P.L. Chebyshev. The Theory of Probability

Transcription

1 P.L. Chbyshv Th Thory of Probability Traslatd by Oscar Shyi Lcturs dlivrd i as tak dow by A.M. Liapuov Brli, 4 Oscar Shyi !" 936 Cotts Itroductio by th Traslator Forword by A.N. Krylov Chaptr. Dfiit Itgrals.. Prliiary Rarks ad th Itgrals of th First Group.. Itgrals of th Scod Group.3. Itgrals of th Third Group {Icorporatig th Eulr Itgrals}.4. Itgrals of th Fourth Group {Icorporatig th Fourir Itgrals} Supplt Chaptr. Th Thory of Fiit Diffrcs.. Dirct Calculus of {Fiit} Diffrcs.. Ivrs Calculus of Fiit Diffrcs.3. Itgratio of Equatios i Fiit Diffrcs Chaptr 3. Th Thory of Probability 3.. Th Laws of Probability 3.. O Mathatical Epctatio 3.3. O th Rptitio of Evts 3.4. Applicatio of th Thory of Probability to th Tratt of Obsrvatios Id of Nas Itroductio by th Traslator. Pafuty Lvovich Chbyshv (Tchébichf (8 894 was o of th two ost it Russia athaticias of th 9 th ctury (th scod, or, rathr, th first o was Lobachvsky. I th thory of probability, h provd th law of larg ubrs i a gral for ad thoroughly studid th coditios for th ctral liit thor (a tr itroducd by Polya i 9 providig, i 887, th cssary frawork for its dfiitiv ivstigatios carrid out by his forr studts, Markov ad Liapuov. Chbyshv also dalt o both of ths ost iportat topics i his lcturs o probability thory that h dlivrd at Ptrsburg Uivrsity fro 86 to 88. Th authoritativ Russia sourcs about his work ar [ 3] ad I yslf publishd a papr [4] discussig his lcturs o probability; its coclusios would b a usful supplt to this Itroductio ad to y ots i th ai tt blow. I 936, Alksi Nikolavich Krylov, a aval architct ad a applid athaticia, publishd Chbyshv s lctur ots as writt dow by Liapuov i (s Krylov s Forword which I a ow prstig i traslatio blow. I 999, y traslatio of th lctur ots appard as a icrofich ditio publishd by Vrlag Dr. Häsl-Hohhaus i thir sris Dutsch Hochschulschrift, No. 665, Eglsbach, but th copyright to ordiary publicatio is i.

2 . This traslatio cotais Chbyshv s ow footots as wll as ots by Krylov ad (y ots ar i curly brackts, all of which ar collctd at th d of th appropriat subsctios, ad a Id of as copild by. Th radrs of th Russia ditio udoubtdly oticd ay dozs of isprits i athatical forulas ad a alost total absc of priods aftr displayd forulas ad (ffctivly w stcs bgiig th with a lowr-cas first lttr of thir first words. Th oly possibl plaatio of this sad stat of affairs ss to b that Krylov, i spit of his tstioy providd i th Forword, had ot rwritt th origial auscript hislf. Th typ-sttr (a apprtic? had cotributd to th wrckig of th forulas; ad hardly ayo rad th proofs. I hav corrctd th isprits without spcial otic but I did ot chck all th forulas; ad wh I writ sothig lik Chbyshv had ot, th fault ca wll li lswhr rathr tha with hi or Liapuov. 3. I had ot iprovd o Chbyshv s styl of oral prstatio ad I attptd to prsrv his athatical triology ad otatio [valu of itgral; actly cotrary vts; athatical pctatio (droppig howvr th adjctiv i 3.3; quatios i fiit diffrcs; li P ] or usd it ithr or oft tha Chbyshv (p [f (] or throughout rathr tha cptioally (i istad of #. Th, I yslf itroducd th otatio C ad!. O th othr had, disrgardig a sigl cptio, I hav ot rtaid Chbyshv s otatio ( 3.3 M L M L p( for th sus of probabilitis. Chbyshv trasford such sus ad arrivd at appropriat itgrals with liits t o ad t (say big fuctios of L ad M rspctivly. Just th sa, I hav ot prsrvd his siilar us of lttr S istad of i Th uratio of th forulas was chaotic; furthror, it did ot abl th author to rfr to his prvious rsults ad h rwrot ay forulas ti ad ti agai. I ordrd his uratio assigig ubrs to thos forulas which wr arkd by astrisks or Grk or Roa lttrs, ad th ubrs ar ow ruig coscutivly through ach chaptr with o sparat systs of th apparig aywhr ayor. I additio, I ubrd ay or forulas dd for rfrcs by usig a paralll syst of Roa urals. Not that th forulas which Chbyshv icludd i his ai syst ar ow pritd i bold typ. I coclusio, I ot that i a fw cass Chbyshv had ot show th cssary itrdiat stps ( 3..5 ad Nots " " ". I hav subsqutly dscribd Chbyshv s work i probability [5, Chapt. 5]. I particular (pp. 5 ad 6, I otd that Tikhoadritsky, i 898, statd that i 887 Chbyshv had rarkd that it was cssary to trasfor th tir thory of probability. I also dscribd a pisod provig that Chbyshv had cosidrd th Ria gotry ad th copl-variabl aalysis as trdy disciplis.. I this rspct Markov followd his tachr. Rfrcs. Brsti, S.N. (945, Chbyshv s work i th thory of probability. (Coll. Works, vol. 4. N.p., 964, pp Trasl. i DHS 656, 999, pp (947, Chbyshv ad his ifluc o th dvlopt of athatics. Uchy Zapiski Mosk. Gos. Uiv., No. 9, pp Trasl. i Math. Scitist, vol. 6,, pp Youshkvich, A.P. (97, Chbyshv. Dict. Scit. Biogr., vol 3, pp Shyi, O. (994, Chbyshv s lcturs o th thory of probability. Arch. Hist. E. Sci., vol. 46, pp (4, History of th Thory of Probability to th th Ctury. Brli.

3 Chaptr. Dfiit Igrals.. Prliiary Rarks ad th Itgrals of th First Group... W shall call dfiit oly such itgrals whos liits ar costat agituds. Thus, w shall ot cosidr dfiit itgrals of th typ d l. May dfiit itgrals ca b dducd fro idfiit itgrals but w shall oly trat such of th which caot b dtrid i a idfiit for. For apl, th itgral p ( d caot b dtrid bcaus it rprsts a trascdtal fuctio ukow to us. At th sa ti, w ca fid its valu if w add th liits ad $; th itgral will th qual #%/. This happs bcaus w do ot hav such a fuctio that shows how th quatity of th ara OAMB chags dpdig o th chag of. This, howvr, dos ot prclud th possibility of dtriig th quatity of all th ara rstrictd by th curv y p (. (i I such ivstigatios it is ipossibil to apply a dirct approach ad w ust prforc choos a idirct way. For this raso th thods usd ar trly divrs ad oft urous. Dfiit itgrals ar sparatd ito svral groups ad spcial tricks ist for ach of ths. I additio, various scitists dtri o ad th sa itgral by diffrt thods. Our thod is this: Issuig fro kow doubl itgrals ad chagig th ordr of itgratio, w shall dtri th dfiit itgrals. W do ot ai at drivig th valu of a crtai dfiit itgral; o th cotrary, w shall rathr dtri various dfiit itgrals fro a giv doubl itgral. Thus, th chag i th ordr of itgratio will b th foudatio of all our coclusios. Such a chag is kow to b oly possibl if th itgral ight b cosidrd as th liit of a su; this, i tur, oly holds if th itgrad rais fiit withi th liits of itgratio. For apl, th itgral d caot thrfor b rgardd as th liit of a su bcaus th itgrad is ifiit at. This is v obvious also bcaus th itgrad taks positiv valus for ay lyig withi th liits of itgatio so that th quality abov is ipossibl. Th sa rark ay b ad cocrig th itgral d l (. Its valu ca b rprstd i a sowhat diffrt way. Sic & i cos& isi&, it follows that &i l (cos& isi&.

