx dy for ll [s, t] [α, β]. (1 Proof. Suppose F is ACG nd f (x, y dy f x 1(x, y dy. Let [s, t] [α, β]. By the second prt of the Fundmentl Theorem, pplied first to F nd then to f (, y, xs y f 1 (x, y dydx F(t F(s y [ f (t, y f (s, y] dy ( y xs f 1 (x, y dx dy. Now ssume (1. Let x (α, β nd let h R be such tht x + h (α, β. Then, pplying the second prt of the Fundmentl Theorem to f (, y gives x x y f 1 (x, y dydx y x x f 1 (x, y dx dy y [ f (x + h, y f (x, y] dy y f (x + h, y dy y f (x, y dy. (3 56 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108

4 And, F 1 (x lim [F(x + h F(x] h 0 h 1 lim h 0 h x x y f 1 (x, y dy f 1 (x, y dydx for lmost ll x (α, β. The lst line comes from the first prt of the Fundmentl Theorem. Repeting the rgument in hows tht x F F(x F(α for ll x [α, β].hence,f is ACG α on [α, β]. The theorem holds for α<β nd < b. Prtil versions of the theorem re given in [, p. 357] for the Lebesgue integrl nd in [,p.63] for the wide Denjoy integrl, which includes the Henstock integrl. From the proof of the theorem it is cler tht only the linerity of the integrl over y [, b] (( nd (3 comes into ply with this vrible. Hence, we hve the following generlistion. Corollry 5. Let S be some set nd suppose f :[α, β] S R.Let f(, y be ACG on [α, β] for ll y S. Let T be the rel-vlued functions on S nd let L be liner functionl defined on subspce of T. Define F :[α, β] R by F(x L[ f (x, ]. Then F is ACG on [α, β] nd F (x L[ f 1 (x, ] for lmost ll x (α, β if nd only if s L[ f 1 (x, ] dx L s f 1 (x, dx for ll [s, t] [α, β]. If f (x, y x α g(x, y dx then f (, y is utomticlly ACG. This gives necessry nd sufficient conditions for interchnging iterted integrls. Corollry 6. Let g :[α, β] [, b] R. Suppose tht g(, y is integrble over [α, β] for lmost ll y (, b. Define G(x x g(x, y dx dy. Then G is α ACG on [α, β] nd G (x g(x, y dy for lmost ll x (α, β if nd only if xs y g(x, y dydx y xs g(x, y d x dy for ll [s, t] [α, β]. ( Combining Corollries 5 nd 6 gives necessry nd sufficient conditions for interchnging summtion nd integrtion. Corollry 7. Let g :[α, β] N R nd write g n (x g(x, n for x [α, β] nd n N. Suppose tht g n is integrble over [α, β] for ech n N. Define G(x x n1 g α n(x dx.thengisacg on [α, β] nd G (x n1 g n(x for lmost ll x (α, β if nd only if June/July 001] 57

5 xs n1 g n (x dx n1 xs g n (x dx for ll [s, t] [α, β]. (5 The Fundmentl Theorem nd its corollry yield conditions sufficient to llow differentition under the integrl. Corollry 8. Let f :[α, β] [, b] R. i Suppose tht f (, y is continuous on [α, β] for lmost ll y (, b nd is differentible nerly everywhere in (α, β for lmost ll y (, b. If (1 holds then F (x f 1(x, y dy for lmost ll x (α, β. ii Suppose tht f (, y is ACG on [α, β] for lmost ll y (, b nd tht b f 1(, y dy is continuous on [α, β]. If (1 holds then F (x f 1(x, y dy for ll x (α, β. Here is n exmple of wht cn go wrong when one differentites under the integrl sign without justifiction. In 1815 Cuchy obtined the convergent integrls x0 { } sin(x cos(x cos(sx dx 1 [ π cos sin ]. He then differentited under the integrl sign with respect to s nd obtined the two divergent integrls x0 x { } sin(x cos(x sin(sx dx s [ π sin ± cos ]. These divergent integrls hve been reproduced ever since nd still pper in stndrd tbles tody, listed s converging. This story ws told in [7]. REFERENCES 1. R.G. Brtle nd D.R. Sherbert, Introduction to rel nlysis, Wiley, New York, V.G. Čelidze nd A.G. Džvršeǐšvili, The theory of the Denjoy integrl nd some pplictions (trns. P.S. Bullen, World Scientific, Singpore, R.A. Gordon, The integrls of Lebesgue, Denjoy, Perron, nd Henstock, Americn Mthemticl Society, Providence, E.W. Hobson, The theory of functions of rel vrible nd the theory of Fourier s series, vol. II, Dover, New York, R.L. Jeffrey, The theory of functions of rel vrible, University of Toronto Press, Toronto, K. Rogers, Advnced clculus, Merrill, Columbus, E. Tlvil, Some divergent trigonometric integrls, Amer. Mth. Monthly, 108 (001 pp ERIK TALVILA holds B.Sc. from the University of Toronto, n M.Sc. from the University of Western Ontrio nd Ph.D. from the University of Wterloo. After visiting position t the University of Western Ontrio nd postdoctorl fellowship t the University of Illinois in Urbn he is now t the University of Albert. His min reserch interests re Henstock integrtion nd its pplictions to hrmonic functions, potentil theory, nd integrl trnsforms. He spends his nonmthemticl moments skiing nd rock climbing (not simultneously nd plying unusul bord gmes. University of Albert, Edmonton AB Cnd T6G E etlvil@mth.ulbert.c 58 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 108