Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D D ( + ( ) ) lim ( + ( ) D) D i i d Cll these lter two itegrls multigrls o Type I d II. (Note: ulike orml products, these products re ot discrete ut cotiuous over itervl). Cosider ech i tur. Multigrls (Type I) By stdrd opertios ( ) e ( l( ( )) ) By ot tkig limits, iite product pproimtio c e otied. For emple, let () rom to. The the Type I multigrl o rom to is: e ( l( ) ) / e This c e pproimted y the sequece [( )( )] ( ).474... 3 3 3 [( )( )( )] ( ).4543... 4 4 4 3 3 4 [( )( )( )( )] ( ).447... 5 5 5 5 4... 999 [( )( )...( )] ( ).3693... etc 999 Which teds to /e.36788 (use Stirlig s Formul to support this).
pi / For ()t() i rdis rom to pi/, t( ) e Ad [ t( p / 6)t( p / 6) ] ( ) [ t( p /8)t( p / 8)t(3 p / 8) ] ( )... etc. 3 The ove pproimtios o the multigrl c e likeed to the mid-poit-rule whe pproimtig stdrd itegrls. Like stdrd itegrls, multiplictive logs o the Trpezoidl Rule d Simpso s Rule c e oud, like: Simpso s Product: ( )» ì[ ( ) ( )] ü D í{ [ ( + D ) ( + 3 D)...] 4 } ý ( ) 3 î {[ ( + D)...] } þ Cosider the ollowig pproimtios: Y e ( l( ) e ( l() - + ) 4 / e.4757765... Multiplictive Alog o. Mid-poit Rule Trpezoidl Rule Simpso s Rule.5 [()()]^(/).44. / [(.5)(.75)]^(/) [()()]^(/4).47999 [.5]^(/) /3 [(7/6)(9/6)(/6)]^(/3).47489668.4564753 [()()]^(/6) [(4/3)(5/3]^(/3).46476345... [()()]^((/)(/3)) [.5]^((4/3)(/)).4784.. /4 [(9/8)(/8)(3/8) (5/8)]^(/4).47343 [()()]^(/8) [(5/4)(6/4)(7/4)]^(/4).467743. [()()]^((/4)(/3)) [(.5)(.75)]^((4/3)(/4)) [.5]^((/3)(/4)).47466559. Like stdrd clculus you c deie multiplictive log o the derivtive ( the m-derivtive), costruct multiplictive versio o the Fudmetl Theorem o Clculus, costruct multiplictive log o Mcluri s Series, etc.
The m-derivtive or Type I multigrls is: ( ) ( ) e ( ) I ( ) The Fudmetl Theorem is: ( ) ( ) ( ) e (( ) ) I ( ) ( ) Compre with the Fudmetl Theorem o Stdrd Clculus: ( ) ( ) - ( ) Smll progrms c e writte to pproimte the ove results y iite products or those who dout. Type I multigrls id pplictio i the re o popultio dymics. With stochstic irth- d deth- rtes, the covetiol pproch is to use mes (ie: epecttios). Without migrtio, me popultios E(P) remi costt i me irth-rtes E() me deth-rtes E(d) uder the stochstic recursive equtio P ( + - + d) P. But, while mthemticlly correct, this result is misledig. I certi circumstces, simultios show tht me irth-rtes c sigiictly eceed me deth-rtes yet MOST popultio trils declie, eve though the me popultio o my trils stys costt. True. Let G() ( p( ) ) where X the rdom vrile o (+-d) d p() is its proility desity uctio. It c e show tht the MODE o popultios (P ) teds to {G() }P s. I geerl G() is < E()E(+-d). Thus whe E()E(d), the mode o P s eve though E(P ) P. Thus the stochstic recursive equtios (where r# is rdom umer etwee d ) P (.78888... r #) P + P ( r # +.7696) P + P ( r # +.544) P + re ll costt (i the log-term mode) ulike P ( r #) P + P ( r # +.5) P +
which re costt i the log-term me ut ted to zero i the mode. (Try simultig usig Ecel i you do t elieve). Now cosider Multigrls (TypeII) Cosider ( + ) which is the limit o the sequece: æ ö ç +.5 è ø æ öæ 3 ö ç + ç +.546875 è 4 øè 4 ø æ öæ 3 öæ 5 ö ç + ç + ç +.57355967 è 6 3 øè 6 3 øè 6 3 ø æ öæ 3 öæ 5 öæ 7 ö ç + ç + ç + ç +.58945565... etc. è 8 4 øè 8 4 øè 8 4 øè 8 4 ø which teds to sqr(e).64877 This is due to the o-stdrd itegrl (which is ot o the orm ( ) ) l( + ).5 d thus ( + ) e ( l( + )) e ( ) e I geerl, ( + ( ) ) e ( ( ) ) provided ( ( ) ) or Î N ³. Fuctios () which il the lter coditio pper to e ew. For istce, ()/ ils this test rom to s ( ) p / 6 (look t the limit deiitio o the itegrl uder equl suitervls to see this). But or most other uctios ( ( ) ) or Î N ³ For Type II multigrls, the m-derivtive is: ( ) II ( ) ( )
