x !1! + 1!2!

Transcription

1 4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio of Beroulli Nubers x B B B B + x + x 3 B + x x 4 + e x (,) -!!! 3! 4! e x - x x x 3 x (.) x!! 3! 4! 5! Maig the Cauchy product of (.) ad (.), B +!! B +!! B B B B +!! x + +!!!3! I order to holds this for arbitrary x, x + B, B B +,!!!! B B B + +,!!!!!3! The coefficiet of x is as follows. B B - B - B B + +!! ( -)!! ( -)!3!!!!( + )! Multiplyig both sides by ( + )!, B ( + )! B - ( + )! B - ( + )! B ( + )! B + +!! ( -)!! ( -)!3!!!! Usig bioial coefficiets, +C B + +C - B - + +C - B C B + +C B Replacig + with, C - B - + C - B - + C -3 B C B + C B (.3) Substitutig, 3, 4, for this oe by oe, C B + C B B - C3 B + C3 B + C3 B B Thus, we obtai Beroulli ubers. The first few Beroulli ubers are as follows. 5 B, B, B4 -, B6, B8 -, B, B -, B3 B 5 B

2 4..3 Calculatio of Beroulli uber () Method of substitutio oe by oe Geerally, it is perfored by the ethod of substitutig, 3, 4, for (.3) oe by oe. Sice it is ecessary to calculate fro the sall uber oe by oe, it is difficult to obtai a large Beroulli uber directly. This ethod is suitable for the sall uber. () Method by double sus. It is calculated by the followig double sus of bioial coefficiets B r + (-) r C r r If the first few are writte dow, it is as follows. B C B C + C - C B C + C - C + C - C + C 3 Usig this, a large Beroulli uber ca be obtaied directly. However, sice the uber of ters icreases with acceleratig speed, this is ot suitable for aual calculatio Beroulli Polyoial () Defiitio Whe B are Beroulli ubers, polyoial B () Beroulli Polyoial. B () x d B () x B - () x ( ) B () x ( ) x that satisfies the followig three expressios is called () Properties The followig expressios follow directly fro the defiitio. x B () x B- () t dt + B ( ) B () x B () B B - x B x - B () B () ( ) B ( x+ ) - B () x x - ( ) B ( -x ) (-) B () x ( ) - -

3 Furtherore, the followigs are ow although ot directly followed fro the defiitio. For arbitrary atural uber ad iterval [, ], B () x B B + () x ( +) B Exaples B () x x-, B () x x -x+, B3 () x x x + x, 6 B 4 () x x 4 -x 3 +x -, B5 () x x x x 3 - x, B 6 () x x 6-3x x 4 - x +, B7 () x x x x x 3 + x, 4 6 B 8 () x x 8-4x x x 4 + x -, B 9 () x x x 8 +6x 7 - x 5 +x x, 5 B () x x -5x x 8-7x 6 +5x x 5 +, Fourier Expasio of Beroulli Polyoial Beroulli Polyoial B () x ca be expaded to Fourier series o x. This eas that Beroulli Polyoial B x - x ca be expaded to Fourier series o x. Forula 4..5 Whe is a atural uber, x is a floor fuctio ad B are Beroulli ubers, B x- x -! s cos sx- x ( s) Accordig to Forula 5.. (" 5 Geeralized Beroulli Polyoials "), the followig expressio holds. B () x -! s ( s) cos sx- x - x B () x o x <, B x- x -! s ( s) cos sx- Sice B O x <, Left: B x+- x+ B x+- x - B x- x Right: -! s ( s ) B x- x -! s cos s( x +) - -! s cos sx- ( s) ( s ) x x < cos sx- x < - 3 -

4 Hereafter by iductio, the followig expressio holds for arbitrary atural uber. B x- x -! s ( s) cos sx- x < + Q.E.D. Exaples If the left side ad the right side are illustrated for,, it is as follows. Blue is ad Red is. Left side Right side - 4 -