4 Sttig hr & % (, w hav d % ( i l (, %i (. This itgral thus has a ifiit st of diffrt valus, all of th iagiary. This ca b plaid by otig that, whil itgratig, w could hav ld through iagiary valus; ay paths of itgratio ar hr possibl which idd plais th idfiitss of th itgral. Not. {I do ot rproduc th appdd figur (which Chbyshv had ot tiod i his tt. Th quatio of th curv thr rprstd was ot spcifid ad it was draw wrogly: at poit B(; th curv (i was ot prpdicular to th ais Oy.}... W bgi with th itgral y β y α y d dy. For th itgrad to rai fiit it is csary that ' > ad ( >. Ths coditios will idd rstrict our ivstigatio. W hav y d y /y, thrfor y d /y. Th β α dy y β l ((/' α y d dy β d α y dy. Howvr, β α y dy β α α β so that α β d l ((/ '. ( This itgral is alost th ost iportat o. W shall idicat o of its applicatios. Supposig that ( ad ', w hav α β d l. Assuig that taks diffrct valus fro to ( ad addig up th prssios obtaid, w gt

5 l [(!] (... ( d or d l [(!] (. This itgral thus allows us to prss th logarith of th product of atural ubrs which is sotis usful. Itgral ( is usually writt dow i a sowhat diffrt for. Suppos that l (/z, th d dz/z, α β d z α z β l z z dz l α β so that z α z l z β dz l('/(. ( W arrivd at this coclusio udr th assuptio that ' ad ( wr positiv agituds which of cours prsupposs that thy wr ral ubrs. Assuig howvr that thy ar iagiary agituds, ad issuig fro th itgral (, w ca scur so otio about th valu of th itgral sic d. W say so otio bcaus our assuptio lads to a o-rigorous coclusio, that, lik all suchlik ifrcs, fails to provid, as w shall s ow, th dsirabl rsults. Suppos that ' ci, ( ci whr c is so ral agitud. Th ic ic d l ( ad, sic [( ic ( ic /i] sic, it follows that sic i d l ( %i so that sic d %/. (ii W hav thus dtrid th valu of this itgral. Howvr, l ( is a idfiit agitud ad %( i should hav b tak istad of %i, ad alrady this circustac idicats that th rsult is doubtful bcaus th itgral ought to hav a sigl ad quit dfiit valu. I additio, th prssio obtaid dos

6 ot dpd o c so that th dtrid valu holds oly for positiv c s whras, accordig to th assuptio ad wh drivig this itgral, w hav oly suuposd that c was a ral agitud. Strictly spakig, w hav ot thus obtaid th rsult sought.... Cosidr ow th itgral β Now dy y sic d. y sic d y y si c y y c cos c d y sic y (c/y y cos c d. Thrfor but y si c d (c/y y cos c d y cos c d y cosc y y y ( c si c d (/y (c/y y si c d. Thus y si c d (c/y (c /y y si c d ad it follows that y c si c d c y, y y cos c d. (3; 4 c y Ths itgrals ar vry iportat bcaus thy abl us to prss th algbraic fractios i (3; 4 through dfiiot itgrals which vry ofdt siplifis th solutio of ay probls. Not that th itgrals (3; 4 ca also b dtrid i th idfiit for. Supposig that y i ths itgrals gradually dcrass tdig to zro, w hav, i th liit, si c d /c, cos c d. It is asy to s, howvr, that th obtaid itgrals possss o dirct ss bcaus ithr si c or cos c approachs ay dfiit liit as icrass to ifiity. Tak by thslvs, ths itgrals ar thrfor idfiit agituds. Nvrthlss, wh cosidrd as th liits of itgrals (3; 4, thy hav quit a dfiit valu as foud by us.

7 ui du p u ui π p. p Substitutig positiv ubrs,, istad of, w gt i th right sid factors p ( p, p ( p, ad, ultiplyig th obtaid itgrals by A, A, rspctivly ad addig togthr th rsults w shall fid (/ [A p ( ui A p ( ui A p ( ui du A p ( ui ] p u π [A p ( ui A p ( ui ]. p Suppos ow that f ( A p ( A p ( (i ad th last itgral will bco f ui ( f ( ui du p u π f (p. (8 p W hav thus drivd th rquird forula. Th o-rigor of this drivatio cosists i that (8 holds oly for such fuctios which ay b padd ito th sris (i whr A, A, ar so costat cofficits, whras, havig o critria for distiguishig btw fuctios that ay, ad ay ot b padd ito such a sris, w cosidr (8 as though valid for ay fuctio. Assuig f ( A p ( /a A p ( /a w shall fid f aui ( f ( aui du p u π f (ap. p Diffrtiatig this quality with rspct to a, w shall obtai ( f ( aui f aui ui du p u π f (a p ad thrfor ( f ( aui f aui i udu p u π f (a p. Dotig f ( * ( w gt

8 ϕ aui i ( ϕ ( aui udu p u π * (ap. (9 If ow f ( (, th itgral (8 will provid ψ ( ψ ( aui aui du p u π (ap. p Itgratig this quality with rspct to a btw th liits ad ' w shall fid ψ ( ψ ( aui α aui du p u π ψ ( α p ψ (. p Howvr, α ψ ( αui ψ ( (aui da ui α, ψ ( ψ ( αui ( aui da ui ad w obtai ψ ( αui ψ ( αui ui or ψ αui i du p u ( ψ ( αui du u( p u π ψ ( α p ψ ( p π ψ ( α p ψ (. ( p Itgrals (8 ( ar to b foud i a cotributio by Abl ad o of th is i Brtrad s writig, but th gral forula that itrsts us was first giv by Cauchy. W coclud hr th study of th itgrals of th scod group..3. Itgrals of th Third Group.3.. Th itgrads of th itgrals that w shall ow study cotai fuctios which could at first sight s to b algbraic whras i ssc thy ar spcial trascdtal fuctios. Ths itgrals ar of th typ λ d whr, is ay ubr. W shall thrfor bgi by sayig a fw words about W call algbraic oly such fuctios that ar th roots of th quatio A o y A y - A y - A - y A ( algbraic fuctios i gral. whr is a positiv itgral ubr ad th cofficits A o, A,, A ar so itgral fuctios of. Fuctios ot fittig i with this dfiitio o logr rprst algbraic fuctios so that, that caot satisfy our quatio at all valus of, is a trascdtal fuctio, but it bcos algbraic as soo as w assu that, is a cosurabl ubr. W shall ow idd cosidr th itgral ( at ay,. But w shall ot first of all that this itgral will hav a fiit valu oly for valus of, withi ad. Idd, for, > th dgr of th product,, will b highr tha ad th itgral, i virtu of a kow thor, will b ifiit. Ad for, <, sttig /z,, µ ad µ >, w hav

9 λ d µ z z dz. This as that th itgral will {agai} b ifiit. For, ± w obtai d (/ [l( ] $; d (/ z (/ z dz [ z ] z z dz $. If ow, ' i ( it will b cssary that ' b cotaid btw cosidr th itgral ( assuig that >, >. ad. Thus, w shall Not. {Chbyshv s tr.}.3.. W hav but λ λ d d λ d λ, ( d, ( 4 6 d,4 d,6 d I gral, howvr,, λ d [ ] λ λ d,, d, d bcaus, udr our coditios rgardig,, (, is always positiv. If, ' (i th li (, li ( ' (i. Sic ( ca b gativ it could s at first sight that (i ca b ifiit, but it is ot difficult to prov that this factor caot cd so boudary. Idd, (i (i l cos (( l i si (( l ad thrfor i ay cas li (, if oly ' is cotaid withi th boudaris idicatd abov. Thus, λ d λ λ 3 λ 5 Th scod itgral is

10 λ d z λ (/ z [ dz ] z λ z z dz. But, bcaus of th abov, th last itgral is qual to λ λ 3 λ 5 Ad so λ λ d ( λ 3 3 λ ( λ 3 λ 5 5 λ 3 λ ( W hav thus prssd our itgral as a sris whos su w still ought to dtri. Th siilarity of this pasio with th dcopositio of ratioal fractios ito partial fractios at oc arrsts our atttio. Idd, w ca prst ( as λ d λ λ λ 3 λ 3 Howvr, ay ratioal fractio f ( /F ( whr F ( has o ultipl roots ca b rprstd as f ( F( f ( F ( f ( F ( f ( 3 F ( 3 3 so that i our cas th roots of F ( ar, 3, 5, 7,,, 3, 5, 7, ad f ( / F ( at ay qual to o of th roots of F (. Ad w fuctio F ( satisfyig ths coditios. shall ow dtri th.3.3. Lt us cosidr th fuctio F ( cos( arccos whr is ay itgr. It is asy to show that F ( is a itgral fuctio Idd, substitutig arccos * ad otig that of th -th dgr. cos * [ *i *i ]/ w shall fid that F (cos * cos * [ *i *i ]/. Howvr, ±*i cos * ± i si * so that F (cos * {[cos * i si *] [cos * i si *]}/ {[cos * i si *] [cos * i si *] }/. Thus F ( cos( arccos (/ [( ( ]. (

