On the use of the variable change w=exp(u) to establish novel integral representations and to analyze the Riemann ς ( s,

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1 On th of th variabl chang p() to tablih novl intgral rprntation and to analyz th Rimann ς (, -fnction, incomplt gamma-fnction, hyprgomtric F-fnction, conflnt hyprgomtric Φ -fnction and bta fnction Srgy K. Skatkii Th variabl chang i applid to tablih novl intgral rprntation of th incomplt gamma-fnction, hyprgomtric F- fnction, conflnt hyprgomtric Φ -fnction and bta-fnction, and to analyz th fnction a ll a th Rimann ς (, -fnction. In particlar, ing th rprntation giv a pdagogically intrctiv proof of th ll knon approimat fnctional rlation for th Rimann ς -fnction and driv Hritz rprntation of th ς (, -fnction.

2 . Introdction In th middl of th XIXth cntry Schläfli d th variabl chang for th t of th Bl intgral J (+) ν ν ( z) p z( ) d πi hich nabld him to tablih om important proprti of Bl fnction;.g. []. Intgral rmbling that givn abov ar d to pr th Rimann ς -fnction: it i ll knon that for R>: ( ) ) ( ) d + ς () (.g. [] for proof and tandard dfinition), and imilarly for R> and mor gnral ) ς ( ) d () a ) ς (, d (3). Similar intgral rprntation i alo a tarting point of th dfinition of th incomplt gamma-fnction [3] γ (, d (4) and actally thy appar in many othr intanc. Hnc of cor it look intrting to try th am variabl chang for analyi of th intgral. Rcntly, thi approach ha bn attmptd for th Rimann ς fnction by Brgtrom [4] (th prnt athor i naar of othr rarch of thi typ). In th prnt ot analyz om rlt of it application to diffrnt analytical fnction.. Application of th variabl chang for th Rimann ς -fnction A variabl chang convrt th qality () into ( ) ) ς ( ) d (5) + Th intgrand i a ingl-vald fnction on th hol compl plan + and th Cachy thorm i applicabl. Thi fnction ha impl pol at ln( π (n + )) + iπ (/ ) hr n i intgr or qal to zro and n, m + m m, ±, ±,... Th corrponding rid ar qal to ( ) n, m m i( ) (n + ) π p( iπ / ) p( iπ m ) (bca

3 d n m m,, n, m d d + C ). Lt apply Cachy thorm to th intgral takn arond th rctanglar contor formd by th point X, ln(π ), ln(π ) + πi, + πi hr i a poitiv intgr and ral X>. Inid thi contor hav pol ith m, and n<+, ho m i qal to S in( π / ) π iπ (n + ). n Important obrvation of Brgtrom i that th intgral ovr X + π i, ln(π ) + π ) i qal to iπ tim th am intgral ovr ( i ( X, ln(π )) and hnc i iπ ( )( ) ) ς ( ) 4πi in( π / ) π (n + ) I + I' n hr, I ', π (6). I ar vrtical intgral takn ovr ( π π + πi) and X + π i, + π ) rpctivly. Writing (6) hav nglctd th ( i trncation rror (intgration ovr ( X, ln(π )) intad of (, ) ) hich i π O( ). For R> intgral I ' i alo trivially ngligibl hn X. π Uing ς ( ) π in( π / ) ) ς ( ), Γ ( ) ) and in π iπ iπ i on can th rrit inπ i( hr ) iπ ς ( ) π + n iπ π in( π / ) 4πi in( π / ) π (n + ) ii (7) π (ln(π ) + i iy I ln( π ) + iy π (8). π (co y+ iin + + W ar intrtd in th rang < σ < (a al σ R ) hnc (7) can b rrittn a hr ~ ς ( n + ) I n ( ) ( ) π (7 iy ~ I π (8. i π (co y+ iin in( π / ) + A mntiond bfor, qation (7-(8 ar not act bt contain an ponntially mall rror trm d to th intgration ovr ( X, ln(π ) intad of (, ) (th limitation of intgr i not principal and can aily 3

4 b rmovd). Hnc, trictly paking, thy ar not intgral rprntation of th Rimann ς -fnction althogh might conidr thm a approimat intgral rprntation. ot alo that thy ar provn hr for R> bt by analytical contination imilar rlation hold for all cpt, of cor,. For R< intgral (8 vanih in th limit of larg th giving th ll knon rlation ( ) ς ( ) (n + ). n For compltn, finih thi Sction rriting qation (7-(8 ing th formal chang : hr ς ( n + ) I n ( ) ( ) π π i i( ) y ~ I π (co y+ iin co( / ) + ~ (7b) (8b).. Proof of th approimat fnctional qation for th Rimann ς - fnction Th natral firt approimation to th val of intgral (8 i imply to tak th intgrand qal to zro for (, π / ) and ( 3π /, π ), hr co y >, iy and qal to for ( π /, 3π / ), hr co y <. In ch a mannr obtain that ~ I 3i ( iπ in( π / ) i ) π / iπ / and hnc from (4) can infr, ignoring for a momnt th prciion of th prion, that ( ) ς ( ) + (n + ). It i convnint to n mak a formal chang th riting ( ) ς ( ) ( + n ) (n + ). Thi i th am a (9) ς ( ) + ( n + / ) (). n By contrction, thi i limitd ith R<, bt again thi rlation dfinitly hold alo for R> giving in th limit th ll knon rprntation [] ( ) ς ( ) ( n + / ). n 4