5 4. Euler-Maclauri Suatio Forula Forula 4.. Whe f( x) is a fuctio of class C o a closed iterval [ a,b ], x is the floor fuctio, are Beroulli ubers ad B () x are Beroulli polyoials, the followig expressio holds. b - b f() a f() x + a r Whe li r! f ( r- ) () r- () (-) + b R! B x- xf ( ) () x a ( ) s ( s) cos sx- b - f ( ) a + R (.) - a b R is a eve uber s.t. () x ( ) f ( ) x are Beroulli polyoials, Whe are Beroulli ubers ad B () x B () x, B () x + B + () x -B + B + () B + () B + Usig these, f() x B () x f() x Here, Ad Therefore,! B () x f() x -! B () x f() x -!! B () x f() x -! B () x f() B () x f! ' () x B () x f ' () B () x f ' () x + x + 3! B 3 () x f " () x ( - ) r- r r! () x f ( r- ) () x + ( ) x - f() - f ( ) () x (.r) (.r') iiu for x[ ] a,b - - B () x f! " () x B 3 () x f "' () x 3! B () x ( f ) () x! - f() f( + ) f( ) (-) r- B ( ) r (-) r- B ( ) r - for r - 5 -

6 (-) r- () x f ( r- ) () -Br f ( r- ) () - f ( r- ) () Substituti these for the above, x B r f() x f( + ) f( ) - r r! Replacig f( x) with f( x + ), f( x+) f( + ) + f( ) That is, + f() x f( + ) + f( ) Accuulatig this fro a to b -, b b - f() x a a f( + ) + f( ) Here, b - f ( ) a b - a r- ( +) - f ( ) f( + ) + f( ) B ( x- ) B r- () + f( ) x- + f ( r- ) () r- () - f ( ) (-)! B () x f ( ) () x - r + - r + - r r! r- ( +) - f ( ) f ( ) r- () (-)! B () x f ( ) ( x+) r! r- ( +) - f ( ) f ( ) r- () (-) +! B ( x-) f ( ) () x r! a b - f ( ) r- ( +) - f ( ) r- () (-) b - + +! a B ( x-) f ( ) () x f ( r- ) () b - f ( r- ) ( a) a +f( ) b b a f() x ( x +) Therefore, b b f() x a f() - a f( a ) +f( ) Trasposig the ters, b - r b b f() a f() x + a f( a ) +f( ) Subtractig f( b) fro both sides, + b + r - r! f ( r- ) () b - f ( r- ) ( a) (-) b! B x- xf ( ) () x a r! f ( r- ) () b - f ( r- ) ( a) (-) b! B x- xf ( ) () x a - 6 -

7 b - b f() a f() x - a f( b) -f( ) a + r - r! f ( r- ) () Sice B -/, icludig the d ter of the right side ito, we obtai b - b f() a f() x + a r R Last, substitutig Forula 4..5 B x- for (.r), we obtai x -! s r! f ( r- ) () b - f ( r- ) ( a) (-) b! B x- xf ( ) () x a r- () (-) + b! B x- xf ( ) () x a ( s) R (-) b a s b - f ( ) a + R (.) cos sx- ( s) Whe R is diverget, the aplitude is alost deteried by eve uber should be chose such that () x ( ) f ( ) x ( cos sx- f ) () x f ( ) () x ( ). Therefore, at this tie, becoes iiu for x [ a,b ]. (.r) (.r') Q.E.D. Reovig the ters of larger odd uber tha i the Forula 4.., we obtai the followig forula. Forula 4.. Whe f( x) is a fuctio of class C o a closed iterval [ a,b ], x is the floor fuctio, are Beroulli ubers ad B () x are Beroulli polyoials, the followig expressio holds. b - b f() a f() x - a f( b) - f( a) Whe li R + r ( r )! f ( ) () b - f ( ) () a + R (.) b R - ( )! B x- xf ( ) () x a (.r) - a b s ( s) ( ) cos( sx), is atural uber s.t. f ( ) () x ( ) f ( ) () x (.r') iiu for x [ a,b] Reovig B 3, B 5, B 7, fro Forula 4.., we obtai (.), (.r). Ad, fro Forula 4..5, - 7 -

