The Frullani integrals. Notes by G.J.O. Jameson

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1 The Frullni integrls Notes b G.J.O. Jmeson We consider integrls of the form I f (, b) f() f(b) where f is continuous function (rel or comple) on (, ) nd, b >. If f() tends to non-zero limit t, then the seprte integrls of f()/ nd f(b)/ diverge t, nd similr comment pplies t infinit. The point is tht under suitble conditions, the integrl of the difference converges. The bsic theorem is s follows. I do not know mn books or rticles where proofs re given: two references (for which I m grteful to Nick Lord) re [Fer, p ] nd [Tr]. FRU THEOREM. Consider the following conditions: (C) f() c s + ; (C2) f() c s ; (C3) there eists K such tht F () K for ll >, where F () f(t) dt. Under conditions (C) nd (C2), we hve Under conditions (C) nd (C3), we hve d, I f (, b) (c c )(). () I f (, b) c (). (2) Proof. We m ssume tht b > : interchnging nd b then gives the cse b <. Let < δ < X. The substitution gives So X δ X δ f() f(b) f() d d X X f() f() d. d f() d I(δ) I(X), bδ X f() f() d d

2 where we write I(r) br r for n r >. Let ε > be given. Under condition (C), there eists δ > such tht if < bδ, then f() c ε. Then for δ δ, we hve I(δ) c () + r (δ), where hence r (δ) r (δ) So I(δ) c () s δ +. X. f() d f() c d, ε d ε(). In ectl the sme w, condition (C2) implies tht I(X) c () s Under condition (C3), integrtion b prts gives hence [ F () I(X) ] bx X + I(X) 2K X + K X X F () 2 d, 3K d < 2 X, so I(X) s X. (Of course, this lso shows tht sttement could be tken s the hpothesis insted of (C3)). f() d converges; this In prticulr, I f (, b) equls (c c ) log b in cse (), nd c log b in cse (2). We record number of prticulr emples which re trnsprentl cses of () or (2), without repetedl writing out the integrl epressions: f() I f (, b) e /( + 2 ) cos e cos sin / tn tnh π(log log b) 2 log log b Among these emples, cos is the onl one stisfing (C3), but not (C2). A simple lterntive proof for this cse is given in [Jm, p. 28]. 2

3 Mn further emples of Frullni integrls re given in [AABM]. We now stte simple etension of the Theorem. FRU2. Let G() n j m jf( j ), where j > for j n nd n j m j. If f stisfies (C) nd (C2), then G() d (c c ) If f stisifes (C) nd (C3), the sme pplies with c replced b. B (), n m j log j. (3) Proof. Write M j m + + m j. B Abel summtion, since M n, j n G() M j [f( j ) f( j+ )]. j G() d (c n c ) M j (log j log j+ ) (c c ) j n m j log j. j Double integrl method for (). For monotonic f, the following is n lterntive method for () (but not (2)). The method hs been in circultion for long time, t lest for the specil cse f() e. For emple, it cn be seen in [Cou, p. 24]. Hence Note tht for, >, b d d f() d d f() f (). [ ] b f () d f() f(b) f(). Since f () is of constnt sign, reversl of the following double integrl is justfied: But So I f (, b) b f () d d [ f () d f() b f () d d. (c c ). b I f (, b) (c c ) d (c c )(). 3

4 Some pplictions of the cse f() e. This is undoubtedl the best known Frullni integrl. Written out eplicitl, the sttement is e e b d. (4) The substitution log trnsforms (4) to So, for emple, (cf. [BM, p. 97]). Now consider the integrl I b d. log log d log 2 2 ( e )( e b ) d. Integrtion b prts gives I A + J, where [ A ( e )( e b ) J ( e + be b ( + b)e (+b)). Since < e < for >, we hve ( e )( e b ) < b, hence A. So b (3), Net, consider the function I J ( + b) log( + b) log b log b. E() e t B reversl of the implied double integrl, one sees esil tht E() d. Now let I 2 t dt. e E() d, where >. Reversing the double integrl nd ppling (4), we find I 2 e e t t e t t dt d e d dt t e t ( e t ) t log( + ). 4 dt,

5 Applictions of the cse f() cos We describe two pplictions of this cse. First, since 2 sin sin b cos( b) cos( + b), we hve, for > b >, Second, consider the integrl sin sin b I 3 d 2 log + b b. sin 5 2 d. Since 6 sin 5 sin 5 sin 3 + sin 5, we hve b (3) 6I 3 [ sin5 + ( cos 5 cos cos 5) d ( log 5 log log 5) 5 log 3 5 log 5. Integrls of this tpe re discussed more generll in [Tr]. References [AABM] [BM] Mtthew Albno, Tewodros Amdeberhn, Erin Beerstedt nd Victor H. Moll, The integrls in Grdshten nd Rzhik. Prt 5: Frullni integrls, Scientic, Series A: Mthemticl Sciences 9 (2), 3 9. George Boros nd Victor H. Moll, Irresistible Integrls, Cmbridge Univ. Press (24). [Cou] R. Cournt, Differentil nd Integrl Clculus, vol. II, Blckie & Son (936). [Fer] W. L. Ferrr, Integrl Clculus, Oford Univ. Press (958). [Jm] G. J. O. Jmeson, Sine, cosine nd eponentil integrls, Mth. Gzette 99 (25), [Tr] J. Trinin, Integrting epressions of the form sinn nd others, Mth. Gzette m 94 (2), updted 6 November 25 5