5 Of cor, () i th approimat fnctional qation (applid for ( ) ς ( ) rathr than ς () bt thi i not principal) hich prciion (rror trm) i long tablihd and i givn by th ll knon thorm (P.77 []) THEOREM. W hav ( ) ς ( ) + niformly for σ > than. n σ, t π / C ( n / ) + O(/ σ ) () <, hr C i a givn contant gratr Thi Thorm i not too difficlt to prov by compl variabl mthod a it i don in p. 8 of Rf. []; hovr, th mthod adoptd thr rqir rathr artificial introdction of th intgral z cot πzdz to obtain, i applying th rid thorm, th mming n (corrpondingly, th intgral + i i z πzdz n> + i tan and mming ( n / ) in or c. For or approach, thi am m appar qit natrally and hnc it old b intrctiv to prov thi thorm bad on th propod variabl chang, at lat by pdagogical raon. Thi i don blo. For implicity oftn th condition of th Thorm rqiring t < 4 / C intad of t π / C A follo from abov, for th proof hold rigoroly timat n> <. th diffrnc btn th actal val of th intgral π (co y+ iin + i3π / iπ / and th val ( ) obtaind hn th intgrand i approimatd i a dicd abov. For thi, lt divid th intgration rang a π / π 3π / π π / π 3π π iy. W ill dnot th intgral rpctivly S, S, S3, S4, and analyz thm on by on. All th intgral ar imilar hnc ill analyz only S 4. (Writing for th intgral S, 3 th diffrnc btn th intgrand at hand and that iy iy iy d for an approimat calclation π (co y+ iin π (co y+ iin + + 5

6 aily it imilarity ith S, 4 ). For S 4 mak a variabl chang 3π / π and obtain 4 π (in ~ y i co ~ y ) + Lt dnot π in ~ y a and plicitly rit th dnominator of th π intgrand a in ~ y (co(π co ~ y ) i in(π co ~ y )) +. Th qar of it modl i a + + a co(π co ~, and no ill dmontrat that alay for ~ y [, π / ] ~ ) a + + a co(π co y a / 4 (). Thi i ay to if a bca thn hav a + + a co(π co a + a ( a ) a / 4, and alo () i trivially flfilld if a (hich i alay tr) and co( π co ~ y ). Whn co( π co ~ may b ngativ? For thi on nd co ~ y < /(4 ) and th y 3 / + ~ y S π / iy ~ i ~. ~ ~ in in y co y /( ). Bt if o, ~ y π a > >> th dmontration of (). Th hav (hr and blo t π / iy ~ π / Ty ~ T ): π hich finih ~ ~ ~ (in ~ co ~ < in ~ < ~ ) 4 + π y i y π y y (). ( T 4 ) / ( 4 ) / ~ y π T π ( ) T 4 T 4 Hr d in ~ y ~ y / π valid for ~ y [, π / ]. For ngativ T-4 () i 3 / clarly of th ordr of O(/). Th factor iπ in front of th intgral S 4 i canclld by th dnominator i π ~ in( π / ) in front of I ( q. 8 and th hav that th rror ariing from th approimation of all intgral S,,3, 4 i of th ordr of O(/). Thi finih th proof: chang from π to arbitrary i tandard for thi Thorm, cf. [], and do not po any problm... Hritz rprntation of th ς (, fnction Similar conidration can b givn for intgral () and (3) lading to knon intgral rprntation of th fnction involvd. A an ampl, lt ho Hritz rprntation of th ς (, [5] can b obtaind by th mthod ndr dicion. Th variabl chang at qtion tranform (3) into Γ ) (, n, m ln( n) + iπ (/ + m a ( ς. Th impl pol ar locatd at d π / π ) ith n intgr and, ±, ±,... th rid ar qal to Ty ~ m Similarly a abov, 6