11 Th trs that iclud odd dgrs of th root will dpd o it; howvr, it is ot difficult to s that thy will fially cacl out so that w shall obtai for F ( a itgral ratioal fuctio. Sic w hav (/ (/ (/ (/ (/ (/ 3 (/ (/ (/ (/ (/ (/ 3 ad th tr [ ] i ( will cotai oly gativ powrs of ; all such trs will fially cacl out with th trs cotaiig gativ powrs of [ ] ad oly th itgral part of this prssio will b lft. Dotig {hr ad i th squl} th tir part {of a fuctio} by E, w ca prss this rsult as F ( E {[[ ] / ]}. Cotributios cocrig th study of such fuctios ar aily du to Chbyshv so that th prssio ( is also calld Chbyshv polyoial. At prst, such ivstigatios ar icludd i ay writigs o itgral calculus, ad, for apl, i Eglad thy ca b foud i ay courss i itgral calculus udr th hadig Chbyshv s works. Zolotarv s studis cocr siilar issus; or prcisly, issus rlatig ot to circular but to lliptic fuctios. Accordigly, thy ar uch or coplicatd, but also lss iportat tha th works of Chbyshv. Suppos ow that whr is a itgr. Th F ( cos ( arcos A o A - A - W shall try to pad th fuctio /F ( ito partial fractios. To attai th roots of th quatio this goal w shall fid F ( or cos ( arcos ad prov that all of th ar diffrt. It is ot difficult to s that this ( k π arccos 4 whr k is ay itgr. Assuig that it taks diffrt valus fro to followig roots of this quatio: quatio is satisfid if (, w obtai th k, cos (%/4; k, cos (3%/4; k, 3 cos (5%/4; ; k µ, µ cos [(µ %/4]; k µ, µ cos [(4 µ %/4]; k, cos [(4 %/4]. Sic th quatio F ( {abov} is of dgr, it caot hav ay othr roots. It is s thrfor that all of its roots ar diffrt ad orovr ral. For this raso our pasio will b of th typ F( F ( F (. F ( µ µ Howvr, si(arccos F(, si{ arccos cos[(µ π / 4]} F( µ si[(µ π / 4] si[(µ π / ]. si[(µ π / 4]

12 But si[% (µ /] ( µ- so that F( µ ( µ-. si[(µ π / 4] Thus, cos(arccos si[(µ π / 4 ] ( cos[(µ π / 4] µ (/ ( µ- si[(µ π / 4] cos[(µ π / 4] whr th sus should tt coscutivly ovr µ,,,. W rarkabl forula ow hav th followig cos(arccos si[(µ π / 4] µ π ( µ- cos[( / 4] fro which, assuig that arccos * ad, w ca driv th kow forula cos * cos *. For cos* w hav cosϕ ( µ- si[(µ π / 4]. (3 cosϕ cos[(µ π / 4] W shall copar ow two trs of this forula, thos whr µ k ad Itroducig, i gral, µ ( k. - (µ ( µ- si[(µ π / 4] cosϕ cos[(µ π / 4] w hav - (k ( k- si[(k π / 4], cosϕ cos[(k π / 4] - ( k ( -k si[(4 k π / 4] cosϕ cos[(4 k π / 4] si[(k π / 4] ( k cosϕ cos[(k π / 4]. It is thrfor s that i our pasio th su of th trs qually distat fro th iddl is - (k - ( k ( k- ( k π si 4 [ ] cosϕ cos[(k π / 4] cosϕ cos[(k π / 4]

13 ( k- si[(k π / ] cos ϕ cos [(k π / 4] ( k- si[(k π / ]. si [(k π / 4] si ϕ] Forula (3 thus bcos cosϕ ( k- si[(k π / ] si [(k π / 4] si ϕ] (3 whr k should b chagd fro to iclusivly. Suppos ow that * tds to zro ad icrass idfiitly i such a way that th product *, which w shall dot by %,/, rais fiit. W shall try to fid th liit of our su udr ths coditios. W hav %,/*, thrfor λπ / ϕ cos( λπ / ( k- si[(k ϕπ / πλ] si [(k πϕ / λπ ] si ϕ ( k- si[(k ϕ / λ], si [(k ϕ / λ] si ϕ π / cos( λπ / ( k- si[(k ϕ / λ] ( λ / ϕsi [(k ϕ / λ] ( λ / ϕsi ϕ si[(k ϕ / λ] ( k- λ{si[(k ϕ / λ]/ ϕ]} [(siϕ/ ϕ]. Ad, sic i gral li [si (N*/*] * N, π / [ ] * ( k- (k / λ cos( λπ / λ[(k / λ] (k ( k- (k λ. Thus, w fid th followig prssio π / cos( λπ / ( k- (k (k λ (4 whr k should tak all th itgr valus fro to $. Evidtly, sic, coditio is hr dtrid by th, (4*/% *, $, it ay tak all possibl valus icludig icosurabl os. If, has valu of th typ (' i(, which is oly possibl if * taks iagiary valus (bcaus is a ral agitud, th obtaid forula is also valid i this cas, bcaus, wh dtriig th liit of th su at * ad $ by as plicatd i th diffrtial calculus (th thod basd o gotric

14 cosidratios that w applid i this cas {? } vidtly will ot prssio. do, w would hav arrivd at th sa Not. Alksv, N.N. (Itgral Calculus. Not. {Cf. Not to.3..}.3.4. W ay rprst th prssio (4 i such a way: π / cos( λπ / λ 3 3 λ 5 5 λ Supposig ow that, is cotaid btw th boudaris ad, ad coparig this quality with (, w shall hav λ d π /, <, <. (5 cos( λπ / W hav thus foud th itgral sought. It is sotis prssd i a z w shall obtai diffrt way. Substitutig π / cos( λπ / λ z z z dz λ z z z dz λ z z z dz. Howvr, λ z z z dz λ z z z dz λz z z dz ad thrfor λ z λ z z z dz π /. (6 cos( λπ / Assuig ow that, (i, w shall gt zβ i z zβ i z dz cos βz dz z z π / cos( βπi / π βπ / βπ /. Thus cos βz z z π / dz. (7 βπ / βπ / Ev or rarkabl is aothr odificatio of th itgral (5 to which substitutig z, d (/ z / dz: w ca arriv by λ / z z λ / / z z (/ z / dz dz π cos(λπ / π /, cos( λπ /.

15 Supposig hr that,/ /, w shall fid z z dz π cos[( / π ] π cos( π / π π siπ. If ow z /( w shall hav fially - ( d π siπ, < <. (8 I such a for this itgral is a particular cas of th Eulr itgral of th,- ( µ- d with, ad µ. W shall ow go o to cosidrig such first kid itgrals. Th Eulr Itgrals { }.3.5. A Eulr itgral of th scod kid is th itgral. d which is usually dotd /(. as a fuctio of.. Ad, as statd abov, th,- ( µ- d. itgral of th first kid is Thr is o spcial otatio for this itgral; howvr, so authors dot it by (,; µ, othrs by B (,; µ. W shall adopt th fist sybol, so that, rplacig. by,, by p ad µ by q, w hav p- ( q- d (p; q, (9 d /(. (3 I ordr to cosidr ths itgrals as liits of so sus it is cssary that th paratrs, p, q b positiv; othrwis, ach itgrad bcos ifiit at o of its liits. W shall thrfor assu that >, p >, q >. Tru, th itgrad i (9 bcos ifiit at ach of th liits, ad i (3, at th lowr liit, wh th paratrs big positiv ar lss tha. Howvr, it ca b provd that i ths cass th itgrals ar fiit. Idd, d d d. Supposig ow that is a positiv propr fractio, w shall fid that or that d < d d < [ /]. This itgral is thus fiit at ay positiv whras d < d < /. As to th Eulr itgral of th first kid, it will also b fiit at th statd valus of th paratrs p ad q. Idd, w shall soo prov that it is coctd wuth th itgral of th scod kid by th rarkabl quatio

