Polygon 2011 Vol. V 81

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1 Polygon Vol. V 8 A NOTE ON THE DEFINITE INTEGRAL ln( ) d. Shakil Dparmn of ahmaics iami Dad Collg, Hialah Campus iami, FL 33, USA mshakil@mdc.du ABSTRACT Th dfini ingral ln( ) d, which involvs logarihmic, ponnial and powr funcions, occurs frqunly in many problms of applid scincs, such as physics, nginring, probabiliy, rliabiliy horis, c. Th purpos of his papr is o giv analyical proof of his ingral hory. ln( ) d, wih som hisorical background and applicaions in probabiliy Ky words: Incompl ingral, gamma funcion, digamma funcion, acdonald funcion, llin ransform, nropy, probabiliy dnsiy funcion. SC : A5, 6A36, 8A5, 33B5, 6G3, 94A5

2 Polygon Vol. V 8. INTRODUCTION Th dfini ingral ln( ) d, whr and is an ingr, (.) occurs frqunly in many problms of applid scincs, such as physics, nginring, probabiliy, rliabiliy horis, c. Rcnly, w cam across his ingral in som problms on nropis of rcord valus. For ampl, h compuaion of Shannon s nropy of an absoluly, ha is, coninuous random variabl wih a probabiliy dnsiy funcion f () f ) H f ( )ln[ f ( )] d (, (.) can b rducd o h valuaion of h ingral (.) and similar ingrals (s, for ampl, Shakil (3), Zahdi and Shakil (6), among ohrs). Afr a horough sarch, no analyical valuaion or any hisorical vidnc of h ingral (.) was found in h availabl liraur. Only som rfrncs appar, for ampl, in Gradshyn and Ryzhik (98), Birns d Haan (867), and Erdlyi (954, Vol. ). In his aricl, an analyical valuaion of h ingral (.) is prsnd, which is basd on som compl mahmaical rsuls. Th organizaion of his papr is as follows. Scion conains hisorical background of h ingral (.). Th analyical valuaion of h ingral (.) is prsnd in Scion 3. Applicaions of h ingral in h compuaions of nropis of rcord valus corrsponding o som coninuous probabiliy disribuions ar discussd in Scion 4. Som concluding rmarks ar givn in Scion 5.. SOE HISTORICAL REARKS As poind abov, h ingral (.) occurs frqunly in h compuaions of nropis of rcord valus corrsponding o coninuous probabiliy disribuions. A formula for h ingral can b found in Gradshyn and Ryzhik (98, pag 576), as givn blow ln d ln, whr R, R. No analyical proof of h ingral is givn in Gradshyn and Ryzhik (98), cp h rfrncs of Birns d Haan (867) and Erdlyi (954, Vol. ). Our sarch coninud. W wr abl o find a 957 rprin of Birns d Haan's 867 "Nouvlls Tabls D'Ingrals Dfinis," publishd by Hafnr Publishing Company, Nw York and London. Th abov ingral appars in Birns d Haan (957, pag 496, Tabl 353, Formula Numbr ) as follows q l. p d p q / pz p lq, whr Z / p is usd for gamma funcion and lq dnos h naural logarihm of q. No analyical

3 Polygon Vol. V 83 proof of h ingral is givn in Birns d Haan's 867 diion, cp a rfrnc o his own 86 publicaion of Tabls D'Ingrals Dfinis, publishd as Volum VIII of h moirs of h Royal Acadmy of Scincs of Amsrdam. I is inrsing o no ha David Birns d Haan (8-895) was a profssor of mahmaics a Lidn Univrsiy. H is known for compiling abls of ingrals. Following h nominaion by D. Birns d Haan and H.G. van d Sand- Bakhuyzn, a docora in mahmaics and asronomy, honoris caus was confrrd upon Thomas Joanns Sils by Lidn Univrsiy in Jun, 884. For mor on h lif and work of D. Birns d Haan, h inrsd radr is rfrrd o Shldon (9), Schrk (955), and Talvila (). In Erdlyi (954, Vol. I, pag 35), h abov ingral appars as h llin ransform of f ( ) ln, R( ), givn by [ ln ] s d s s s ln, Rs R, whr s dnos a compl numbr. Bu no drivaion or proof of h ingral is givn in Erdlyi (954, Vol. I). ln( ) 3. ANALYTICAL EVALUATION OF THE DEFINITE INTEGRAL d In his scion, an analyical valuaion of h abov ingral is providd. For his, w considr a mor gnral class of an incompl ingral givn by b a ( ), ; b [ln ] d, whr, a, b,, (3.) and is an ingr. Th ingral (3.) is usd in h sudy of h gnralizd invrs Gaussian disribuion, whos probabiliy dnsiy funcion is givn by f ( ), whr, a, b, ; b b a. (3.) For dails on (3.), s, for ampl, Chaudhry and Zubair (), and rfrncs hrin. In