8 B x- x -( )! s ( s) -(-) ( )! s Substitutig this for (.r), we obta (.r'). cos( sx-) cos( sx) ( s) Whe R is diverget, the aplitude is alost deteried by atural uber should be chose such that Forula 4..' f ( ) () x ( ) f ( ) () x ( ). Therefore, at this tie, becoes iiu for x [ a,b ]. Whe f( x) is a fuctio of class C o a closed iterval [ a,b ], x is the floor fuctio, are Beroulli ubers ad B () x are Beroulli polyoials, the followig expressio holds. b b f() a f() x + a f( b ) + f( a) Whe li R + r ( r )! f ( r- ) () b - f ( r- ) () a + R (.') b R - ( )! B x- xf ( ) () x a - a b s ( s) ( ) cos( sx), is atural uber s.t. f ( ) () x ( ) f ( ) () x Q.E.D. (.r) (.r') iiu for x [ a,b] Addig f( b) to both sides of Forula 4.. (.), we obtai the desired expressio iediately. Q.E.D

9 4.3 Su of Eleetary Sequece 4.3. Su of Arithetic Sequece Forula ( a +d ) a +( -) d (.) Fro this, S - a, a+d, a+d, a+3d,, a+( -) d f( x ) a+xd () x d, f () () x f () 3 () x f () Substitutig these for Forula 4.., - ( a +d ) ( a +xd) - ( a +d) - ( a +d) xa x d +!! B + { d - d } +! r - d + R a d d R!! R - ( )! Thus, we obtai the desired expressio Su of Geoetric Sequece Forula r r - s B x- x { - } + R ( r )! B s ( log r) s- + R s! (.) ( ) R (-) + log r! B x- x r x (.r) - r r - s B s ( log r) s- r - s! r - (.') S - r, r, r, r 3,, r - Fro this, f( x ) r x f() x r x log r r -r log r - 9 -

10 f ( s- ) () x r x ( log r) s- ( s,,+) Substitutig these for Forula 4.., - r r -r + log r s B s r s! ( log r) s- - r ( log r) s- + R R Icludig the st ter of the right side ito, we obtai (-) +! B x-[ x ] r x ( log r) - r r - s B s ( log r) s- + R s! (.) R (-) ( log r)! B x- x r (.r) This is the su of geoetric sequece by Euler-Maclauri Suatio Forula. Here, substitutig x log r for s s Fro this, s Furtherore, li B s ( log r) s! s log r e log r - B s ( log r) s- s! r - ( log r)! The R. Thus, - r r - s B s x s x s! e x - log r r - B s ( log r) s- r - s! r Su of iteger powers of atural ubers Forula ( Jacob Beroulli ) ( defiitioal idetity of Beroulli ubers ) r r +-r (3.) + B + () - B + () (3.') Fro this, S -,,, 3,, ( -) f( x ) x f() x x

11 f ( r- )! () x ( -r+ )! Substitutig these for Forula 4.., i.e r R r + r ( )! x -r+ ( r,,+)! -r+ - -r+ + R r! ( -r+ )!! -r+ - -r+ + R r! ( -r+ )! ( + )! B r!( +-r )! r +-r - +-r + R - + B x-[ x] - + r R Here, B ( - ) +! B x-[ x] x-[ ] The R. Therefore - x r - + r Furtherore + r! x! + r +-r - +-r + R B () x + r + r + + B + - The, (3.w) is rewritte as follows r + Expressig this with the Beroulli polyoial, - + B + () - B + () r +-r - +-r (3.w) +-r (3.) +-r - +-r (3.') Exaple: 3, B 3+ ( ) -B 3+ ()

12 4.3.4 Su of alterative iteger powers of atural ubers Forula (-) - + r Whe is a eve uber, i.e. + + r ( -) - +-r r B + () - B B + - B ( -) (-) - r - + r r r Addig ( +) to both sides, + (-) - + r - + r r r Let +, + (-) - r Whe, (-) - r Replacig with +, - (-) - r Applyig Forula to this, - r r r Usig these, - + r r +. The r - + r r - + r r r - r - + r - r r - + r r r - + r +-r + + r +-r + - r B + () - B + () B + - B + () - -