16 /( p/( q (p, q. (3 /( p q Howvr, bfor provig this rlatio, w shall idicat so proprtis of th Itgrals of th first ad th scod kid bgiig with th lattr. Wh itgratig by parts, w fid that i.., that d [ /( (/ /(. Thrfor, ] d (/ d, /( /(. (3 I th sa way /( ( /(, /( 3 ( /(, /( ( /(. Multiplyig ths qualitis w obtai /( ( ( ( /(. Assuig that ad otig that /( d w fid for ay itgral th followig quality /(!. (33 It is s ow that /( /( so that btw ad th fuctio / ought to hav a iiu (th scod drivativ of /( is always positiv. Cosidr ow th itgral (p; q p ( q d whr p is supposd to b ay positiv ubr ad q, a positiv itgr gratr tha. Itgratig by parts, w fid that (p; q [( p /p ( q ] (/p p (q ( q d so that (p; q [(q /p] (p ; q. Hc (p ; q [(q / (p ] (p ; q, (p ; q [(q 3 / (p ] (p 3; q 3, (p k; q k [(q k / (p k] (p k ; q k if oly w adit th iquality q k >.

17 Multiplyig ths qualitis w obtai (p; q ( q ( q...( q k p( p...( p q (p k ; q k. Assuig that q k w gt (p; q ( q! p ( p...( p q (p q ;. This quality will b valid for ay positiv p ad positiv itgr q. Howvr, (p q ; pq d. p q Thrfor, w obtai, for such valus of p ad q, (p; q ( q!. (34 p ( p ( p...( p q If w assu ow that p is also a itgr, th it follows that (p; q ( q!( p! ( p q! /( q/( p. /( p q Ad so, th quatio (3 is provd for itgr valus of p ad q. Not. {Hr, ad i ay cass i th squl, Chbyshv as though avoids irratioal ubrs.}.3.6. I ordr to prov th validity of (3 i th gral cas, w cosidr th doubl itgral I p y y pq y d dy. Itgratig first with rspct to, ad th to y, ad substitutig y z, w shall hav p y d (/y p z p z dz /( p, p y p q y I /(p y dy /(p/(q. p y Itgratig ow i th othr ordr ad assuig that y ( u, w fid that y pq y ( dy ( p q u pq u du ( p q /(p q, ( p p q /(p q d /(p q ( p p q d. Thrfor ( p p q /( p/(q d. (i /( p q

18 Supposig that p q ad usig forula (8 w ay icidtally rarkabl forula driv th followig /(p /( p π si pπ (35 which is vry iportat i th practical ss for copilig tabls of th valus of th fuctio /. It shows that, withi th itrval (;, it is sufficit to calculat th valus of / oly for th argut ot cdig /; th othr valus will b dtrid by forula (35. W rtur ow to th itgral (i. Assuig that z / ( z so that /( z ad d dz/( z, w obtai ( p p q d z ( z p p z p ( z q dz (p; q, ( z pq dz ( z hc (3. This quatio shows that (p; q is a sytric fuctio of p provd dirctly by sttig z y: ad q which ca also b (p; q z p ( z q dz y q ( y p dy (q; p. ( y p y q dy Thus, th validity of th quatio (3 is provd for all valus of p ad q for which th doubl itgral at th vry bgiig of this subsctio ight b cosidrd as th liit of a su. This last is tru for ay positiv p ad q bcaus, wh itgratig i th first ordr, w arriv at th product /(p/(q, ad, accordig to th abov, ach of ths two factors is fiit for all positiv valus of p ad q Assuig that p / i (35, w hav /(/ #%. (ii But, i accord with (9, p ( d (/ /(/. Th sa rsult ca also b obtaid dirctly by sttig p ( d z (/ z ½ dz (/ (/ /(/. z. Th, idd, z / z dz.3.8. A rarkabl quatio coctig th valus of th / fuctio ca b drivd fro quatio (3. Supposig that p q w hav hr (p; p /( p/(p. /( p Trasforig th lft sid of this quality w obtai

19 (p; p p ( p d {(/4 [ (/] } p d (/4 p Sttig z w arriv hr at (p; p p 4 ( z p (/ dz p 4 [ ( ] p d [ ( ] p d. ( z p dz bcaus th itgrad is a v fuctio. Assuig ow that z u, w fid that (p; p p 4 Thus, p ( u p (/ u ½ du u ½ ( u p du (/; p. p /( p/(p /( p p /(/ /( p /( p / ad, o th strgth of (ii, w obtai π /(p/(p / p /(p. (36 This rarkabl forula was first discovrd by Lgdr Lt us ow go ovr to itgrals prssd by th logarith of gaa. I.. w foud that for ay positiv ad itgr ( l [(!] d ad o th strgth of forula (33 this is qual to l /(. W shall prov that this forula is valid for ay valus of for which /( gral, forula (3. Diffrtiatig it with rspct to, w obtai d /( / d l d. But, i virtu of forula (, w hav z z l dz. z Thrfor d /( / d z z dz d z z z dz d. z Itgratig hr with rspct to, w arriv at ( z z d z ad, sttig ( z t, w obtai ( z d ( z d t t dt /( ( z ( z d ca b tak. W hav, i

20 so that d /( / d hc d /( d /( [ z ( z dz ]/(, z d l/( d [ z ( z dz ]. z It follows that d l/( l /( l /( ( z dz z [ z ( z dz ] d, z ( z [ ] l( z dz. z Thus l /( [( z ( z ( z l( z dz ]. z Notig ow that /(, w hav [ z ad, cosqutly, ( z ( z l( z dz ] z [( z ( ( z ( z l( z dz ] z. Subtractig this quality fro th prcdig o w shall hav l /( ( z ( z ( [( z l ( z ad, assuig that l ( z, w fid that l /( [ ( ( ] Fially, it follows that [( ( z ] d. dz z d ] l /(. ( Rplacig by ( i (37 w hav l /( [ ( d ]. (38 This forula is trly iportat i athatics ad, spcially, i th thory of probability whr it is applid for itgr ad vry larg valus of. W shall dwll o it also bcaus rarkabl corollaris follow fro it. If is a vry larg ubr, it will b or covit to rprst this forula i such a way that its right sid cosists of two parts, o of th icludig all th fiit ad vry larg trs ad th othr o srvig as a supplt ad icludig oly vry sall trs ad vaishig at $. This is cssary for apl wh calculatig l /( for vry larg valus of. W shall idd driv this logarith. Th itgrad i (38 ca b rprstd as

21 [ But ( ] [ ]. 3 /! / 3!... 3 /! / 3!... * ( whr * ( is th su of all th othr trs of th pasio. It is ot difficult to s that * ( icluds oly positiv powrs of bcaus its tr cotaiig i th lowst dgr is /, so that * ( / will hav a fiit liit at. W thus hav [ ϕ ( Howvr,. * ( ( ] [ ( ( ] [ ] ad thrfor l /( [ (/ [ ( d ] ] d. Or, supposig that th two itgrals ar dotd by F( ad ( rspctivly, w obtai l /( F( (. W thus rducd our prssio to th dsird for: th fuctio ( is rprstd by a itgral of a fuctio takig fiit ad vry sall valus at all valus of cotaid withi ad icludig th liits of itgratio, ad, orovr, takig vry sall valus at vry larg valus of. This fuctio has actly thos proprtis which, accordig to our assuptio, should b possssd by th scod part of th prssio sought. It is oly lft to dtri th typ of th fuctios F( ad (. To fid th first of th w diffrtiat this {yt ukow} fuctio with rspct to : F ( It follows that ad F ( [ ( d ] [ (/ d ]. d (/ F( l d (/ d l (/ (/ d l (/l C (iii whr C is a costat that for th ti big w lav idfiit. Thus, F( C l (/ l.