4 Polygon Vol. V 84 (3.), ; b is givn by b a b ; b d K ab a, (3.3) whr a, b, and K is acdonald funcion, (s, for ampl, Gradshyn and Ryzhik (98), and Chaudhry and Zubair (), among ohrs). Th chi-squar, ponnial, Erlang, gamm log-normal, Raligh, and Wibull probabiliy dnsiis ar h spcial cass of h probabiliy dnsiy funcion (3.), and can b asily drivd by simpl ransformaions of h variabl or by aking spcial valus of h paramrs b, and. A closd form soluion of h incompl ingral, ; b of, ; b, whn as givn in (3.) abov is no known. Howvr, a soluion and A soluion of an incompl ingral of h form, is givn in Gradshyn and Ryzhik (98, pag 34). ( ) b a b a [ln ] d, (3.4), a, b and, can b found in Gradshyn and Ryzhik whr, (98), for, whr,,, 3,. A soluion for and Chaudhry and Ahmad (99). Th incompl ingral, ; b h following funcional rcurrnc rlaion, ; b, ; bb, ; a b a, whr, a, b, is givn in as givn in (3.) saisfis b a, ; b [ln ], (3.5) and, (s, for ampl, Chaudhry (994), among ohrs). If, h abov rcurrnc rlaion (3.5) holds ru a, b, and. Howvr, for, h following funcional rcurrnc rlaion holds

5 Polygon Vol. V 85, ; b, ; b b, ; a b a, whr, b,, (3.6), ; b a and, (s, for ampl, Chaudhry (994), among ohrs). In ordr o valua h ingral ln( ) d, whr and is an ingr, no h sinc, subsiuing, a, b, and in (3.6), w obain h following rcurrnc rlaion, (3.7) whr, ;,, ;,, ;,, ;, ln( ) d From (3.7) and (3.8), w hav. (3.8) ( ) ln( ) d ( ) ln( ) d [ln( )] d (3.9) ( ) ln( ) d No ha h scond ingral on h righ-sid of (3.9) rprsns h clbrad gamma funcion d ( givn by ). Hnc (3.9) bcoms d. ( ) ln( ) d ( ) ln( ) d ( ), (3.) whr is an ingr and which implis ha. Combining (3.7), (3.8) and (3.), i follows ha, ;,, ;, ( ), (3.)

6 Polygon Vol. V 86, (3.) ;,, ;, ( ), From (3.) and (3.), h following rcurrnc rlaion follows ( ), ;,, ;, ( ) ( ). Procding in his mannr, w hav ( ) ( ) ( ) 3, ;,, ;, () ( )! ( ) ( ) ( ) ( () () ( ) ( ), ;,, (3.3)!! ( )!! ) Bu, ;, ln( ) d ln( ), (3.4) whr lim ln( m) m 3 m is Eulr s consan, (s, for ampl, Gradshyn and Ryzhik (98, p. 573), among ohrs). Applying (3.4) in (3.3), and simplifying, w hav!, ;, ) ln( ) ( 3! ln( ) 3. (3.5) Noing h following propris of digamma funcion ( ) and ( ), 3 and applying in (3.5), w hav is an ingr,