Ad the Fudmetl Theorem is: ( ) ( ) ( + II ( ) ) ( + ) ( ) ( ) Higher order m-derivtives c e lso used, like: ì ( ) ü í ý ( ) ( ) ( ) II ( ) ( ) ( ) ( + II ( ) ) ( ( ) ) î þ + - ( ) ( ) II ( ) ì ( ) ü ( ) ( ) í ý î ( ) þ Ad so o. I geerl, the Fudmetl Theorem ecomes more complicted or higher order m-derivtives, ulike (sy) polyomils with stdrd clculus. For istce, II ( ) Ad thus é ù ( ) ( ) æ ( ) ö æ ( ) ö - - ç + ç ( ) ( ) ç ( ) ç ( ) è ø è ø é ( ) ( ) ù - ( ) ( ) é ù ì ( ) ( ) æ ( ) ö æ ( ) ö ü ( ) ( ) - - ç + ç é ù ( ) ( ) ( ) ( ) ç ç è ø è ø - ( ) ( ) ( ( ) ) + í + II é ( ) ( ) ù ý - ( ) ( ) ( ) ( ) é ù - ( ) ( ) î þ For emple, let ()l(+) the '()/(+), ''()-/(+)^, '''()/(+)^3 d ()l(), ()l(3), etc. The II II ( ) ( )
( + ( ) ) ì é /( + ) 3 /( + ) æ - /( + ) ö ææ/( + ) ö öù ü ( ) + ( ) - ç ( ) + ç /( ) l( ) /( ) ç l( ) + + è + ø èè + ø ø í + ý é - /( + ) æ/( + ) öù ( ) - /( + ) ç l( + ) î è ø þ é ù l { ( + ) + l( + ) - } ( ) + l( + ) + () () é () () ù - () () () () - () () (/3)(( + / l(3))/( + / l())).53474447... Approimtig usig N suitervls gives: N pproimtio.5963.598.537 Whcko! Like stdrd clculus you c chge vriles i the stdrd wy: ( ) ì du ( + ( ) ) ( + u ) ( ) ( - ( u )) rom ílet u() the du () Þ u() Þ u() - îdu/ ()du/( ( ( u))) Ad thus, or emple: ( + ) ( + (( - ) + )( - ) ) Product d Quotiet Rules or Type I d II multigrls re:
Type I Derivtive ( ) e ( ) ( ) Product Rule ( g) g Quotiet Rule ( ) g g Type II ( ) ( ) ( g) + g ( ) - g g Surprisigly Type II multigrls hve the sme sort o Mcluri s Product s Type I. It is () æ () ö æ () ö e ç ç ()! è () ø 3! è () ø 3 ( ) () ( + + +...) Ad the two types o multigrl c e relted y ( ) ( + l( ( )) ) or cceptle (). Other Types o Multigrl With type II multigrls, prolems rise or uctios like ()/ due to the ct tht ( ( ) ) ¹ or Î ³. But sometimes relted multigrls c e evluted usig certi thet uctios. For istce, 3 5 ( + ( ) ) 5( + ( ) )( + ( ) )... cosh( p )».59... d ( - ( ) 4) - cosh( p )» -.59... d é ì 3 ü ù ( ) 3 íg ý cosh( p 3) ( ( ) 3) (4 3 ) î þ + + 7 p d the like. G( ) However, these type III multigrls hve certi uusul properties like
( + ( ) )) ( + ( ) )) k k ( + ( ) )) ( + ( ) ))... etc. So tke cre whe plyig roud with. Type IV Multigrls Surprisigly the multigrl e e e.5 ( + ( )) e ( - + -...)»... eists! ( e -) ( e -) ( e 3-) This is thks to the o-stdrd stdrd itegrls o e ( ).95957... ( e -) k e ( k / ) ( ) ( e k - ) These type o multigrls re more restricted (i rge) th type I d II, ut c still e used to derive certi stochstic limits such s ì ü e mod íå( r # i) ý».95957... î i þ ( e -) ì ü æ e e e.5 ö mod í ( + ( r # ) ) ý e ç - + -...»... i î i þ è ( e -) ( e -) ( e 3 -) ø Where mod is the mode d r# is rdom umer etwee d. Uswered Questios. How my types o multigrls re there? Do they ll hve m-derivtives, Fudmetl Theorems, logs o Simpso s Rule, Mcluri Series, etc?. Wht do multigrls do i the comple ple? Aswers plese. Hppy multigrtig! All commets welcome. Plese sed to: everythiglows@hotmil.com e