22 It ought to b otd hr that th diffrtiatio applid by us dstroyd th costat C i whos dtriatio all th difficulty rally cosists. W wr abl to calculat th fuctio F( i such a idfiit for without ay troubl actly bcaus of this diffrtiatio. Lt us ow ivstigat th fuctio (. Hr, w caot carry th itgratio to its coclusio ad hav to prss ( by a sris. W hav / / / / so that th itgrad bcos / / (/ [ / / ]. Th product of th first two factors is hr a v fuctio {s blow} so oc that its pasio ito powrs of will b of th for that it is possibl to say at ad A A A 3 4 W hav ± / ± / (/! (/ ± (/3! (/ 3, / / [ (/! (/ ] / / [(/ (/3! (/ 3 ] / / / / /8... / / Thrfor (/ [ ] (/ [ ] 6 Thus, A /. Takig ito accout a largr ubr of trs i th urator ad th doiator ad carryig out th divisio, w shall obtai, i th sa way, A, A 3, Th ubrs that prss ths cofficits ar vry closly rlatd to th Broulli ubrs; at prst, vry ay athaticias ar studyig th. So, w hav ( A [A A 3 A 3 5 ] d d A d A 3 4 d Notig that, i gral, l l z d d/ (/ l l z l z /(l dz l ad that, for ay itgr ad positiv valus of l, /(l (l!, w obtai ( (!/ A (!/ 3 A (4!/ 5 A 3 (39

23 W hav thus prssd ( by a sris dpdig o th cofficits A, A, A 3, Thrfor, to say sothig gral about this quality, it is cssary to kow th law of thir copositio. Howvr, th thod of thir dtriatio applid abov dos ot provid this law ad w shall driv{othr} prssios for ths cofficits, awkward for calculatio but vry covit for our prst purpos..3.. Sttig / &i w obtai / / θ i / / θ i θ i θ i cosθ i siθ ctg θ. i But, i gral, si & & [ (& /% ] { [& /(% ]} { [& /(3% ]} so that l si & l & l [ (& /% ] l { [& /(% ]} l{ [& /(3% ]} Diffrtiatig this quality w hav θ ctg & θ π θ θ (π θ θ (3π θ, θ (/ [ctg& (/&] π θ θ ( π θ θ (3π θ Howvr, sic i gral [/( '] ' ' ' 3 ' l ad ( π θ ( π ( θ/ π, l α α it follows that ( π ( π θ θ ( π θ ( π l l 4 θ ( π 4 ( π. ( π θ l θ l (π Thrfor θ ( π θ θ ( π θ ( π l 3 l 4 3 θ ( π 4 ( π θ 5 θ ( π 6. l θ l ( π Assuig that taks hr diffrt valus bgiig with ad addig w shall hav θ θ θ (&/% [ (/ π θ (π θ (3π θ (/3 ] (& 3 /% 4 [ (/ 4 (/3 4 ] (& l /% l [ (/ l (/3 l ] th thus obtaid qualitis

24 l 3 θ { l 4 π π θ (/ [ctg& (/&]. [(π θ ] l 4 } l 4 3 [(3π θ ] But & /i i / so that / / (/ [ / / (S 6 /% 6 ( i/ 5 { l π π [( i/ l3 ] ] i (S /% ( i/ (S 4 /% 4 ( i/ 3 / 4 whr th otatio itroducd is obvious. Hc / / (/ [ / / (S 6 / 5 % 6 5 ( l ( l [ l 3 l π π It is s ow that l [(π ] (S /% (S 4 / 3 % 4 3 Sl l l l π / 4 l [π / 4] A (/% S (/% [ (/ (/3 ], A (/ 3 % 4 S 4 (/ 3 % 4 [ (/ 4 (/3 4 ], A 3 (/ 5 % 6 S 6 (/ 5 % 6 [ (/ 6 (/3 6 ], / 4] ] l3. } ad i gral, that l ( A l l l π l ( S l l l π [ l ]. l 3 W thus hav ( {A A 3 A 3 5 A l l l ( l 3 l π [ ] l3 } (d/ l π / 4 [(π / 4] ad, takig ito accout th quality (39, ( (A / A [/(3 / 3 ] A 3 [/(5 / 5 ] A /(l l l l ( [ ] l d. l 3 l π π / 4 Ad so, w hav prssd ( by a sris whos trs ar altrativly positiv ad gativ ad th sig of whos additioal tr {raidr} is opposit to that of its prcdig tr. Such sris ar calld liitativ. Thy possss a rarkabl proprty: thir additioal tr is always urically lss tha th tr subsqut to th last o cosidrd. Idd, lt us tak th sris X u u u 3 ± u R, X u R ; or, u u R ; or, u u u 3 R 3, tc. It is s that

25 R < u ; R < u 3 ; ; R < u. Thrfor, such sris always abl us to dtri th rror takig plac wh calculatig thir {approiat} su ad this is thir advatag ovr othr sris. Howvr, thy also suffr fro a srious shortcoig: thy provid o clu for dtriig wh should w stop i ordr to obtai th ost prcis valu of th su. Idd, i such sris ach tr ca b {urically} ithr gratr or lss tha its prcdig tr. Wh brakig off at so tr ± u k, w will ot that i o cas th additio of o or tr, u k, ca lowr, ad i aothr cas it ca hight th dgr of approiatio. I gral, as it is s fro what was said about ths sris, w should stop at such a tr u l whos subsqut ighbor u l is last; th su, calculatd i such a way, will b th ost prcis ad diffr fro th ral su lss tha by u l. Forula (39 shows that at first th trs of th sris prssig ( will dcras, but that, bgiig with so tr, thy will icras. Thrfor, i ordr to dtri, i this cas, th tr at which w should brak off, w ought to hav u l > u l or u l / u l > ad to dtri th last valu of l satisfyig this iquality. W shall show how to acquir so otio about th last boudary of l by as of this iquality. W hav u l A l /(l / l, u l A l /(l 3 / l 3, ul u l A /(l l 3 l l A l /(l 3 (A l /A l (l (l 3 (/ whr oly th urical valus of A l ad A l ar tak ito accout. W shall thus hav a cojctural iquality (A l /A l- (l (l 3 > or (/ % (/ (/ l (/ 3 (/ 3 l l l (l (l 3 >. (4 Now w ay coclud that iquality (l (l 3 > % also ists so that (l > (% ad l > %, l > %. (4 Thus w s that l should ot b lss tha %. Howvr, th thod of obtaiig this lowr boudary dos ot provid th possibility of sig whthr, for l > %, u l will rally b th last, havig th last valu satisfyig this iquality. W wt ovr fro iquality (4 to (4, but, vidtly, w had o right to go i th othr dirctio. It is thrfor s that % provids oly a approiat otio about how grat should l b. W hav rarkd (.3. that th cofficits A, A, ar vry closly rlatd to th Broulli ubrs B, B, Th dpdc btw th is prssd by th quatio B A l ( l l (l! so that A B /!, A B /4!, A 3 B 3 /6!, tc.

26 Th Broulli ubrs play a rathr iportat part i athatics, ad, ar prssly dvotd to studyig thir proprtis. accordigly, vry ay tratiss.3.. Lt us ow go ovr to th dtriatio of th costat C i th prssio (iii. Takig th logariths of both sids of th Lgdr forula (36 /( /[ (/] (#% / /( w hav l /( l / [ (/] l #% l / ( ( l. Howvr, l /( C l (/ l ( ad thrfor l /( C ( l ( (/ l ( (, l /[ (/] C [ (/] l [( (/] (/ (/ l [ (/] [ (/], l /( C ( l ( (/ l ( (. W thus obtai such a quatio: C [ (/] l ( l [ (/] (3/ ( ( / l #% C [ (/] l ( ( ( l so that C l #% [ (/] l ( [ (/] l ( l [ (/] (/ ( l ( [ (/] (. Howvr, i gral w hav l ( l { [ (/]} l l [ (/] l (/ (/ (/ ad thrfor l ( l (/ (/8 l ( l (/ (/, l [ (/] l ( l l (/ (/8 Ad so or C l#% [ (/] [l l (/ (/8 ] [ (/] [l (/ (/ ] [l (/ (/8 ] (/ ( l ( ( [ (/] C l #% (/ l {[ (/] /} {[ (/]/}