7 ( ), m n m ( ) πnai m+ m i( ) (n) π p( iπ / ) p( iπm)p(( ) π nai ). Application of th am contor a abov bt ith th vrtic at X, ln(π + π ), ln(π + π ) + πi, + πi giv ( iπ πi(π ) ) ) ς (, p( iπ / ) n n ( p(πnai) + p( πnai) p( iπ)) I. + I' (To avoid mindrtanding, ndr th m ign plicitly rit th m π of rid for th pair m, for any n). Uing again ) and in π ) in π iπ i iπ aftr trivial algbra on can th rrit ς (, )(π ) in( π / ) n in(πna + π / ) (3). n hnc for R< lim I. hr took th limit Rlation (3) i, of cor, Hritz formla again, it i obtaind hr in a dirct and traightforard ay ithot introdction of any artificial contor, cf. [5]. In th am ay an approimat fnctional qation for ς (, alo can b aily obtaind. 3. Intgral rprntation for th incomplt gamma-fnction A a alra mntiond, incomplt gamma-fnction i dfind, hnvr appropriat, a γ (, d [3] and by analytical contination othri. W mak th am variabl chang and not that th intgrand contain no pol. o tak th mor gnral rctanglar contor formd by th point X, ln, ln + πni, X + πni hr n ±, ±,... and for a momnt i poitiv. Uing th am notation a i n abov, obtain ( π ) γ (, I + I' hr I πn (ln + i πn iy i ln + iy i. (co y+ iin Th hav a nmbr of rprntation ( i iπn ) γ (, i πn co y πn iy (co( y in (co y + i in y ) + i in( y in ) (4). 7

8 hich rprnt incomplt gamma-fnction if th val of n ar nonintgr and hich, by analytical contination, ar valid for arbitrary compl val of. Of cor, th rprntation ar cloly connctd ith th knon rprntation [3] π coϑ γ (, co( ϑ + inϑ) dϑ in π (5) hich i ally provn in a rathr indirct ay ing Hankl gnralization of Γ (z) a a contor intgral,.g. p. 44 of Rf. [6]. At th am tim, th rprntation (5) can b aily and dirctly obtaind by or mthod a follo. Lt introdc an ailiary fnction ~γ (, d, mak th am variabl chang and conidr a rctanglar intgration contor formd by th vrtic X π i, ln πi, ln + πi, + πi ith ral X. It i ay to that along th lin ( πi, ln πi) or intgral z z dz i qal to iπ iπ d γ (, ) (ln hil along th lin + π i, + πi) it i qal to i π γ (,. Th, rpating all hat hav bn don abov (not that th intgral I ' i again ngligibl for R>), hav: π iy (coy+ iin coy i (co( y + iπ iπ ( ) γ (, i in π π. Thi i actly th rprntation (5), and ch a rprntation alo can b gnralizd by taking th rctanglar contor formd by th point X nπ i, ln nπi, ln + nπi, + nπi hr n i an intgr. Thi, if n i not an intgr, giv πn coϑ γ, co( ϑ + inϑ) dϑ in πn ( (6). REMARK. Whn analyzing th Rimann ς (, -fnction, a mor gnral rctanglar contor formd by th point X, ln(π ), ln(π ) + πni, + πni, hr n ±, ±,... alo can b takn. Qit imilarly, on can alo rpat th am hint hich a d dring th analyi of th incomplt gamma-fnction and tak th contor ith th vrtic at X π ni, ln(π ) πni, ln(π ) + πni, + πni, hr n,,... Unfortnatly, thi do not lad to any n intrting rlation. 8

9 4. Intgral rprntation for th conflnt hyprgomtric Φ - fnction, bta fnction and hyprgomtric F-fnction Or nt ampl dal ith th intgral rprntation of conflnt hyprgomtric Φ -fnction. W hav for poitiv and Rc>Ra> (Rf. [5] P. 55): Φ a ca ( a, ( ) d c Or variabl chang convrt thi into Φ a ca ( a, ( ) d c (7) (8). o tak imilar rctanglar contor a abov ith th vrtic at X,, πi, + πi hr ral X and indnt th cornr, πi of th contor. Clarly, for Rc>Ra th contribtion of intgral ovr th infinitimally mall qartr-circl rlatd ith th indntation vanih, and for Ra> th vrtical intgral I ' alo vanih. Inid th contor hav no pol and or intgrand i a ingl-vald fnction, hnc obtain th folloing n intgral rprntation of th conflnt hyprgomtrical fnction: π π i iy ia aiy iy ca ( ) Φ( a, ( ) ( πia ) Φ( a, π c i co y ) ) a c a (9). Similarly a abov, taking mor gnral contor. Thi can b rrittn a (co( in y + a + i in( in y + a)( co y i in n ±, ±,... obtain alo th rprntation ( nπia ) Φ( a, πn ca X,, nπi, + nπi ith i co y ca (co( in y + a + i in( in y + a)( co y i in c (). By analytical contination (9) and () ar valid for all compl, a, c providd Rc>Ra and hn ± na ar not intgr; of cor, th pol of th gamma-fnction alo hold b avoidd a it tak plac alo for (8). Th am hint that d in th prvio ction can b d to obtain mor ymmtric intgral rprntation. For thi introdc an 9