7 Polygon Vol. V 87!, ;, ) ( ) ln( ). (3.6) ( Combining (3.8) and (3.6), i follows ha, ;, ln( ) d! ( ) ln( ) Consqunly, in (3.7), rplacing by ( ), w obain, ;. (3.7) ( )!, ln( ) d = ( ) ln( ) ( ) Noing h following propris of digamma and gamma funcions ( ) ( ) and ( )! ( ), and applying in (3.8), h rquird prssion for h ingral (.) is obaind as follows. (3.8), ;, ln( ) d ) ln( ) ( ) whr is an ingr,, and ( ) and ( ) Subsiuing in (3.9), w g, in paricular, h following ingral, ;, ln( ) d ( ) ( ) (, (3.9) dno gamma and digamma funcions., (3.) whr is an ingr. No ha h ingral on h lf-sid of (3.) can b dducd by diffrniaing boh sids of h following clbrad Eulrian ingral ( ) d, wih rspc o h paramr so ha

8 Polygon Vol. V 88 / ( ) Th drivaiv of ln ( ) givn by ln( ) d,. is calld digamma funcion or psi funcion, dnod by ( ), and is ( ) d d / ( ) ( ) ln ( ) d, whr is Eulr s consan. Thr iss a vas liraur on gamma and psi funcions. Hisorically, h gamma funcion was firs inroducd by Eulr (77 783) in his sudis of gnralizing h facorial! for any nonand nam ingr valus of. Lgndr (75 833) was firs o inroduc h noaion ( ) i as gamma funcion, whil Gauss ( ) usd h noaion ( ), which rprsns ( ). For dails on hisory, analysis, propris and applicaions of gamma and psi funcions, s, for ampl, Sibagaki (95), Davis (959), Arin (964), Srivasava and anocha (984), Tichmarsh (986), Nikiforov and Uvarov (988), Gauschi (998), Gourdon and Sbah (), Chaudhry and Zubair (), and rfrncs hrin. 4. SOE APPLICATIONS IN THE COPUTATION OF ENTROPY In his scion, applicaions of h ingrals (3.9) and (3.) in h compuaions of nropis of rcord valus, corrsponding o coninuous probabiliy disribuions, namly, uniform and ponnial disribuions, ar prsnd. For nropis of rcord valus, corrsponding o Wibull, Paro, normal, gamm b and Cauchy disribuions, and hir propris, s, for ampl, Shakil (3), and Zahdi and Shakil (6), among ohrs. L,, b i. i. d. obsrvaions from an absoluly coninuous disribuion funcion () F, wih a probabiliy dnsiy funcion f (). L R ( ) dno h im (ind) a which h h rcord valu is obsrvd, and l h h rcord valu R( ) b dnod by ( ). A rcord valu occurs a im,,,, whr n, if, i, largr han ach of h prvious valus,., is i ha is, Th firs obsrvaion is always a rcord valu, and if a rcord valu occurs a im, hn will b calld a rcord valu. Th

9 Polygon Vol. V 89 oin probabiliy dnsiy funcion of h rcord valus,, is givn by g (), (),, ( ) (,, ) f ( r ) f ( ) r { F( )}, (4.) r and h marginal probabiliy dnsiy funcion (pdf) of h h rcord valu ( ) is givn by g ( ) ( ) ln{ F( )} ( ) f ( ), (4.) For dails on (4.) and (4.), s, for ampl, Ahsanullah (4), among ohrs. 4.. UNIFOR DISTRIBUTION Dfiniion. A coninuous random variabl is said o hav a uniform disribuion ovr an inrval ) (,, ha is, ~ uniform (, ), if is pdf f () and cdf F( ) P( ) rspcivly, givn by and f ) I (, ( ); ( ) ( ) ar,, F ( ), ;, I ( whr (, ) ) rprsns h indicaor funcion dfind ovr h inrval (, ), and. In wha follows, wihou loss of gnraliy, for simpliciy of compuaions, w only considr h nropy of a rcord valu corrsponding o a uniform (, ), from which h nropy of a rcord valu corrsponding o any gnral uniform (, ) can asily b obaind by a simpl ransformaion of h paramr. L,, b a squnc of i. i. d. random variabls from a uniform parn disribuion in (, ) wih h cdf F and pdf givn by f, rspcivly,