27 {[( (/]/8 } {[ (//] } (/8 ( ( [ (/]. Assuig ow that i this quatio or idtity (it should hold for all valus that ($, w fid that of $, ad otig Thus C l#% (/ l l#%. l /( l#% l (/ l ( (iv so that /( π / (. But ( (/ (/ ad w thus fially obtai /( π / [ (/ ]. (4 If is a itgr w shall gt! π / [ (/ ]. This forula that abls us to calculat th approiat valu of th product to Stirlig. of atural ubrs is du Not. {I 73, D Moivr drivd this forula idpdtly fro ad siultaously with Stirlig; th lattr oly couicatd to hi th valu of th costat. D Moivr also publishd a tabl of lg! for ( 9 with 4 dcials of which or wr corrct.}.3.3. I cocludig th sctio o th Eulr itgrals w shall driv th faous Gauss quatio coctig {various valus of} gaa. W ay always suppos that /( /[ (/] /[ (/] /[ ( /] F (; /(. Lt us ow try to dtri th fuctio F (;. W hav /(a /[a (/] /[a (/] /[a ( /] / /(a F (a;. Substitutig hr (a istad of a ad rcallig quality (3 w shall fid that ( a i / /( a i / i F (a ; /( a or a ( a / ( a /...[ a ( / ] ( a ( a... a/( a /(a /[a (/] /{[a ( /]} F (a ;,

28 a ( a ( a...( a ( a ( a... a F (a; F (a ;, F ( a ; F( a; ( a a ad cosqutly F( a ; ( a F ( a;. a Thus, assuig that i gral k F (k; & (k, w hav & (a & (a so that & (a & ($. Thrfor /( a/( a / /( a /.../[ a ( / ] /( a whr a li [& (] $ & ( Thrfor /( /( / /( /.../[ ( / ]. /( l & ( l /( l / [ (/] l /{ [( /]} l l /(. But bcaus of (iv l & ( l l π [ (/] l ( ( π [ (/ (/] l[ (/] (/ [ (/] l π [ (/ (/] l[ (/] (/ [ (/] l π { (/ [( /]} l{ [( /]} [( /] { [( /]} l π [ (/] l ( l (. Notig that i gral l ( r l [ ( (r/] l l[ (r/] l r/ w fid that l & ( ( l π [ (/] [l (/ ] ( [ (/ (/] {l [ (/]/} [ (/] [ (/ (/] {l [ (/]/} [ (/] { (/ [( /]} ( / [l ] { [( /]} l ( {[( ( ]/} [ (/] [l (/ ] ( ad

29 l & ( ( l π l l l (/ l [ ( ]/ (/ l... ( [ ( ]/ l ( [ (/] { [( /]} (. Thrfor l & ( ( l π (/ l W hav rtaid oly two trs bcaus all th rst of th vaish at $. Ad so li [ & (] $ ( l π l # which as that ad li [l & (] $ # (% ( / ; F(a; a/ (% ( / /( a/( a / /( a /.../[ a ( / ] /( a a / (% ( /. Thus w hav /(, /[, (/] /{, [( /]}, / (% ( / /(,. (43 This quatio which is valid for ay, ad ay itgral ad positiv is a gralizatio of quatio (36. Assuig hr that, ad supposig that is a itgral positiv ubr, w obtai /[ (/] /[ (/] /{ [( /]} / (% ( / /(. It follows that (! /(//(/.../[( /] / (% ( / /( ad i virtu of th quality (33 /(/ /(/... /[( /] (% ( / /. ( W dfid /(, as th valu of th dfiit itgral {s (3}, d. Accordigly, w hav cssarily cosidrd th gaa fuctio oly for such valus of, for which this itgral had ss, i.., for which it was th liit of so su. W saw that ths valus wr cotaid withi ad $. Thr ists, howvr, aothr or gral dfiitio of th gaa {fuctio}. Th valus of this fuctio ar tdd oto such cass whr th variabl also taks gativ valus. Eactly this last fact gav th occasio for th w dfiitio ad i ordr to go ovr to it w ot that for ay itgral ad positiv, /(, (!/( λ ( λ! λ $. This quality ight howvr b prssd i th fors

30 /(, /( /( λ /( λ λ $ λ (! λ ( λ ( λ... ( λ $. (45 Th last quality is idd adoptd as th dfiitio of gaa ad it is also tdd valus of,. oto fractioal ad gativ.3.5. W ar ow goig ovr to itgrals that hav a vry clos coctio with th Eulr itgrals ad ar usually drivd fro ths lattr by assuig iagiary valus for. Howvr, so as to follow quit a rigorous path, w shall dtri th idpdtly fro th Eulr itgrals. W shall cosidr th itgral I z cos ( z µ d dz. Itgratig at first with rspct to, ad th to z, w fid, wh substitutig z y, so that cos( z µ d cos y (y/z µ dy/z z µ y µ cos y dy I y µ cos y dy z µ z dz /( µ y µ cos y dy whr µ is supposd to b positiv ad lss tha. Itgratig ow i th othr ordr, w hav, i accord with forula (4 of.. z cos ( z dz. Th, o th strgth of forula (5 w obtai µ - I ad thrfor d π cos[( µ - π/] π y µ cos y dy. Γ(- µ cos [( µ - π/] Assuig that µ is hr positiv ad usig forula (35 w hav y µ cos y dy /( µ si µπ cos [( µ - π/] Γ( µ si µπ/ cos µπ/ cos [( µ - π/] /(µ cos (µ%/. Itgratig this quality by parts w obtai a siilar forula cotaiig a si: ( µ so that y µ cos y dy [y µ si y] (µ y µ si y dy y µ si y dy Γ( µ cos (µ%/. - µ Substitutig µ, w arriv at y µ si y dy y, si y dy Γ ( λ cos[(, %/] /(, si (,%/. λ W hav rplacd [/(, /,] by /(,; i this cas, howvr, sic µ is cotaid withi th boudaris ad,, taks valus btw ad whras (v

31 th substatiatio of th forula (3 hld oly for positiv valus of bcaus this rstrictio follows fro th dfiitio of th gaa fuctio that udrlis th proof. So as to rov this difficulty w ca prov th obtaid forula also for gativ valus of ad w shall ow hav to us th dfiitio of th gaa fuctio icludd i forula (45. Notig that li [/(,] $ w ay assu that λ λ (! (! [ ] $ [ λ ( λ ( λ... ( λ λ ( λ... ( λ ( λ so that λ (! /(, [ ] $,,/(, /(, λ ( λ... ( λ ad y, si y dy /(, si (, %/. Ad so, forula (v is justifid. W drivd it for gativ valus of, but it will also hold for positiv, s bcaus it ca b obtaid i th sa way as th forula cotaiig th cosi. W thus hav th forulas µ cos d /(µ cos (µ %/, < µ < µ si d /(µ si (µ %/, < µ < W ar cocludig hr th sctio o th Eulr itgrals. ] $.4. Itgrals of th Fourth Group.4.. W ar ow goig ovr to th itgrals whos charactristic fatur is that thir liits ar agituds havig spcial sigificac for such agls as, %/, %, %, tc. W bgi by cosidrig th itgral π π * i * i d* (46a (46b ad w shall prov that its valu is ithr or % dpdig o whthr ad othr. I th first cas w obtai by itgratio ar diffrt or qual to ach π π ( ϕ i π ( * i d* [ ] ( i π i si[( π ] si[( π ]. ( i ( π i ( ( i π i It is s howvr that, wh ad ar itgrs, ad, i additio, diffrt, th itgral vaishs. If, th forula taks a idfiit for /. Wh dtriig its ral agitud i accord with th ruls of diffrtial calculus, w shall hav %. Th sa rsult ca also b gott dirctly: i this cas, th itgral will b π π Thus π π d* %. * i * i d*, if π if A larg ubr of athaticias studid, ad ar studyig, itgrals of this kid. Thir iportac for {athatical} aalysis is basd o th fact that, through thir proprty cosistig i {th istc of} qualitis siilar to (47, ay fuctio ca b padd ito powrs of o of th factors of th itgrad. Thus, by as of (47