10 a ca ailiary intgral ( + ) d, th am variabl chang and th contor ith th vrtic at X π ni, πni, πni, + πni ith ral X. Or variabl chang tranform th intgral into a c a ( + ) d, and it i ay to that along th lin ( X πni, π ni ) it i qal to i na iπ na a ca ( ) d hnc π a ca ( ) d hil along th lin ( π ni, X + πni ) it i qal to in( πan) Φ( a, πn i co y ca (co( in y a iin( in y a)( + co y + iin c πn (). Hr ± na ar not intgr and again th pol of th gamma-fnction hold b avoidd. Qit imilarly th intgral qality dfining th bta fnction (R>, Ry>) [Rf. 5 P. 9]: B y (, ( ) d can b tratd. Thi lad to th rprntation πn () niπ y ( ) B(, i (co( ϑ) + i in( ϑ))( coϑ i inϑ) dϑ (3). By analytical contination, (3) i valid if Ry> and ± n i not an intgr. ot th diffrnc btn (3) and th intgral apparing aftr th impl variabl chang inθ (or imilar) in (): B(, π / in y ϑ( inϑ) coϑdϑ (4). Again, th am hint that a d for an incomplt gamma-fnction to obtain mor ymmtric intgral rprntation can b rpatd by y introdcing an ailiary intgral ( + ) d and conidring th contor formd by th vrtic nπ, ith ral X : X π ni πni, πni, + πni y in( πn B(, (co( ϑ) + i in( ϑ))( + coϑ + i inϑ) dϑ (5) nπ

11 hr n i not an intgr. Th am approach can b d to obtain an intgral rprntation for th hyprgomtric fnction F ( a, b; z) tarting from th knon rprntation F cb a ( a, b; z) ( ) ( z) d b) c b) b valid for R c > R b >, arg( z) < π ; [5] P. 4. For th condition or tandard rctanglar contor ith indntd cornr hich d to t th Φ ( a, fnction again contain no pol and hav ( nπib i c (6). ) F( a, b; z) πn (co( b + i in( b)( co y i in cb ( z co y iz in Hr nb i not an intgr and th pol of gamma fnction hold b avoidd. Similarly a abov, mor ymmtrical rprntation can b obtaind introdcing an ailiary intgral b cb a ( + ) ( + z) d and conidring th contor formd by th vrtic c, ith ral X : X π ni πni, πni, + πni in( πnb) F( a, b; z) nπ nπ (co( b + i in( b)( + co y + i in cb ( + z co y + iz in a (7). a 5. Conclion Th hav hon that th variabl chang i fl for an analyi of intgral rprnting th Rimann ς fnction, incomplt gamma-fnction, conflnt hyprgomtric Φ -fnction and bta fnction. Thr ar no dobt that othr intrting and yt nplord application of thi variabl chang can b fond. And no lt mak th final rmark. Might it b raonabl to propo th introdction of th incomplt Rimann ~ ς -fnction hich may b dfind for poitiv and R> a ) ~ ς (, d or imilarly and for othr paramtr by an appropriat analytical contination? Thi fnction

12 dpn th analogy btn th Rimann fnction and gamma fnction and might b fl for om phyical application. For ampl, it i ll knon that th pctral dnity of thrmal bo radiation nrgy i prd in 3D via th Rimann fnction 3 4) ς (4) d (and via othr rlatd fnction in othr nmbr of dimnion and may b in fractal alo (?)). Th hight poibl frqncy of th ytm can b limitd by diffrnt phyical raon (.g. by Planck lngth and tim cal if nothing l) hich apparntly mak th introdction of th corrponding incomplt fnction qit raonabl: in 3D th ral pctral dnity of th 3 blackbo radiation i 4) ~ ς (4, d, and o on. REFERECES []. Waton, A trati on th thory of Bl fnction, Cambridg, Cambridg Univ. Pr, 958, P. 76. [] E. C. Titchmarh and E. R. Hath-Bron, Th thory of th Rimann Zta-fnction, Oford, Clarndon Pr, 988. [3] A. Erdlyi, Highr trancndntal fnction, Vol. II, McGro and Hill, -Y., 953, P. 37. [4] A. Brgtrom, Proof of Rimann' zta-hypothi, arxiv:89.5, 8. [5] A. Erdlyi, Highr trancndntal fnction, Vol. I, McGro and Hill, - Y., 953, P. 6. [6] E. T. Whittakr and G.. Waton, A cor of modrn analyi, Cambridg, Cambridg Univ. Pr, 965. S. K. Skatkii, Laboratoir d Phyiq d la Matièr Vivant, IPSB, BSP 48, Ecol Polytchniq Fédéral d Laann, CH 5 Laann- Dorigny, Sitzrland. rgi.katki@pfl.ch

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero. SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain