Power Means Calculus Product Calculus, Harmonic Mean Calculus, and Quadratic Mean Calculus

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1 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Powr Ms Clculus Product Clculus, Hrmoic M Clculus, d Qudrtic M Clculus H. Vic Do Mrch, 008

2 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Ech Powr M of ordr 0 Abstrct r r r r r, ( ) ( r) is ssocitd with Powr M Drivtiv of ordr r, D. W dscrib th Arithmtic M Clculus obtid if r, Gomtric M Clculus obtid if r 0, Hrmoic M Clculus obtid if r, Qudrtic M Clculus obtid if r Kywords Clculus, Powr M, Drivtiv, Itgrl, Product Clculus. Gmm Fuctio, Mthmtics Subjct Clssifictio 6A06, 6B, 33B5, 6A4, 6A4, 46G05,

3 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Cotts Itroductio.. 9. Arithmtic M Clculus.... Th Arithmtic M of f () ovr [ b...,]. M Vlu Thorm for th Arithmtic M....3 Th Arithmtic M of () f ovr [, d] Arithmtic M Drivtiv 3.5 Th Arithmtic M Drivtiv is th Frmt-Nwto-Libit Drivtiv. 3.6 Th Arithmtic M Drivtiv is Additiv Oprtor Th Arithmtic M Drivtiv is ot Multiplictiv Oprtor 4. Th Product Itgrl..5. Growth problms 5. Th Product Itgrl of rt () ovr [,] b Th Product Itgrl of f ( ) ovr [ b...6,].4 Itrmdit Vlu Thorm for th Product Itgrl.7.5 Th Product Itgrl is Multiplictiv Oprtor 8 3. Gomtric M d Gomtric M Drivtiv Th Powr M with r 0 is th Gomtric M Th Gomtric M of f ( ) ovr [ b.....0,] 3.3 M Vlu Thorm for th Gomtric M Th Gomtric M of f ( ) ovr [, + d] Gomtric M Drivtiv 3

4 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do (0) Dlog G( ) D G( ).. DG( ) (0) G ( ) D G() Th Gomtric M Drivtiv is o-dditiv oprtor Th Gomtric M Drivtiv is multiplictiv oprtor Gomtric M Drivtiv Ruls Gomtric M Clculus Fudmtl Thorm of th Product Clculus Tbl of Gomtric M Drivtiv, d Product Itgrls Product Diffrtil Equtios Product Diffrtil Equtios...6 dy d 5. Product Clculus Solutio of Py ( ) dy d 5.3 Py ( ) + Q ( ) my ot b solvd by Product Clculus y'' P( ) y' + Q( ) y my ot b solvd by Product Clculus Product Clculus of si Eulr s Product Rprsttio for si Covrsio to Trigoomtric Sris Gomtric M Drivtiv of si 6.4 Scod Gomtric M Drivtiv of si 6.5 Product Itgrtio of si 6.6 Eulr s d Product Rprsttio for si

5 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 6.7 Gomtric M Drivtiv of Eulr s d product for si Product Clculus of si Eulr s Product Rprsttio for si Gomtric M Drivtiv of si Th Wllis Product for π Product Clculus of cos Eulr s Product Rprsttio for cos Gomtric M Drivtiv of cos Product Clculus of t Product Rprsttio for t Gomtric M Drivtiv of t Product Clculus of sih Product Rprsttio of sih Gomtric M Drivtiv of sih Product Clculus of cosh Product Rprsttio of cosh 46. Gomtric M Drivtiv of cosh Product Clculus of th Product Rprsttio for th Gomtric M Drivtiv of th Product Clculus of 3. Product rprsttio of 3. Gomtric M Drivtiv of

6 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3.3 Gomtric M Drivtiv of 3.4 Gomtric M Drivtiv of k Gomtric M Drivtiv by Epotitio (0) cot D si 5 (0) D Product Clculus of Γ () Eulr s Product Rprsttio for Γ () Gomtric M Drivtiv of Γ () Γ () Γ ( + ) Γ ( ) Product Rflctio Formul for Γ () Γ()( Γ ) π si π Γ () π ( ) 5.8 ( ) Γ Products of Γ () 64 Γ ( + ) 6. Γ ( + w ) Γ ( + w ), whr w + w Γ() Γ ( + i) Γ( i) sih...65 Γ ( + ) Γ ( + ) 6.3 Γ ( + w ) Γ ( + w ) Γ ( + w ) 3 Γ ( + ) Γ ( + )... Γ ( + ) 6.4 Γ ( + w ) Γ ( + w )... Γ ( + w ) k l, whr + w + w + w

7 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 7. Product Clculus of J ( ) ν Product Formul for J () 67 ν 7. Gomtric M Drivtiv of J ().67 ν 8. Product Clculus of Trigoomtric Sris Product Itgrl of Trigoomtric Sris Ifiit Fuctiol Products Gomtric M Drivtiv of Eulr Ifiit Product Pth Product Itgrl Pth Product Itgrl i th Pl Gr s Thorm for th Pth Product Itgrl Pth Product Itgrl i 3 E Stoks Thorm for th Pth Product Itgrl.7. Itrtiv Product Itgrl..73. Itrtiv Product Itgrl of f (,) t..73. Itrtiv Product Itgrl of rtd (, ) Hrmoic M Itgrl.74. Hrmoic M Itgrl Hrmoic M d Hrmoic M Drivtiv Th Hrmoic M of f ( ) ovr [ b...75,] 3. M Vlu Thorm for th Hrmoic M Th Hrmoic M of f ( ) ovr [, d] Hrmoic M Drivtiv 77 7

8 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3.5 D...77 ( ) H( ) DH( ) 4. Hrmoic M Clculus Th Fudmtl Thorm of th Hrmoic M Clculus Tbl of Hrmoic M Drivtivs d Itgrls Qudrtic M Itgrl Qudrtic M Itgrl Cuchy-Schwrt Iqulity for Qudrtic M Itgrls Holdr Iqulity for Qudrtic M Itgrls 8 6. Qudrtic M d Qudrtic M Drivtiv Qudrtic M of f () ovr [ b...83,] 6. M Vlu Thorm for th Qudrtic M Th Qudrtic M of f () ovr [, + d] Qudrtic M Drivtiv..84 D () Q () D Q () ( ) / 7. Qudrtic M Clculus Th Fudmtl Thorm of th Qudrtic M Clculus Tbl of Qudrtic M Drivtivs d Itgrls..86 Rfrcs 88 8

9 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Itroductio W dscrib grlid clculus tht ws suggstd by Michl Spivy s [Spiv] obsrvtio of th rltio btw th Gomtric M of fuctio ovr itrvl, d its product itgrl. W will s tht ch Powr M of ordr r 0, ( ) r + r +... r r is ssocitd with Powr M Drivtiv of ordr r, ( r) D. Th Frmt/Nwto/Libit Drivtiv () d D D d is ssocitd with th Arithmtic M , which is Powr M of ordr r. Th Gomtric M Drivtiv (0) D is ssocitd with th Gomtric M ( ) /... which is Powr M of ordr r 0 [K]. 9

10 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Product Itgrtio is oprtio ivrs to th Gomtric M Drivtiv. Both r multiplictiv oprtios, tht pply turlly to products, d i prticulr to Γ (), th lytic tsio of th fctoril fuctio Th Hrmoic M Drivtiv ( ) D is ssocitd with th Hrmoic M which is Powr M of ordr r. Th Qudrtic M Drivtiv () D is ssocitd with th Powr M of ordr r, ( ) Th ivrs oprtio, th Qudrtic M Itgrtio trsforms fuctio to its L orm squrd. W procd with th dfiitio of th Arithmtic M Drivtiv.. 0

11 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Arithmtic M Clculus. Th Arithmtic M of f () ovr [ b,] Giv fuctio f () tht is Rim itgrbl ovr th itrvl [,] b, prtitio th itrvl ito sub-itrvls, of qul lgth b Δ, choos i ch subitrvl poit c, d cosidr th Arithmtic M of f (), i f ( c) + f( c) +... f( c ) ( f ( c) + f( c) +... f( c )) Δ b As, th squc of th Arithmtic Ms covrgs to whr b Fb () F () fd ( ) b, b t F ( ) ftdt ( ). t 0 Thrfor, th Arithmtic M of f () ovr [ b,] is dfid by b fd ( ) b

12 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do. M Vlu Thorm for th Arithmtic M Proof: Thr is poit < c < b, so tht Sic t t 0 b f ( d ) fc ( ) b. F ( ) ftdt ( ) is cotiuous o [,] b, d diffrtibl i ( b,), by Lgrg Itrmdit Vlu Thorm thr is poit so tht Tht is, < c < b, Fb () F () b b f () c. f ( d ) fc ( ) b..3 Th Arithmtic M of f () ovr [, + d] Th Arithmtic M of f ( ) ovr [, + d] is th Ivrs oprtio to Itgrtio Proof: By., thr is so tht < c < +Δ, t +Δ f () tdt fc () Δ t Lttig Δ 0, th Arithmtic M of f () t, quls f ().

13 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do t +Δ lim f ( tdt ) f ( ) Δ 0 Δ t. Thus, th oprtio of fidig th Arithmtic M of f ( ), is ivrs to itgrtio. This lds to th dfiitio of th Arithmtic M Drivtiv..4 Arithmtic M Drivtiv of Th Arithmtic M Drivtiv of t F ( ) ftdt ( ) t t 0 t F ( ) ftdt ( ) t 0 t is dfid s th Arithmtic M of f () ovr [, + d] () Δ 0 Δ t +Δ D F ( ) lim f ( t ) dt t.5 Th Arithmtic M Drivtiv is th Frmt- Nwto-Libit drivtiv () D F( ) df( ) d t + d () Proof: D F( ) Stdrd Prt of f( t) dt d Stdrd Prt of t F ( + d) F ( ) d 3

14 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do df( ) d DF( )..6 Th Arithmtic M Drivtiv is Additiv Oprtor ( ) D F( ) + F ( ) DF( ) + DF ( ) Thus, th Arithmtic M Drivtiv pplis ffctivly to ifiit sris..7 Th Arithmtic M Drivtiv is ot multiplictiv oprtor ( ( ) ( )) ( ( )) ( ) + ( )( ( )) D F F DF F F DF Thus, Arithmtic M Drivtiv dos ot pply sily to ifiit products. 4

15 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Th Product Itgrl. Growth Problms Th Arithmtic M Drivtiv is usuitbl wh w r itrstd i th quotit Prst Vlu Ivstd Vlu Similrly, ttutio or mplifictio is msurd by Out-Put Sigl I-Put Sigl Th d for multiplictiv drivtiv oprtor motivtd th crtio of th product itgrtio... Th Product Itgrl of rt () ovr th itrvl [ b,] A mout A compoudd cotiuously t rt rt () ovr tim dt bcoms rtdt () A. Ovr qul sub-itrvls of th tim itrvl [ b,,] Δ t, b w obti th squc of fiit products rt ( ) Δt rt ( ) Δt rt ( ) Δ t [ rt ( ) + rt ( ) rt ( )] Δt A... A. 5

16 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do As, th squc covrgs to Th mplifictio fctor t b A t rtdt () is clld t b th product itgrl of t rtdt () rt () ovr th itrvl [ b,] d is dotd t b rtdt (). t Thus, th Product Itgrl of rt () ovr th itrvl [ b,] is t b t b rtdt () rtdt () t t.3 Th Product Itgrl of f ( ) ovr th itrvl [ b,] Giv Rim itgrbl, positiv f ( ) o [ b,,] prtitio th itrvl ito sub-itrvls, of qul lgth b Δ, choos i ch subitrvl poit c, d cosidr th fiit products, i 6

17 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Δ Δ Δ f ( c ) f( c )... f( c ) Δ Δ Δ l[ fc ( ) fc ( )... fc ( ) ] [l f ( c ) + l f( c ) l f( c )] Δ. As, th squc of products covrgs to b ( l ( )) f d > 0. W cll this limit th product itgrl of f ( ) ovr th itrvl [ b,,] d dot it by b d f ( ). Thus, th Product Itgrl of f ( ) ovr th itrvl [ b,] is f ( ) b ( l ( )) b f d d.4 Itrmdit Vlu Thorm for th Product Itgrl Thr is poit < c < b, so tht t Proof: Sic ( ) ( l ( )) t 0 b d ( b ) f ( ) fc ( ) ϕ f t dt is cotiuous o [ b,,] d diffrtibl i ( b,), by Lgrg Itrmdit Vlu Thorm thr is poit 7

18 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do so tht Hc, b < c < b, ( ) ϕ ϕ ( ) l f ( ) d ( b) ( ) l f( c) ( b ). b ( l fd ( )) ( ) f () c l fc ( ) ( b ) ( b )..5 Th Product Itgrl is multiplictiv oprtor If < c < b, b c b d d d f ( ) f ( ) f ( ) c Proof: If < c < b, c b ( l f( ) ) d+ ( l f( ) ) d d ( ) c d d f f ( ) f ( ) b c b. c Th ivrs oprtio to product itgrtio is th Gomtric M Drivtiv. 8

19 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3 Gomtric M d Gomtric M Drivtiv 3. Th Powr M with r 0 is th Gomtric M Proof: Lt r 0 i r r r ( r ) (... 3) r 0 r r r r r r... r r ( ) ( ) + + log... log + +. Th, th pot log( r r... r + + ) log r d by L Hospitl, its limit is r 0 r r r { log( ) log } Dr lim Dr Thrfor, r r 0 r r r r r r ( ) lim l + l +... l ( ) l l... l ( ) l.... ( r r r l(... ) ) r (... ) r 0. is of th form 0 0, 9

20 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3. Gomtric M of f() ovr [ b,] Giv itgrbl fuctio f () tht is positiv ovr [,] b, prtitio th itrvl, ito sub-itrvls, of qul lgth b Δ, choos i ch subitrvl poit c, d cosidr th Gomtric M of f () ( ( ) ( )... ( ) ) i / ( l f ( c) + l f( c) +...l f( c )) Δ fc fc fc b. As, th squc of Gomtric Ms covrgs to whr Thrfor, b b Gb G ( ) ( l f( ) ) d (), t G ( ) t ( l ( )) f t dt b b ( l ( )) f d is dfid s th Gomtric M of f ( ) ovr [ b.,] 3.3 M Vlu Thorm for th Gomtric M Thr is poit < c < b, so tht b ( l f( ) ) d b f() c 0

21 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Proof: By.3, d Th Gomtric M of f ( ) ovr [, + d] Th Gomtric M of f ( ) ovr [, + d], is th Ivrs Oprtio to Product Itgrtio Proof: By 3.3, thr is so tht < c < +Δ, t +Δ l f( t) dt Δ t f() c. Lttig Δ 0, th Gomtric M of f () t quls f ( ). l f( t) dt Δ lim t f( ) Δ 0 t +Δ Thus, th oprtio of fidig th Gomtric M of f ( ) ovr [, + d], is ivrs to product itgrtio ovr [, + d]. This lds to th dfiitio of th Gomtric M Drivtiv 3.5 Gomtric M Drivtiv Th Gomtric M Drivtiv of t G ( ) t ( l ( )) f t dt t is dfid s th Gomtric M of f ( ) ovr [, + d]

22 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do (0) D G( ) lim Δ 0 t +Δ ( ) l f( t) dt Δ t 3.6 Proof: (0) (0) log ( ) D G D G( ) D G( ) lim Δ 0 t +Δ ( ) l f ( t) dt Δ t t +Δ 0 Δ Δ t ( ) lim l f ( t ) dt Stdrd prt of t + d ( ) l f ( t) dt d t ( ) ( + ) Stdrd Prt of G d d G ( ) log ( ) log ( ) + Stdrd Prt of d G d G Dlog G( ). 3.7 DG( ) (0) G ( ) D G( ) 3.8 Th Gomtric M Drivtiv is o- dditiv oprtor Proof: (0) ( ( ) + ( )) G( ) + G( ) +. D ( G G )( ) DG G

23 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3.9 Th Gomtric M Drivtiv is multiplictiv oprtor (0) Proof: D G ( ) G( ) Dlog[ G ( ) G ( )] DG ( ) ( ) ( DG + G ( ) G ( ) ) DG( ) DG( ) G( ) G( ) ( D (0) G )( (0) ( ) D G( ) ). 3.0 Gomtric M Drivtiv Ruls ( ) (0) (0) Dl G( ) D l G( ) D G() D ( ) (0) D l G( ) D G( ) ( ) (0) g ( ) Dg ( ) gd ( ) l f ( ) D f( ) f( ) df dg dg d (0) ( ( )) D f(()) g f g 3

24 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 4 Gomtric M Clculus 4. Th Fudmtl Thorm of th Product Clculus t (0) dt D f() t f( ) t Proof: t D f t D ( l ( )) t f t dt (0) dt (0) () t t t t ( l f( t) ) dt t t D ( l f( t) ) dt t ( l f( t) ) dt t t D l f( t) dt t t t ( l f( t) ) ( ) dt t ( l ( )) D f t dt t f (). 4. Tbl of Gomtric M Drivtivs d itgrls 4

25 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do W list som gomtric M Drivtivs, d Product Itgrls. Som of ths r giv i [Spiv]. f f f (0) (0) () D () I f( ) () / (l ) / (l ) / (l ) / (l ) / log l( ld ) l / (l /)/ + /( ) + si cot cos t t /si l(si d ) l(cos d ) l(t d ) si cos cos cos si si 4 5

26 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 5 Product Diffrtil Equtios 5. Product Diffrtil Equtios A Product Diffrtil Equtio ivolvs powrs of th Gomtric M Drivtiv oprtor d o sums, oly products. (0) D, A ordiry diffrtil qutios ivolvs sums of powrs of th Arithmtic M Drivtiv oprtor D. Such qutio is ot suitbl to th pplictio of product diffrtil qutio. (0) D, d dos ot covrt sily ito [Doll] ttmpts to writ th solutios to ordiry diffrtil qutios i trms of product itgrl, but big uwr of th Gomtric M Drivtiv, it fils to produc o product diffrtil qutio. [Doll] dmostrts tht products itgrls r ot turl solutios for ordiry diffrtil qutios. Th ttmpt md i [Doll] to itrprt Summtio Clculus i trms of th Product Itgrl lo, big oblivious to th product Clculus drivtiv, dos ot ld to bttr udrstdig of diffrtil qutios, or to y w rsults. 6

27 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Oly th bsic qutio dy Py ( ) d my b covrtd to product diffrtil qutio, d b solvd s such. dy 5. Product Clculus Solutio of Py ( ) d. Dividig both sids by y (), y ' P ( ) y. y ' y P ( ) (0) ( ) D y P y ( ) t Ptdt () Ptdt () t 0. t 0 t dy 5.3 Py ( ) + Q ( ) my ot b solvd by Product Clculus d Proof: W do t kow of Product Clculus mthod to solv th qutio dy Py ( ) Q ( ) d +. 7

28 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do I Arithmtic M Clculus, w multiply both sids by t 0 Th, t t t y ' + yp ( ) Q ( ) P() t dt P() t dt P() t dt t 0 t 0 t 0 t Ptdt (). t t d ( y ) Q ( ) d Ptdt () Ptdt () t 0 t 0 y t t u Ptdt () u Ptdt () t 0 t 0 u 0 Q( u) Writig this s y u Ptdt () u 0 Qu ( ) t t 0 t u t 0 Ptdt () y u t Qu ( ) u 0 t t 0 t 0 Ptdt () Ptdt () dmostrts why th qutio cot b covrtd ito product diffrtil qutio, d cot b solvd s such: I product Clculus w d to hv pur products. No summtios. 8

29 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 5.4 y'' P( ) y' + Q( ) y my ot b solvd by Product Clculus Proof: W my writ s first ordr systm y'' P( ) y' + Q( ) y y' ' P( ) + Q( ) y Thus, i mtri form, d y 0 y d Q( ) P( ). But w d th mthods of summtio Clculus, to obti two idpdt solutios y (), d y () tht sp th solutio spc for th qutio. 9

30 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 6 Product Clculus of si 6. Eulr s Product Rprsttio for si si For y compl umbr, cos cos cos Proof: si cos si cos cos si 4 4 cos cos cos si cos cos cos... cos si 4 8 ( ) si cos cos cos...cos. 4 8 Thrfor, for y compl umbr 0, si si cos cos cos...cos 4 8 Lttig, si cos cos cos This holds lso for 0. Hc, it holds for y compl umbr. 6. Covrsio to Trigoomtric Sris 30

31 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Products of Cosis c b covrtd ito summtios, d th ifiit product my b covrtd ito Trigoomtric Sris. For istc, ( ) cos αcos βcos γ cos( α + β) + cos( α β) cos γ cos( α + β)cos γ + cos( α β)cos γ cos( α + β + γ) + cos( α + β γ) cos( α β + γ) + cos( α β γ) Gomtric M Drivtiv of si cos t t t si Proof: Gomtric M Diffrtitig both sids of 6., (0) si (0) (0) (0) D ( D cos )( D cos 4)( D cos 8)... si cos cos D cos D D D 3 si cos cos cos 3... Tht is, t t cos t si 3 3 cos t t t si 3

32 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 6.4 Scod Gomtric M Drivtiv of si + si cos cos cos Proof: Scod Gomtric M Diffrtitio of 6. givs cos si ( ) ( t t ) D D D (0) (0) (0)... + si ( ) cos cos cos Tht is, +... si 4 6 cos cos cos 3 Th lst sris c b obtid by trm by trm srisdiffrtitio of Product Itgrtio of si si B 3 B4 5 B6 7 B8 9 log d ( ) + ( ) ( ) + ( ) ! 8 5! 7! 6 9! Whr th B, B4, B6,... r th Broulli Numbrs. 3

33 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Proof: Product itgrtig 6., si log d logcosd logcosd logcosd By [Grob, p.3, 8b], for < π, up to costt, si log d log si d log d B 3 B4 5 B6 7 B8 9 ( ) + ( ) ( ) + ( )..., 4 3! 8 5! 7! 6 9! whr th B, B4, B6,... r th Broulli Numbrs. By [Grob, p.3, 9b], for < π, up to costt, log cos d log cos d 4 6 ( ) B 3 ( ) B4 5 ( ) B 6 7 ( ) ( ) ( ) ! 8 5! 7! log cos d 4 log cos d ( ) B 3 ( ) B4 5 ( ) B6 7 4 ( ) ( ) ( ) ! 8 5! 7!.. Comprig th cofficits of 3, 5, 7,... o both sids, dos ot yild y w rsult. 33

34 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 6.6 Eulr s d Product for si For y compl umbr si cos 4cos 4cos Proof: Usig th tripl gl formul, [Zid, p.57], w writ si 3 si 4 si ( ) si 3 4[ cos ] 3 3 ( 4cos ) si 3 3 ( )( ) si 3 3 si 4 si, 4cos 4cos si ( ) ( ) 4cos... 4cos si cos 4cos si Thrfor, for y compl umbr 0, Lttig, si si si 4cos cos 4cos 4cos

35 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Sic this holds lso for 0, it holds for y compl umbr icludig Gomtric M Drivtiv of Eulr s d Product Prsttio for si 4si 4 si 4 si cos si 3(4 cos ) 3 (4 cos ) 3 (4 cos ) Proof: Gomtric M Diffrtitig 6.6, 4cos (0) si (0) 3 (0) 3 4cos D D D D 4cos 4cos si 4 cos 4 cos 4 cos D D D si 4cos si 4si 4si (4cos ) 33(4cos ) cos 3(4cos ) si ( ) si 4si 4si cos si 3(4cos ) 3 (4cos ) 3 (4cos )

36 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 7 Product Clculus of si 7. Eulr s Product Rprsttio for si For y compl umbr, si... π ( π) (3 π) Th product covrgs bsolutly i y disk < R. Proof: si si cos ( ) si cos si + π 4 4 ( π π ) ( ) si si + si π π π ( ) ( ) ( ) si si + si + cos π + π π ( ) ( ) ( ) si si si cos + π + π π π ( ) ( ) ( ) si si si si + + π + π π ( ) ( ) ( ) si si si si + π + π π ( ) ( ) ( ) si si si si + π ( ) ( π + π3 ) ( ) ( 3 π ) si si si si

37 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Now, + π + π π si si si si + π ( ) ( π + 3π ) ( ) ( 3 π ) si si si si... + ( ) π ( ) π... si si ( + + )( + + ) + π π π π + + π π si si si cos si cos + π ( si cosπ )( siπ + π cos ) ( si si π )( si π si ) + Ad, π si si Thrfor, + π π π si si si si si si cos ( si π si ) Tht is, ( ) ( ( ) si π si... si π si ) si π ( ) si ( ) si π si... si π si cos si si Lttig 0, ( ) ( ) 37

38 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do π π ( ) π si si... si Dividig by this lst qutio, si si cos.. si si si si π si π ( ) π si si si si si cos.. si π si π ( ) π si Lttig, for y fid turl umbr m w hv si m π mπ si mπ mπ mπ si si ( ) ( ) Cosqutly, th ifiit product Covrgs to si.... π ( π) (3 π) Th covrgc is bsolut i y disk < R, bcus th ifiit sris π ( π) (3 π) covrgs bsolutly i y disk < R. 38

39 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Idd, i < R, π ( π) (3 π) π 3 π π 6 < 6 R. 7. Gomtric M Drivtiv of si cot... π ( π) (3 π) Proof: (0) (0) (0) (0) (0) D si D D D D... π ( π) (3 π) Thus, cos ( ) (3 ) si π π π... cot... π ( π) (3 π) 7.3 Th Wllis Product for π π

40 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Proof: Wllis product for π follows from th product formul for si π, [Brt, p. 44]. si π π π π

41 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 8 Product Clculus of cos 8. Eulr s Product Rprsttio for cos, For y compl umbr, cos... π 3π 5π Th covrgc is bsolut i y disk < R. Proof: cos si si... π π 3π 4π 5π... π π 3π 4π 5π... π 3π 5π 8. Gomtric M Drivtiv of cos t π ( ) (3 π) ( ) (5 π) ( ) 4

42 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Proof: cos... π 3π 5π (0) (0) (0) (0) D D D D Thus, si cos π ( ) (3 π) ( ) (5 π) ( ) t π ( ) (3 π) ( ) (5 π) ( ) 4

43 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 9 Product Clculus of t 9. Product Rprsttio for t For y compl umbr, t... π ( π) (3 π)... π 3π 5π Th covrgc is bsolut i y disk < R. Proof: 7. d Gomtric M Drivtiv of t 8 + si π π ( ) 8 + ( π) (3 π) ( ) (3 π) (5 π) ( ) Proof: 43

44 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do D (0) t (0) (0) (0) (0) D D D D... π π 3π (0) (0) (0 D D ) D π 3 π... 5 π Thus, si ( ) (3 ) π π π π ( ) (3 π) ( ) (5 π) ( ) 8 + si π π ( ) 8 + ( π) (3 π) ( ) (3 π) (5 π) ( ) 44

45 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 0 Product Clculus of sih 0. Product Rprsttio of sih sih π ( π) (3 π) Th covrgc is bsolut i y disk < R. Proof: sih isii, d us Gomtric M Drivtiv of sih coth... + π + + ( π) + + (3 π) + + Proof: (0) (0) (0) (0) (0) D sih D D + D + D +... π ( π) (3 π) Thus, Dsih ( ) (3 ) sih π + π + π +... coth... + π + + ( π) + + (3 π)

46 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Product Clculus of cosh. Product Rprsttio of cosh cosh π 3π 5π Th covrgc is bsolut i y disk < R. Proof: cosh cosi, d pply 8.. Gomtric M Drivtiv of cosh th π + ( ) (3 π) + ( ) (5 π) + ( ) Proof: (0) (0) (0) (0) D cosh D + D + D +... π 3π 5π Thus, Dcosh ( ) (3 ) ( ) (5 ) ( ) cosh π + π + π th π + ( ) (3 π) + ( ) (5 π) + ( ) 46

47 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Product Clculus of th. Product Rprsttio for th For y compl umbr, th π ( π) (3 π) π 3π 5π Th covrgc is bsolut i y disk < R. Proof: 0. d.. Gomtric M Drivtiv of th 8 + sih π + π + ( ) 8 + ( π) + (3 π) + ( ) (3 π) + (5 π) + ( ) Proof: 47

48 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Now, D (0) th (0) (0) (0) (0)... D D + D + D +... π π 3π (0) (0) (0) D + D + D + π 3π 5π Thrfor, cosh sih Dth cosh. th sih sih cosh sih cosh Thus, sih π+ ( π) + (3 π) π+ ( ) (3 π) + ( ) (5 π) + ( ) sih π + π + ( ) 8 + ( π) + (3 π) + ( ) (3 π) + (5 π) + ( ) 48

49 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3 Product Clculus of 3. Product rprsttio of + 3. Gomtric M Drivtiv of (0) D Proof: D (0) (0) + D + ( D + ) Thus, (0) D. 49

50 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 3.3 Gomtric M Drivtiv of (0) D (0) (0) Proof: D + D + D( + ) (0) Thus, D. 3.4 Gomtric M Drivtiv of k k k (0) k D 50

51 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 5 Proof: (0) (0) k k D D + + ( ) k k D + + k k k + k k k + k k Thus, (0) k k k D.

52 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 4 Gomtric M Drivtiv by Epotitio Gomtric M Drivtiv c b obtid by usig th Product Clculus of th potil fuctio. W dmostrt this mthod by mpls. 4. (0) cot D si Proof: Sic log si log si si lim ( + ), w pply th Gomtric M Drivtiv to log si ( + ). D (0) log si log si fctors (0) log si (0) log si D +... D + fctors 5

53 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do log si D( + ) log si + cot log si + cot logsi / + cot. Thus, (0) cot D si. 4. Proof: Sic (0) D w pply D (0) D to log log lim ( ) +, log ( + ). (0) log (0) log + D + 53

54 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do log D( + ) log + + log log + + log log + + log Thus, (0) D. 54

55 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 5 Product Clculus of Γ ( ) O th hlf li > 0, Eulr dfid th rl vlud Gmm fuctio by t t 0 t Γ ( ) t dt. I th hlf pl R > 0, th compl vlud itgrl t t 0 t t dt covrgs, d is diffrtibl with t t t t t 0 t 0. D t dt t ltdt Thus, th compl vlud itgrl tds th Eulr itgrl ito lytic fuctio i th hlf pl R > 0. It is dotd by Γ (). This fuctio c b furthr tdd to product rprsttio tht is lytic for y, cpt for simpl pols tht it hs t 0,,, 3,... Th fuctio / Γ ( ), tht is giv by th ivrs product, is lytic for y, with simpl ros t 0,,, 3,... 55

56 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Thrfor, th turl clculus for Γ () d for / Γ ( ) i th compl pl is th product Clculus. 5. Eulr s Product Rprsttio for Γ () Γ () ( + ) ( + ) ( + ) 3 ( + )( + )( + ) Proof: t t Γ () t dt t 0 t t lim t 0 t dt Uiform covrgc llows ordr chg of limit, d itgrtio t t lim t 0 t dt Th chg of vribl, u t /, du dt /, givs u u 0 ( ) lim u u du c b writt s product ( + ) ( + ) ( ) ( ) ( )...( ) 3 ( + )

57 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Itgrtig by prts with rspct to u, kpig, d fid u u u u u du u d ( ) ( ) u 0 u 0 u u u u ( u) d( u) u u 0 u 0 u 0 u( u) du u + u ( u) d + u 0 u + u ( u) d + + u 0 u u ( u) d u 0. u + ( ) u... ( u) d u 0 + ( ) u u u 0 57

58 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do ( ) ( ) ( + ) ( + ) ( + ) ( + ) Thrfor, u u 0 ( ) lim u u du lim {( ) ( ) ( )...( ) 3 ( + ) ( + ) ( + ) ( + ) ( + ) lim ( + ) ( + ) ( + )...( + ) 3 ( + )( + )... ( + )( + ) ( + ) ( + ) ( + ) 3 ( + )( + )( + ) Gomtric M Drivtiv of Γ () Γ '( ) lim log... Γ ( )

59 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Proof: D ( + ) ( + ) ( + ) ( + ) ( + ) ( + ) (0) (0) (0) D D D (0) 3 Γ ( ) (0) (0) (0) (0) D D D D D D(3/) D(4/3) (3/) (4/3) lim ( + )... lim lim log lim log Sic Γ'( ) (0) Γ( ) D Γ (), 59

60 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Γ '( ) lim log... Γ ( ) Γ () Proof: ( + )( + )( + )... 3 Γ () ( )( )( ) Γ ( + ) Γ ( ) Proof: Γ ( + ) ( + ) ( + ) ( + ) ( + )( + )( + ) ( + ) ( ) ( + ) ( + )( + )( + )( + )... 3 ( + ) ( + ) ( + ) ( ) ( ) ( ) ( ) ( + ) ( + ) ( + )... 3 ( )( )( )( ) Γ ()

61 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 5.5 Product Rflctio Formul for Γ () Γ()( Γ ) ( )( )( )... 3 Proof: Γ()( Γ ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )( )( ) ( + )( + )( + )... 3 ( + )( )( + ) ( ) ( ) ( ) ( ) ( + )( + )( + ) ( )( )( )( ) ( + )( )( + )( )( + )( ) ( )( )( ) Γ()( Γ ) π si π Proof: By 5.5, Γ()( Γ ) ( )( )( )

62 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do π π ( π) ( ( π) )( ( π) )( ).. π ( π) (3 π) π. si π Γ () π 5.7 ( ) Proof: Substitutig π i Γ()( Γ ), w obti si π Tht is, ( ) ( π Γ Γ ). si π () Γ π. This c b obtid dirctly through th Wllis Product for π. 5.8 ( ) Proof: By 5.5, ( ) Γ Γ

63 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Wllis Formul of 7.3, follows from 5.7, d

64 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 6 Products of Γ () Γ ( + ) 6. Γ ( + w ) Γ ( + w ), whr w + w ( )( ) ( ) ( )( ) ( ) w w w w Γ ( + ) ( + w)( + w) ( w ( ) ) ( w) 3... Γ + Γ Proof: Γ ( + ) Γ ( + w ) Γ ( + w ) ( + ) ( + ) ( + ) ( + )( + )( + )( + ) 3 + w + w + w ( + w )( + )( + )( + )... 3 ( + ) ( + ) ( + ) + w + w + w w + w + w ( + w )( + )( + )( + ) ( + ) ( + ) ( + ) + w + w + w... 3 ( + ) ( + ) ( + ) ( + )( + )( + )( + )

65 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do w w w ( + w )( + )( + )( + ) ( + ) ( + ) ( + ) w w w... 3 w w w ( + w )( + )( + )( + ) ( + ) ( + ) ( + ) w w w... 3 w w w w ( + w)( + )( + ) ( + w 3 )( + )( + 3 ) ( + )( + )( + )( + ) ( + w)( + w) ( ) ( )( ) ( ) ( )( ) ( ) w w w w Γ() Γ ( + i) Γ( i) sih Proof: Γ() Γ ( + i) Γ( i) ( i )( i i i i i )( )( )( )( ) ( )( )( ) sih. Similrly, w obti 65

66 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do Γ ( + ) Γ ( + ) 6.3 Γ ( + w ) Γ ( + w ) Γ ( + w ) 3, whr + w + w + w3. Γ ( + ) Γ ( + ) Γ ( + w ) Γ ( + w ) Γ ( + w ) 3 ( + w)( + w)( + w3) ( )( ) w w w3 ( + )( + )( + ) ( )( ) w w w3 ( + )( )( + 3 ) ( + )( + ) Mor grlly, 6.4 If w + w w, k l Γ ( + ) Γ ( + )... Γ ( + ) Γ ( + w ) Γ ( + w )... Γ ( + w ) k l my b simplifid A wkr rsult tht rquirs tht k 0], d i [Ri, p. 49]. l, is sttd i [Ml, p. 66

67 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 7 Product Clculus of Jν () For compl umbr ν, th Bssl fuctio Jν () solvs Bssl s diffrtil qutio dw dw ν + + ( ) w 0. d d For ν rl, Jν () hs ifiitly my rl ros, ll simpl with th possibl cptio of 0. For ν 0, th positiv ros j ν,k r mootoic icrsig squc ν, < ν, < ν,3 <... j j j 7. Product Formul for Jν () [Abrm, p.370] J ν ν ()... Γ ( ν + ) j j j ν, ν, ν,3 7. Gomtric M Drivtiv of J () DJ () ν... J j j j ν ν( ) ν, ν, ν,3 ν 67

68 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do D J D D ν (0) (0) (0) Proof: ν () ν Γ ( ν + ) D D D... (0) (0) (0) j ν, j ν, jν,3 D D j ν, D jν, D j ν ν,3 / j ν ν, / jν, / jν,3 ( / ) ( / ) ( / ) 0... ν j j j ν, ν, ν,3... Thus, ν j ν, jν, jν,... DJ ( ) ν... J j j j ν ν( ) ν, ν, ν,3 68

69 Gug Istitut Jourl, Volum 4, No 4, Novmbr 008 H. Vic Do 8 Product Clculus of Trigoomtric Sris If π π f ( ) 0 + cos + b si L L th Gomtric M Drivtiv c b pplid to π π π π f( ) 0 cos + bsi cos + bsi L L L L Product Itgrl of Trigoomtric Sris o [0, π ], Thrfor, si si si 3 π π si si 3 si Product Itgrtig both sids, Hc, cos cos 3 π cos 3... cos cos 3 π cos

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