32 quality (47 w ay pad F ( i * ito powrs of i *. Not that w ay attai th sa goal by diffrtial calculus ad it would s that this lattr thod is prfrabl bcaus o difficultis ca b coutrd thr, but actually this is ot so: diffrtiatio prsts o troubl oly wh w dtri drivativs of kow ordrs prssd by ubrs, ad bcos as difficult as itgratio is as soo as w dsir to calculat th prssio for th -th drivativ. I additio, it is vry oft iportat to dtri th tr at which w should brak off, ad this probl is rducd to fidig out how th gral tr of th pasio chags with th chag of its ubr. I studis of this kid th trs prssd by itgrals, v wh ths caot b calculatd, provid uqustioabl advatag ovr prssios dpdig o drivativs of a kow ordr. It is for this raso that th itgrals of th cosidrd kid ar iportat for aalysis. By applyig th proprtis of ths itgrals it is vidtly possibl to solv covrs probls as wll: Giv a pasio, to dtri th valu of th itgral that prsss its gral tr. Idd, lt us suppos that F ( * i A o o * i A * i A * i A * i Now, ultiplyig both sids of this quality by i * ad itgratig withi th as of th forula (47, π π so that F ( * i * i d* A % π A π F ( * i * i d*. π W thus obtai a forula that abls us to driv both th gral tr of th th pasio. Th cofficit of th gral tr ca also b rprstd as liits % ad % w hav, by pasio, ad th itgrals giv A F ( (/!. Substitut ow F( f (, so that F ( i * f (, i *. Th F ( df(/d df (,/d,f (,, F( d F/d, df (,/d, f (,,, F ( (, f ( (, ad thrfor A, f ( (/!. W thus arriv at th forula π π f (, * i * i d* (%/! f ( (,. (48 W ca sowhat graliz this forula by sttig f (z & (a z so that thus obtai π π & (a, * i * i d* (%/! & ( (a,. (49 Not. {A datd cocpt.} f ( (z & ( (a z. W or.4.. Suppos ow that f (, l (, so that f (,,, /, 3 3 /3, f (, * i, * i (, / * i (, 3 /3 3* i f (, * i (,/ (cos * i si * (, / (cos * i si * It is s ow that w ought to assu hr that, <. Oly udr this coditio th sris will always covrg: th sris of its cofficits, (,/, (, /, (, 3 /3, will always covrg oly if, < bcaus, for such valus of,, th sris (,,, 3 will rprst a ifiitly dcrasig gotric progrssio.

33 Ad so, supposig that, < ad applyig forula (48, w fid that π I π l (, * i d* Trasforig this itgral, w hav π I l (, * i d* l (, * i d* I I. π Suppos ow that *, th π I l (, i d l (, * i d*, π π I I π l (, * i d* l [(, * i (, * i ] d* ad thrfor, for, <, π π I l [, ( * i * i, ] d* l [, cos *, ] d*. If, /R whr R >, w shall fid that π π l [ R cos * R ] d* l R d* % l R. W do ot rplac l R by l R bcaus R ca b gativ so that th forula would hav providd a idfiit rsult whras it should b quit dfiit. W thus co to such a itgral: π l [ cos * ] d* if < ad % l if > (5 which is usually attributd to Poisso Issuig fro th itgral (5 w ca driv svral rarkabl itgrals. Sttig w shall fid that π l ( cos * d*. W ay adit this as a liitig quality bcaus ach of th forulas (5 lads to it π so that π l (si */ d*, l si */ d* π π Assuig hr that * w gt π / l (si */ d* l d* % l. l si d (%/ l. (5 If si th d d / ad l π l d. (5 Itgratig th prssio (5 by parts, w gt at. Ad so π / π / π / l si d [ l si ] π / (/si cos d ψ dψ. tgψ Sic li ( l si w hav

34 π / ψ dψ tgψ π l. ( W shall show ow th applicatio of th forula (49 to dtriig th uppr boudaris of drivativs, but bfor that w shall say wh this forula ay b usd without ay dagr of coutrig cotradictios. Th drivatio of this forula was basd o th possibility of padig th fuctio F ( * i ito powrs of * i, but it is kow that ot ay fuctio ay b padd ito powrs of its idpdt variabl; i othr words, that th sris obtaid will ot covrg always. It is s ow that th covrgc of th sris A o A * i A * i which, accordig to our suppositio, prsss th fuctio F ( * i, should b forulatd as th coditio for th validity of forula (49. Sic w rplac F by a idtical fuctio f [, ( * i ] padd ito a sris of th kid A o A, *i A, *i (vi th covrgc of this sris is cssary for th possibility of th istc of th forula (49. But (vi ca b rprstd as a su of two sris arragd i th ordr of cosis, ad sis, of ultipl arcs. It follows that (vi will always covrg if oly th sris of th urical valus of th cofficits A o, A,, A,, of its trs covrgs. W ay for a opiio about th covrgc of this sris oly if, < bcaus, udr this coditio, as it is ot difficult to s, th sris will always covrg if oly A k dcrass, or at last rais fiit with a icrasig k. W thus s that th forula (49 ay b adoptd oly for such fuctios th cofficits of whos pasio ito powrs of th variabl always rai fiit for th valus of, <. Aftr ths rarks w procd to solv th issu ow itrstig us. W hav, i gral, od (k k < od k od k Thrfor, cosidrig th itgral as th liit of a su, w fid that od - (u du < od - (u du. Fro forula (49 ad fro th o just obtaid, sic π f od ( a λ! ( π od f (a, * i * i d*, π it follows howvr that th lft sid is lss tha π π od f (a, * i * i d*. But, sic od * i, od [f (a, * i * i ] od f (a, * i od * i od f (a, * i. Lt us assu that w hav sohow dtrid th uppr boudary R of th odulus, od f (a, *i : od f (a, *i 3 R. (vii Th od hc π f ( a λ! ( π < π R d* % R, od f ( (a, < R!. Supposig ow that both th fuctio f ( (a ad, appropriat urical valus, w obtai th forula ar ral ad stipulatig that ths prssios oly dot th

35 f ( (a < R! /, (54 whr R is dtrid by th coditio (vii. To provid a apl of applyig forula (54 lt us tak F ( * i /[k i * ] with od k >. Th th sris F ( * i (/k (/k * i (/k 3 * i will b covrgt. Assuig that a w fid that f (a, * i ϕ i k λ whr, i accord with th rark abov, w ought to suppos that, <. I ordr to dtri R w ot that, i gral, od (A Bi ( A Bi ( A Bi so that (od k λ ϕ i ( ϕ i k λ. ϕ i ϕ i k kλ ( λ It is s ow that th odulus will b aial at * so that w should assu that R k kλ λ ad, sic k >,, R /(k,. W thus obtaid d [ /( k ]! < d. ( k λ λ Ad so, w dtrid th uppr boudary of th -th drivativ of th fuctio /(k at. Not that it is advatagous to driv th last valu of th uppr boudary which th studid agitud caot cd; ad sic th iquality obtaid by us taks plac for ay valus of,, w ought to fid such of its valus that will iiiz th dtrid uppr boudary. This probl rducs to th drivatio of th aial valu of th fuctio (k,, k,,. For calculatig this valu w hav th quatio k, (, ad, k/(. If th thus dtrid valu of, will b lss tha, w ight us it, ad th k k,, ( ( k ( k (. Thrfor, if k < [ (/], w shall hav d [ /( k ]!( < d k. Th Fourir Forulas { }.4.5. W ar ow goig ovr to ultipl itgrals ad to drivig th Fourir forula that was prviously cosidrd vry iportat; rctly, howvr, it is vr or losig its sigificac. This happs bcaus w ar uabl, whil drivig this forula, to provid th coditios dtriig th fuctios for which it rais valid.