10 Polygon Vol. V 9 and f F whr ) ( ; ) I (, ) ( ), (4..) ( ; ) I (, ) ( ), (4..) I ( rprsns h indicaor funcion dfind ovr h inrval (, ). (, ) L ( ) dno h h obsrvd rcord valu. Using (4..) and (4..) for f () and F () g rspcivly in (4.), ), h pdf of ( ) ( ( ), is givn by g ln ) ( ) ( ) ( I (, ) (, is an ingr, and whr )., (4..3) L h nropy of h h rcord valu ( ) from h uniform (, ) parn disribuion b dnod by H ( ), or simply H ( ). Thn, using (4..3) in h prssion (.) for Shannon s nropy, w hav H ( g ( ) g ( ) lng ( ) E ln ( ) ( ) ( ) ) d ln ln ln ( ) ( ) d. (4..4) Subsiuing ln, which givs d ( ) d, as, and as, h abov prssion (4..4) for h nropy of h h rcord valu ( ), rducs o

11 Polygon Vol. V 9 H ( ) ( ) ln ( ) d ( )ln( ) ln{ ( )} d ( ) ln{ ( )} ln( ) d ( ) ( ) d, from which, using Eulrian ingral and h ingral (3.), h nropy of h h rcord valu ( ) from h uniform (, ) parn disribuion is obaind as follows ( ) ( ) ( ) H ( ) ln( ) ln, whr is an ingr,, and ( ) is digamma funcion. 4.. EPONENTIAL DISTRIBUTION Dfiniion. A coninuous posiiv random variabl is said o hav an ponnial disribuion, wih man, if is pdf f () and cdf F( ) P( ) ar, rspcivly, givn by and f / ( ; ) I (, ) ( ), (4..) whr ) / I ( ), F ( ; ) (, ) (4..) I ( rprsns h indicaor funcion dfind on (, ). (, ) L,, b a squnc of i. i. d. obsrvaions from an ponnial ) ( parn disribuion. L ( ) dno h h obsrvd rcord valu. Using (4..) and (4..) for f () and F () rspcivly in (4.), g ( ), h pdf of ) ( ) (, is givn by g ( ) ( ) ln( F ( )) f ( ) ( )

12 Polygon Vol. V 9 ln{ ( ( ) )} / = / I(, )( ) = ( ) / (, ) ( ) I,,, 3,...,. (4..3), No ha ( ) ~ Gamma ( shap paramr, scal paramr ). Th nropy of a gamma disribuion is wll-known, bu, for h sak of complnss and o show h applicaions of ingrals (3.9) and (3.), w driv h nropy hr. L h nropy of h h rcord valu ( ) from h ponnial ) ( parn disribuion b dnod by H ( ), or simply H ( ). Thn, using (4..3) in h prssion (.) for Shannon s nropy, w hav H ( g ( ) g ( ) lng ( ) E ln ( ) ( ) ( ) ) d / / ln ( ) ( ) d ln( ) ln{ ( )} ( ) / d ( ) / d ( ) / ln( ) d. (4..4) Thus, using Eulrian ingral and h ingral (3.9) in (4..4), h nropy of h h rcord valu ( ) from h ponnial ( ) parn disribuion is givn by ( ) ( ) ( H ( ) ln( ) ln ), whr is an ingr,, and ( ) is digamma funcion. 5. CONCLUDING REARKS In his papr, w hav ampd o giv a nw analyical proof, wih som hisorical background, of h dfini ingral

13 Polygon Vol. V 93 ln( ) d ( ) ln( ) ( ), (6.) whr is an ingr,, and ( ) and ( ) dno gamma and digamma funcions. This ingral appars o b wll-known. I is usful in h sudis of probabiliy hory, nropis of rcord valus, ordr saisics, numrical analysis, physics, and hory of spcial funcions. Howvr, vn afr an nsiv sarch, w wr no abl o find any proof or drivaion of his formula in any book or ournal, or, a las, h nam of h discovrr of his formula. On mus no forg o acknowldg h conribuion of h mahmaician, who wo hundrd yars ago, provd or cam up wih such a gra formul which now has such rmndous applicaions in nropy, informaion, probabiliy and rliabiliy horis. Th accomplishmns of h pas and prsn mahmaicians can srv as pahfindrs o hir conmporary and fuur collagus. Th achivmns of many mahmaicians, and hir conribuions, boh small and larg, hav bn ovrlookd whn chronicling h hisory of mahmaics. By dscribing h acadmic hisory of hs prsonaliis wihin mahmaical scincs, w can s how h ffors of individuals hav advancd human undrsanding in h world around us. Hisory bars simony o hir achivmns, abiliis and accomplishmns. I should b h rsponsibiliy of h prsn mahmaical world o highligh h achivmns of h pas mahmaicians bhind such gra formulas and horis. A his poin i is clar o pos h following opn problm. Though h ingral (6.) appars o b wll-known, i will b inrsing if w wr abl o find h nam of is discovrr and ohr proofs of h ingral. REFERENCES. Abramowiz,., and Sgun, I. A. (97), Handbook of ahmaical Funcions, wih Formulas, Graphs, and ahmaical Tabls, Dovr, Nw York.. Ahsanullah,. (4). Rcord Valus Thory and Applicaions, Univrsiy Prss of Amric Lanham, D. 3. Arin, E., Th Gamma Funcion (964), Nw York, Hol, Rinhar and Winson. 4. Birns d Haan, D. (86), "Epos d la h ori, ds propri s, ds formulas d ransformaion ds m hods d' valuaion ds in grals d finis," C.G. Van dr Pos,

14 Polygon Vol. V 94 Amsrdam. 5. Birns d Haan, D. (867), Nouvlls abls d'in grals d finis, P. Engls, Lidn. 6. Birns d Haan, D. (957), Nouvlls abls d'in grals d finis, Hafnr, Nw York. 7.Chaudhry,. A., and Zubair, S.. (), On a class of Incompl Gamma Funcions wih Applicaions, Chapman & Hall/CRC, Boca Raon. 8.Chaudhry,. A. (994). On a Family of Logarihmic and Eponnial Ingrals occurring in Probabiliy and Rliabiliy Thory, J. Ausral. ah. Soc., Sr. B, 35, pp Chaudhry,. A., and Ahmad,. (99). On som infini ingrals involving logarihmic, ponnial and powrs, Procdings of h Royal Sociy of Edinburgh, A, 5..Davis, P. J. (959), Lonhard Eulr s Ingral: A hisorical profil of h gamma funcion, Amr. ah. onhly, 66, Erd lyi, A.(954), Tabls of Ingral Transforms, Vol., cgraw-hill, Nw York..Gauschi, W. (998), Th incompl gamma funcion sinc Tricomi, in: Tricomi s Idas and Conmporary Applid ahmaics, Ai Convgni Linci, 47, Accad. Naz. Linci, Rom, Gourdon,., and Sbah, P. (), Inroducion o h Gamma Funcion, World Wid Wb si a h addrss: hp:// numbrs.compuaion.fr.fr/consans/consans.hml. 4.Gradshyn, I., and Ryzhik, I. (98), Tabls of Ingrals, Sris and Producs, Acadmic Prss, N. Y. 5.Nikiforov, A. F., and Uvarov, V. B. (988), Spcial Funcions of ahmaical Physics, Birkhausr Vrlag, Basl, Grmany. 6.Schrk, D.J.E. (955), David Birns d Haan, Scripa ah.,, pp Shakil,. (3), Enropy Sudy of Rcord Valu Disribuions obaind from Som Commonly Usd Coninuous Probabiliy odls, asrs Thsis, Florida Inrnaional Univrsiy, Florid USA. 8.Shldon, E.W. (9), Criical rvision of d Haan's Tabls of Dfini Ingrals, Amr. J. ah. 34, pp Sibagaki, W. (95), Thory and applicaions of h gamma funcion, Iwanami Syon, Tokyo, Japan..Srivasav H.., and anoch H. L. (984), A Trais on Gnraing Funcions, John

15 Polygon Vol. V 95 Wily and Sons, Nw York..Talvil E. (), "Som Divrgn Trigonomric Ingrals," Amr. ah. onhly 8, No. 5, Tichmarsh, E. C. (986), Th hory of h Rimann Za-funcion, Oford Scinc publicaions, scond diion, rvisd by D.R. Hah-Brown. 3. Zahdi, H. and Shakil,. (6). Propris of Enropis of Rcord Valus in Rliabiliy and Lif Tsing Con. Communicaions in Saisics, 35: 997.