g () n = g () n () f, f n = f () n () x ( n =1,2,3, ) j 1 + j 2 + +nj n = n +2j j n = r & j 1 j 1, j 2, j 3, j 4 = ( 4, 0, 0, 0) f 4 f 3 3!

Transcription

1 Higher Derivative o Compositio. Formulas o Higher Derivative o Compositio.. Faà di Bruo's Formula About the ormula o the higher derivative o compositio, the oe by a mathematicia Faà di Bruo i Italy o about 5 years ago seems to be the begiig. Ad it is called Faà di Bruo's Formula. Formula.. ( Faà di Bruo ) Let j, j,, j are o-egative itegers. Let g, are derivative uctios ad B,r,, are Bell polyomials such that g g, (,,,) j! B,r,,, Σ j! j! j!!!! j + j + + j r & j +j + +j ( ) r g( ) Σ g B,r,,, (.) j The, the higher derivative uctio with respect to o the compositio g j is epressed as ollows. Eample g( ) First, j, j, j, j such that j +j +j +j, j +j +j +j are j, j, j, j (,,, ) The, d, The,! B,!!!!!!!! j, j, j, j such that j +j +j +j, j +j +j +j are j, j, j, j (,,, ) j, j, j, j (,,, ) B,!!!!!! +!!!!!!!!!!! rd, j, j, j, j such that j +j +j +j, j +j +j +j are j, j, j, j (,,, ) The, Last,! +! B,!!!!!!!! 6 j, j, j, j such that j +j +j +j, j +j +j +j are j, j, j, j (,,, ) - -

2 The, Thus,! B,!!!! g ( ) g! B,!,,, + g!! B,,,, + g B,,,, + g B,,,, g + g g + g As uderstood rom this eample, obtaiig j, j, j, j such that j + j + + j k, j +j +j + + j, j k k,,, is equivalet with solvig the ollowig idetermiate equatio. It is ot easy. j +j + + ( ) - j -k, j k k,,, Althogh the coditio is j k k,,, i Formula.., i j k > k,, is adopted, Formula.. ca be epressed as ollows. Formula..' Let,,, ad are atural umbers, g, are g g ( ), The, the higher derivative uctio with respect to o the compositio g g ( ) Σg ( + ++ ) p! Algorithm o calculatio o the Formula..'! p q t q! t!!!! ( ) is epressed as ollows. p +q ++t I the ormula o a compositio is epressed i this way, (.') ca be calculated with a easier algorithm. This algorithm o author desig is suitable or the computer. However, sice it is easy, maual calculatio ca also be perormed. () Obtai all the products p q t such that p +q ++t as ollows. Step Put o the st row st colum, ad arrage rom the d colum to the th colum. Step Look or the irst umber that is smaller tha the umber o the st colum as or or more sequetially rom the d colum. I Whe such a umber eists i the k th colum, make a ew sequece ( ew row ) as ollows. i From the d colum to the k th colum, arrage the umber that is greater tha the umber i the k th colum o the old sequece ( the row just above ). ii From the k + th colum to the th colum, copy the umbers o old sequece ( row just above ) iii To the st colum, write the complemet that the total o each colum becomes. II Whe such a umber does ot eist, iish calculatig ad go to Step. Step Repeat Step or the ew sequece ( ew row ). Step Let the umber i the each sequece be the order p ad the repetitio requecy be the degree. Ad geerate the product p q t. (.') - -

3 For eample, the combiatio o the 7 th order is geerated as ollows. Seq Blue is a complemet () The coeiciet o each term p q t Coeiciet o : Coeiciet o : Red is smaller tha the st colum by or more. is calculated as ollows, or eample. 7!!!!!!! 7!!!!! 5 () The multiplier g s o each term p q t is calculated as ollows, or eample. Multiplier o : g s g ( ++ ) g () Thereore, the 7th order derivative z 7 is as ollows, or eample. g + 7! 6 5 6!! +!! 5!! +!!!! g!! + 7! 5 5!! +!!!!! + +!!!!!!!!!!! + 7!!! +!!!!! +!!!!! g!! 7! 7 z 7 7!! g - -

4 + 7!!! +!!!!!! 7! 5 +!! 5!5! 7 7! g 6 +! g 7 7! g 5 g 7 + g g g g g g 7 7 Higher dieretiatio up to the 8th order I the higher derivative z up to the 8th order o compositio z g( ) it becomes the ollowig. z g z g + g z g + g + g is calculatedby such a way, z g + g + + 6g + g z 5 g 5 + g 5 + +g +5 +g + g 5 5 z 6 g 6 + g g + 5 z 7 g 7 + g g g 5 + g g g g g g 7 7 z 8 g 8 + g g g g g g g 8 8 Eample Whe z g( ) si g cos, g -si, g -cos, 6, 6 - -

5 The, z cos z cos 6 - si z cos 6 - si 6 - cos.. Hoppe's Formula Aterwards, it was discovered that Formula.. is epressed with a double series o g r ( ) ad s. The iished type is Hoppe's ormula metioed as ollows. Formula.. (Reihold Hoppe) Let be a atural umber, g, are g g ( ),. The, the higher derivative uctio with respect to o the compositio g g ( ) Σ r g r r r! Σ s r s ( ) is epressed as ollows. r-s (- ) s (.) Although this ormula is beautiul, the calculatio is ot easy. Whe the dieretiatio is actually perormed accordig to (.), it is as ollows. g( ) r r r-s Σ r Σ (- ) s s + g r! r s g! (-) + g! Σ s Σ s g! s s (-) -s s + (-) -s s (-) + (-) (-) + (-) + (-) g +! (-) + + g! g! g! g +! g! g + g.. Higher order dieretiatio usig Mathematica A uctio BellY[,r,{,,, -r+ }] is implemeted i a ormula maipulatio sotware Mathematica sice, By usig this, we ca easily geerate the Bell polyomials B,r For eample, B 8,,,, 8 is as ollows.,,,

6 Further, replacig the table {,,, -r+ } with a uctio, we ca geerate Formula.. itsel. For eample, let The, the above z 8 is easily geerated as ollows. O course, i a speciic uctio is give to g r or k, a desired higher order derivative uctio ca be obtaied. The eamples are show i the ollowig chapters... Higher Derivative i case the core uctio is the st degree Although the higher dieretiatio o a geeral compositio is complicated i this way, I the core uctio ( ) is the st degree, it becomes remarkably easy. Formula.. Whe g g ( ), (,,, ), i ( ) uctio with respect to o the compositio g g ( ) g ( ) is epressed as ollows. is the st degree, the higher derivative (.) Proo I Formula.., whe ( ) is the st degree,. The, the Bell polyomials B,r,, are as ollows. B,r B, Thereore, g!,,, Σ j! j! j!!,,, Σ!!! ( ) Σ j!! j! j j + j + + j r < & j +j + +j r g B,r - Σ g g r B,r,,,!,,, + g! B,!,,, - 6 -

7 Eample Whe z si( a+b) z cos( a+b) a a si a+b + z -si( a+b) a a si a+b + z a si a+b + The last epressio is cosistet with liear orm i Formula 9.. i " 9. Higher Derivative "

8 . Higher Derivative o Some Compositio There are quite a lot o combiatios eve i the compositio is limited to the elemetary uctio. We caot calculate these oe by oe. So, i this sectio, we calculate oly a easy ad iterestig thig... Higher Derivative o e Placig with g e ( ) i Formula.., we obtai the ollowig ormulas immediately. e e Σ B,k,,, (. + ) k e -( ) e -( ) Σ ( ) k I (. + ) is writte dow to the 5th order, it is as ollows. e e e e + e e + + e e + + e 5 e 5 + e - -, -, -, Eample The, e - - r B,k,,, (. - ) e - + ( -)(-) + (-) e - - si Eample e si, cos, -si, -cos, si The, e e si si + -cos +si - 6si cos + cos si Eample e -cos 8 We calculate usig a ormula maipulatio sotware Mathematica accordig to (. - ). So, cos, cos+,,, - 8 -

9 .. Higher Derivative o ge Placig with e i Formula.., we obtai the ollowig ormulas. g Σ e r g B,r g (-) Σ e - e,, e r g B,r e -,, e - I the irst epressio is writte dow to the 5th order, it is as ollows. e g e g e g e g e 5 g g e g e + g e g e + g e + g e g e + 7g e + 6g e + g e g e + 5g e +5g e +g e + g 5 e 5 Coeiciets, (, ), (,, ), (,7,6, ), (,5,5,, ), o these right sides are called Stirlig umbers o the d kid ad is give by the ollowig epressio. r S(,r ) r! Σs (-) s r s ( r -s) (.s) Usig this otatio, the above ormulas are more briely epressed as ollows g Σ S(,r) g e g (-) Σ S(,r) g e - r e r (. + ) r e -r (. - ) Eample tae The, g ta e +, g ta e +ta e, g 6ta e +8ta e + ta e ta e +e +ta e +ta e e +6ta e +8ta e +e e ta e + +6e ta e + ta e +e ta e +ta e + Eample e So, e - 5 g e, g e e e- (,,,) - 9 -

10 Note e e - 5 -e e - e - + 5e - +5e - +e - + e -5 The horizotal total o Stirlig umbers o the d kid is called Bell Number. That is B Σ S(,r) The irst ew Bell umbers or,,, are,, 5, 5, 5,, 877,. This Bell umber is give by the ollowig epressio that is called Dobiski's ormula, too. B e Σr r r! Σ.. Higher Derivative o log ( ) r -r r! Σ s (-) s s! g r (-) r- ( r- )! -r rom g log. Substitutig this or Formula.., log ( ) Σ (-) r- ( r- )!B,r,,, -r (.) I these are writte dow to the 5th order, it is as ollows. log ( ) log ( ) log ( ) log ( ) log ( )! -! - -! -! - -! - +! -! - -! + - +!6 - -! - 5! 5 - -! ! ! - +! 5-5 Eample ( log si ) si, cos, -si, -cos, So, ( log si ) cos si cos ( cos ) -! +! si ( si +! cot + ( cot ) ) ( si ).. Higher Derivative o glog r (-) r- ( r- )! -r rom log. Substitutig this or Formula..,!! g( log ) Σ g, -,, (-) - ( -)! r B,r I these are writte dow to the th order, it is as ollows. (.) g( log ) g! g( log ) - g! - g (! ) - -

11 g( log ) g( log )! - g!! + g (! ) g - g! - g!!+(! ) +6g!(! ) -g (! ) Eample ( cos log ) Sice g -si, g -cos, g si, g cos, ( cos log ) - 6g - g + 6g - g cos log - (-6si + cos + 6si - cos ) - Eample ( /log ) Sice g! -, g!, g! -, ( /log ) g - g + g - Eample ( log log ) Sice g ( log ) 6 + ( log )!, g! -, g! ( log log ) g - g + g + + log ( log ) ( log ) 6 + ( log ),!!! - - -!!!

12 . Higher Derivative o Gamma Fuctio & the Reciprocal Formula.. ( Masayuki Ui ) Whe ( z) is the gamma uctio, z is the polygamma uctio ad B,r,, are Bell polyomials, the ollowig epressios hold. d dz z z Σ k d dz Proo Whe z B,k z, z,, - z (. + ) z Σk( -) k B,k z, z,, - z log z, d log z dz z d dz z z d dz - z - z z (. - ) Substitutig these or (. + ), (. - ) i Formula.., we obtai e log z e log z Σ B,r z, z,, - z (. + ) e -log z e -log z Σ (-) r B,r z, z,, - z (. - ) Eample z We calculate usig a ormula maipulatio sotware Mathematica accordig to (. + ). Whe this table deiitio is adopted, the result is slightly hard to see, but the dieretial coeiciet is calculable. Eample /( z) We calculate usig a ormula maipulatio sotware Mathematica accordig to (. - ). Whe this table deiitio is adopted, the result is easy to see, but the dieretial coeiciet is icalculable. - -

13 Note O December 9, 6, I received a mail rom Mr. Ui livig i Yokohama city. I the mail, it was writte that the coeiciets o the Bell polyomials appear i the higher order derivative o the gamma uctio. I was vary surprised. Because, it meas that the gamma uctio is a composite uctio. I act, it was a too simple composite uctio. ( Mr. Ui seems to have oticed it soo, but I eeded days to otice it. ) As ar as I get to kow, the discoverer o Formula.. is Mr. Ui. This sectio is what I added a simple proo with the coset o Mr. Ui. - -

14 . Possibility to super dieretiatio o compositio Faà di Bruo's Formula was as ollows. g ( ) Σ Hoppe's Formula was as ollows. g ( ) r g B,r Σ r g r r r! Σ s,,, (.) r s r-s (- ) s (.) We caot make the upper limit o i both ormulas. It is because the umber o terms i the right side caot become larger tha the umber o orders i the let side. So, it is hopeless to eted the domai o (.) or (.) rom the atural umber to the real umber p. That is, the super dieretiatio o the geeral compositio g( ) is impossible ow. However, whe the core uctio ( ) is the st degree, accordig to Formula.., g( ) g So, there is o obstacle i etedig this domai rom the atural umber to real umber p. Thus, the ollowig epressio holds or a real umber p >. p g( ) g p p This is the grouds or which we have used "Liear orm" sice " Super Derivative " as a ait accompli. (.) (.) updated. Alie's Mathematics K. Koo - -