36 Lt us tak th itgral P $ f ( cos [y ( ']d dy. Itgratig with rspct to y, w fid that cos [y ( '] dy si[( α y] α }. It is s ow that th valu of this itgral is idfiit so that istad of th liits $ ad $ w first assu A ad A, ad oly th, i th fial rsult, w st A $. Cosqutly, w hav A A cos [y ( '] dy si[ A ( α] α, P A f ( si[ A ( α] d. α Substitutig ow A( ' z (viii so that d dz/a, w fid that P A f [' (z/a] [(si z/z] dz. Assuig hr A $, w arriv at P $ f (' [(si z/z] dz f (' [(si z/z] dz ad, o th strgth of forula (5, P $ % f ('. W thus driv th faous Fourir forula f ( cos [y ( ']d dy % f ('. (55 Th o-rigor of this drivatio cosists i that, havig obtaid th liits $ ad $ for z fro forula (viii or fro ' z/a, ad kowig that ths ar th liits for, w {vrthlss} ar ot always abl to forulat th ivrs statt: Assuig that A $ i th fial rsult for z, w caot go ovr to th liits $ ad $ for. It follows that w would b uabl to go back, i.., to pass to th itgral fro th prssio % f (', wh drivig this forula. It is s ow that i gral th forula (55 will ot b valid for ay fuctio uivrsally. Lt for apl f ( p (. By forula (55 w would hav foud p ( ' π p ( cos [y ( '] d dy. W shall itgrat so as to chck this rsult. W hav I p ( cos [y ( ']d si (y ' p ( si (y d. cos (y ' p ( cos (y ] d Th scod itgral vaishs bcaus its itgrad is a odd fuctio; th itgrad i that i accord with forula (3 I cos (y' p ( cos (y d p ( y /4 cos (y'. Oc or i virtu of this forula w hav % p ( '. p ( cos [y ( '] d dy p ( y /4 cos (y ' dy W thus arrivd at th sa rsult as wh applyig th Fourir forula W shall ow odify forula (55. W hav th first itgral is v, so

37 cos * * i i si * so that i f ( cos [y ( '] d dy f ( si [y ( '] d dy. f ( i y ( ' d dy But th scod itgral cotais a odd fuctio with rspct to y ad thrfor Cosqutly f (' f ( cos [y ( '] d dy π f (' whr f ( y i ' y i d dy, f ( y ( ' i d dy. * (y ' y i dy (56 * (y π f ( y i d. Equatio (56 solvs a particular cas of dtriig a fuctio satisfyig th B A * (y F ('; y dy f ('. vaishs. W hav quatio Such issus wr studid aog othrs by Abl. I gral, it ought to b otd that thir solutio lads to vry rarkabl rsults. Util ow {howvr} thy wr copltly solvd oly for th particular cas i which A $, B $, F ('; y i ' y W shall ow odify th quatio (56 ad apply it, i its w for, i th squl. Assuig that ' u i, f (u i F (u, so that f (u F ( u i, w fid that f (u i * (y u y dy F (u, * (y Ad so, w us th followig quatios: π F ( i y i d. F (u * (y u y dy, * (y π F ( i y i dy. (57 Issuig fro th, w ca driv forula (8. Idd, o th strgth of (5 w hav cos a π d a, a i i a d π a. a Sttig i th scod quality z, y, ultiplyig both its sids by * (y dy ad itgratig with rspct to y fro $ to $, w obtai i i ϕ ( y dy d a π a a y * (y dy, (/[ y z i * (y dy dz y z i * (y dy] a z

38 π a a y * (y dy ad, bcaus of (57, F zi ( F( zi a dz z π a F ( a. Now w ca procd to th drivatio of th clbratd Dirichlt forula cocrig ultipl itgrals. To this d, w ot that, i gral, so that. '. d., '. d Th, w hav (t/' t (d t/' /( / ', '. y µ ' y. z. ' z. d dy dz Γ( λ Γ( µ Γ( ν... y µ ' y. dy. λ µ ν... α s,µ. ' s Γ( λ µ ν... ds λ µ ν... α ad cosqutly Γ( λ Γ( µ Γ( ν... Γ( λ µ ν..., y µ. z. ' (yz d dy dz s,µ. ' s ds. Multiplyig both sids of this quality by * (' d' ad itgratig with rspct to ' withi th liits $ ad $, w obtai Γ( λ Γ( µ Γ( ν... Γ( λ µ ν..., y µ. z. [ s,µ. ds Cosqutly, o th strgth of (57, w gt Γ( λ Γ( µ Γ( ν... Γ( λ µ ν... * (' ' (yz d] d dy dz * (' ' s d'., y µ. z. F [ ( y z ] d dy dz s,µ. F( s ds ad, substitutig F ( t f (t, w idd arriv at th Dirichlt forula Γ( λ Γ( µ Γ( ν... Γ( λ µ ν..., y µ. z. f ( y z d dy dz s,µ. f (s ds. ( Forula (58 ca b sowhat gralizd by itroducig w paratrs whos particular valus would hav ld to it. Suppos that au, y bv, z cw, ad that a, b, c, ar positiv so that th liits of itgratio prsist. Notig that, cosqutly, a costat factor a, b µ c. will b icludd i th lft sid of th quality ad dividig both its sids by this factor, w obtai

39 a λ b µ u, v µ. w. f (au bv cw du dv dw Γ( λ Γ( µ Γ( λ µ... s,µ. f (s ds. Sttig ow u, v y, w z p, w arriv at, y µ. z p. p f (a by cz p 4 d y dy pz p dz p a λ b µ Γ( λ Γ( µ Γ( λ µ..., y µ. z p. f (a by cz p d dy dz s,µ. f (s ds. Substitutig fially, ', µ (, p. 5, ad dividig both sids of th quality by p, w gt p... a α / ' y (. z 5 f (a by cz p d dy dz Γ( α / Γ( β / Γ( γ / p... β / γ / p b c... Γ( α / β / γ / p... s '/(/5/p f (s ds. (59 Hr, th paratrs,, p, ar of cours supposd to b positiv; othrwis, ad $ would ot b th liits of th w itgral. Forula (58 ca b drivd as a particular cas of (59 wh substitutig i th lattr a b c, p, ',, ( µ, 5 6,.4.9. Suppos that f (t is a fuctio satisfyig th coditios f (t, t L ad, t > L. Th, assuig {also} th coditio a by cz p L (i whr L is a giv positiv agitud ad applyig forula (59, w rduc th dtriatio of th itgral to th calculatio of ' y (. z 5 d dy dz ( p... a α / Γ( α / Γ( β / Γ( γ / p... β / γ / p b c... Γ( α / β / γ / p... L s '/(/5/p ds sic i this cas L s '/(/5/p f (s ds But th first of ths itgrals is α / β / γ / p... L α / β / γ / p... s '/(/5/p ds L s '/(/5/p ds. ad, o th strgth of forula (3, w fid that

40 Itgral ( quals α / β / γ / p... Γ( α / Γ( β / Γ( γ / p... L. (6 α / β / γ / p p... a b c... Γ( α / β / γ / p... This forula thus solvs th probl of dtriig th itgral ( tdd ovr all th positiv valus of th variabls, y, z, coctd by coditio (i..4.. Probls about th dtriatio of aras ad volus as wll as thos touchig o th attractio of bodis of a kow for ar asily solvd by forula (6. For apl, w shall calculat th ara of a llips; th coditio coctig th variabls ad y will thrfor b a by 3 whr a /A, b /B with A ad B big th si-as of th llips. I this cas,, ' ( so that forula (6 provids d dy ab Γ(/ Γ(/ L 4Γ( whr th itgral is tdd ovr all th positiv valus of th variabls ad y obyig th coditio a by 3 L. I our cas, L ; ad /(/ #%, /( so that d dy %/4 ab (%/4 AB. As pctd, w thus obtaid th agitud of a quartr of th ara sought: th variabls ad y ar positiv oly for th quartr of th llips dtrid by th quatio /A y /B. Lt us also cosidr a tripl itgral that, for ' ( 5 ad udr th coditio of th typ a by cz p L, rprsts so volu. Assu also that p. Th forula (6 provids d dy dz abc Γ(/ Γ(/ Γ(/ L 8Γ(5/ 3/. Howvr, /(5/ (3/ /(3/ (3/ (/ /(/ 3#%/4 ad d dy dz (%/6 abc L 3 /. Suppos ow that th quatio trasford to th oral for a by cz L blogs to a llipsoid. Cosqutly, L, a /A, b /B, c /C si-as ad whr A, B ad C ar th llipsoid s d dy dz (%/6 ABC will b th prssio for /8 of its volu..4.. W dducd th Fourir forula i th for (55. Now, w shall ipart a or gral for to it by choosig so agituds L ad M (L < M as th liits of itgratig with rspct to. I ordr to accoplish this w ight hav actd i th followig way. Sic f ( is a absolutly arbitrary fuctio (rstrictd howvr by coditios idicatd wh drivig forula (55 that ca v b discotiuous, w ay assu that f (, $ < < L; f ( * (, L < < M; f (,