13 - (-) - + r Exaple: 3, + + r +-r r B + () - B B + - B + - (-) B 3+ ( ) - B B B Especially i the case of, the followig iterestig forula holds. Forula 4.3.4' (-) - (-) - (-) - Whe is a eve uber, i.e ( -) - ( +) - ( +) - - ( 3+) ( -) + -( -) -( + ) - ( 3+ ) - ( 5+ ) - - ( -) ( -) - ( +++) ( -) - ( ) -( ++3++) (-) - - Whe is a odd uber, i.e. ( +) ( -) - - ( 3-) ( -) - -(-+) - (-3+) - (-5+) - - (- +) { } - ( +++) ( -) (-) - ( +) Cobiig both, we obtai the desired expressio

14 Exaple: (-) (-) 999- ( ) Su of Trigooetric Sequece Forula 4.3.5s si si - - ( cos -) + (-) r r (-) + R ( )! B x- - si si - - ( cos - ) cot - si si / si + R r! (5.s) ( ) x six (5.sr) (5.s') (5.s") S - si, si, si3,, si( -) Fro this, f( x ) si x f() x [-cosx] -cos + cos -cos + f ( r- ) () x (-) r- cosx ( r,,) f ( ) () x (-) si x Substitutig these for Forula 4.., i.e. - si -cos + cos - ( si - si ) R - ( )! + r (-) r- { cos -cos} + R ( r )! B x- x (-) si x - si B si - - ( cos -) + (-) r r r ( r )! + R (5.s) (-) R ( )! B x- x six (5.sr) This is the su of sie sequece by Euler-Maclauri Suatio Forula. Here, let. The, B li (-) r x- + r cot, li r! ( ) ( )! x six - 4 -

15 Therefore, (5.s) becoes as follows. - si si - - ( cos - ) Moreover, usig forulus of trigooetric fuctios, cot (5.s') i.e. - si - cot ( cos - ) - si - cos ( cos - ) - si si / si si si - - si si si / si / si (5.s") This is cosistet with the result of havig applied Trigooetric Additio Forulas to - si directly. Forula 4.3.5c - cos si R + r B (-) r r - ( cos -) ( r )! (5.c) (-) + ( )! B x- x cosx (5.cr) - cos si cot - ( cos -) - cos si / si (5.c') (5.c") S - cos, cos, cos3,, cos( -) Fro this, f( x ) cosx f() x [ si x] si - si si f ( r- ) () x (-) r si x ( r,,) f ( ) () x (-) cosx Substitutig these for Forula 4.., - cos si - ( cos - cos) R + r ( -) ( r )! r si - (-) r si + R + R ( r )! si - ( cos - cos ) + si (-) r r (-) + ( )! B x- x cosx - 5 -

16 i.e. - cos si R + r B (-) r r - ( cos -) + R ( r )! (5.c) (-) + ( )! B x- x cosx (5.cr) This is the su of cosie sequece by Euler-Maclauri Suatio Forula. Here, let. The, B li (-) x- + Therefore, (5.c) becoes as follows. B cot, li ( )! ( )! x cosx - cos si cot - ( cos -) Moreover, usig forulus of trigooetric fuctios, - - cos cos si / si (5.c') (5.c") This is cosistet with the result of havig applied Trigooetric Additio Forulas to - cos directly

17 4.4 Su of Haroic Sequece & Euler-Mascheroi Costat 4.4. Su of Haroic Sequece Forula 4.4. Whe is Euler-Mascheroi Costat, - + log - -r R Where, < B x- x x + + R (.) r r (.r) Fro this, S -,,,, 3 - f( x ) x h f() x [ log x] h log h - log < < h, h, are itegers f ( r- ) () x (-) r- ( r- )!, f ( ) x r Substitutig these for Forula 4.., h - Fro this, - log h - log - h - h - R - h h - - B - h -r x- x x + - log h + log + - h () x r + r ( )! x + - h r r r Whe h, sice r, h - li h - log h, li h, li h h h r The, - + log - -r R - B x- - r r R x x + Reversig the sig of the reaider ter, we obtai the desired expressio. + R - h r r - R 7

18 Exaple: Whe, / + log - - B B This all digits (4 digits below the decial poit) are sigificat digits. Furtherore, if the is calculated to 8, the sigificat digit reaches 34 digits below the decial poit. c.f. If Forula 4.. is applied straight to f( x ) /x, it is as follows. - log -log - R - - x + B x- x - r r - r r + R However, the degree of the approxiatio of this forula is very bad. If is calculated to 3 i the above exaple, the sigificat digit reaches digits below the decial poit. Ad, it is the best approxiatio of the above exaple by this forula Calculatio of Euler-Mascheroi Costat Trasposig the ters of (.), we obtai as follows. - - log + +r R - B x- x x + + R (.) r r Where, : (uber of sigificat digits +?) / (.r) That is, we ca calculate Euler-Mascheroi Costat coversely fro (.). Whe, the uber of sigificat digits is roughly give by -?. Exaple: - - log + + r r r This all digits (8 digits below the decial poit) are sigificat digits. (.) is a quite good approxiate expressio of. However, regrettably expasio. li R for defiite. That is, (.) is oly a asyptotic 8

19 4.5 Su of Zeta Sequece & Zeta Fuctio 4.5. Su of Zeta Sequece Forula 4.5. Whe ( p) is the Riea Zeta Fuctio ad ( p,q) is the beta fuctio, the followig expressio hols for p. - -p ( p ) + p -p r r -p-r + R (.) B R (,p) x- x x p+ (.r) Where, is a eve uber s.t. p <. S - -p, -p, 3 -p,, ( -) -p ( p >) Fro this, f( x ) x -p h f() x x -p - p f ( r- ) () x -(-) r ( p +r-) x -p-r () p Substitutig these for Forula 4.., Here, h - Usig this, h -p - -p B - p - p (-) r r r r! B (-) r r r! h - h h -p - -p < < h, h, are itegers - p ( r,,+) ( p +r-) h -p-r - -p-r + R () p ( p +r- ) (-) r (-) r +r+( p-) () p ( +r) p - +( p -) (-) r +r+( p-) - -p ( +r) +( p -) - ( - ) r p-+r -p p- -p + - -p r B r r h -p - -p + p - p -p r R -! b B x- x a -p r ( p +) ( p) h -p-r - -p-r + R x -p

20 i.e. h - Fro this, - p -p r -p r h -p-r - -p-r + R B x- x R - (,p) h - p h - h h - - p p - p -p r R - (,p) h x p+ -p r h -p-r - -p-r - R x- x B x p+ Here, let h. sice p >, li h h p--r. Therefore, - ( p ) + p -p r R - (,p) -p r B x- x x p+ Reversig the sig of the reaider ter, we obtai the desired expressio. -p-r - R (.) (.)r Exaple: /. Whe., - (. ) r -. r -.-r This all digits (9 digits below the decial poit) are sigificat digits. O practical use, it sees that this is eough. Whe 6, 4 digits (3 digits below the decial poit) are sigificat digits. c.f. If Forula 4.. is applied straight to f( x ) x -p, it is as follows. - p -p r -p r R - (,p) -p-r - + R B x- x x p+ However, the degree of the approxiatio of this forula is very bad. If is calculated to 4 i the above exaple, the sigificat digit reaches digits below the decial poit. Ad, it is the best approxiatio of the above exaple by this forula Calculatio of Riea Zeta Fuctio Trasposig the ters of (.), we obtai as follows. - -

21 - ( p ) - p -p r R - (,p) -p r B x- x x p+ Where, : uber of sigificat digits +? -p-r + R (.) That is, we ca calculate Riea Zeta Fuctio ( s) coversely fro (.). Whe, the uber of sigificat digits is roughly give by -?. Exaple: (.3) Whe, r -.3 r -.3r This all digits (3 digits after the decial poit) are sigificat digits. (.) is a quite good approxiate expressio of ( p ). However, regrettably li R for defiite. That is, (.) is oly a asyptotic expasio. (.r) - -

22 4.6 Su of real uber powers of atural ubers Forula 4.6. Whe ( p) is the Riea Zeta Fuctio ad ( p,q) for p -. - p +p ( -p) + +p r r +p-r + R B R (,-p) x- x x -p+ Where, is a eve uber s.t. p <. Reversig the sig of p i Forula 4.5., we obtai the desired expressio. is the beta fuctio, the followig expressio hols Exaple :. Whe, -. ( -. ) + +. r +. r +.-r This all digits (9 digits below the decial poit) are sigificat digits. O practical use, it sees that this is eough. Whe 6, 5 digits ( digits after the decial poit) are sigificat digits. Exaple : 3 Whe, - 3 ( -3) + +3 r This result is cosistet with the exaple by Forula r +3-r 555 Exaple 3 : -. Whe, - -. (. ) + -. r This reduce to the exaple of Forula r c.f. If Forula 4.. is applied straight to f( x ) x p, it is as follows. - p +p +p r r R - (,-p) + p-r - + R B x- x -.-r x p- - -

23 However, the degree of the approxiatio of this forula is very bad. If is calculated to 6 i Exaple the sigificat digit reaches 4 digits below the decial poit. Ad, it is the best approxiatio of Exaple by this forula

24 4.7 Su of alterative real powers 4.7. Su of alterative positive powers of atural ubers Forula 4.7. Whe ( p) is the Riea Zeta Fuctio ad ( p,q) for p -. - (-) - p - +p ( -p) +p + +p r B r r R (,-p) B x- x x -p+ Where, is a eve uber s.t. p <. is the beta fuctio, the followig expressio hols +p-r - +p - +p / The followig equatio was obtaied i the proof of Forula r - (-) - r - r - + r x -p+ +p-r B x- x This equatio holds eve if the atural uber is exteded to the real uber p. That is, - (-) - p r Applyig Forula 4.6. to this, The - r p - p+ r - r p ( -p) + r +p r - r r p ( -p) + +p R (,-p) R (,-p) r +p r - r p +p r +p-r + R B x- x x -p+ B x- x / x -p+ - (-) - p ( -p) + +p r +p-r + R +p + R r +p-r - +p +p +p-r ( -p) + +p r r + R +p r +p-r - +p - +p ( -p) + +p r + R - +p R +p-r - 4 -

25 i.e. - (-) - p - +p ( -p) +p + +p r r B R (,-p) x x -p+ x- Where, is a eve uber s.t. p <. +p-r - +p - p+ / x -p+ +p-r B x- x + R Exaple : p.6, I this case, 4 gives the best approxiatio ( digits below the decial poit). c.f. - If Forula 4.. is applied straight to ( -) - p, it is as follows. - (-) - p +pr R - (,-p) +p r + p-r - + p B x - x x -p+ + p-r - +p / -- + p + R B x - x x -p+ However, the degree of the approxiatio of this forula is very bad. If is calculated to 6 i the above exaple, the sigificat digit reaches 3 digits below the decial poit. Ad, it is the best approxiatio of the above exaple by this forula Su of Eta Sequece ad Dirichlet Eta Fuctio Forula 4.7. Whe ( p) is the Riea Zeta Fuctio, ( p) is the Dirichlet Eta Fuctio ad ( p,q) is the Beta Fuctio, the followig expressio hols for p -. - (-) - p - -p () p + -p r -p r -p-r - -p -p-r + R - 5 -

26 R (,p) B x- x x p+ Where, is a eve uber s.t. p <. Especially, whe, ( p ) - -p () p - -p / Reversig the sig of p i Forula 4.7., we obtai the desired expressio. B x- x x p+ Whe, sice the rage of the itegral is fro to, R for ay. Exaples : p.7,, Referece Let - f(), The the followig forula also holds. - (-) - p - r f() x -x p cos x, E () x r- r! s e -tx t dt, ( x ) e -t t x- dt +p E -p (- i ) + E -p ( i ) - E -p (- i ) + E -p ( i) r - (-) R! B x - x s ( +p) ( -) p-s + r--s cos s ( +p -s) ( +p) x p-s -s cos s ( +p -s) x + However, this is too coplicated ad does ot have a great erit, either.. ( r --s) + R ( -s) Reewal Alie's Matheatics K. Koo - 6 -