Super Derivative (No-iteger ties Derivative). Super Derivative ad Super Differetiatio Defitio.. p () obtaied by cotiuig aalytically the ide of the differetiatio operator of Higher Derivative of a fuctio f( ) to a cople plae [,p ] fro a atural uber iterval [, ] is called Super Derivative of f( ). f () Eaple ( si ) () p p si + + c p () c p () is a arbitrary fuctio... Super Differetiatio Defiitio.. We call it Super Differetiatio to differetiate a fuctio f with respect to a idepedet variable o-iteger ties cotiuously. Ad it is described as follows. Eaple d p d p f() d d f() d d d d : p pieces d p p d p cos cos+..3 Fudaetal Theore of Super Differetiatio The followig theore holds fro Theore 7..3 i 7.. Theore..3 Let f () r r [, p] be a cotiuous fuctio o the closed iterval I ad be arbitrary the r-th order derivative fuctio of f. Ad let a( r) be a cotiuous fuctio o the closed iterval [, p ]. The the followig epressio holds for a( r ), I. d p d p f() f () p d p p - () () + d p f r Especially, whe a( r ) a for all k[, p ], Proof r a( p -r) d p d p f() f () p d p p - () () + d p f r ( ) () a r ( +r) Theore 7..3 i 7. ca be rewritte as follows. f <> p () a( p) a( ) f() d p + r p - f < p-r> a( d r (.) a() p p-r) -a r (.) a( p -r) a( a() p p-r+) d r - -
Especially, whe a( r ) a for all k [, p ], f <> p () a a f() d p - + f < p-r > ( -a) r () a r ( +r) Differetiatig these both sides with respect to p ties, d p d p f <> p () f <> d p p - () + d p f < > r p-r a( p -r) d p d p f <> p () f <> d p p - () + d p f < p-r > ( -a) r () a r ( +r) a( a() p p-r) Shiftig by -p the ide i the itegratio operator <> ad replacig <> by differetiatio operator (), we obtai the desired epressio. Costat-of-differetiatio Fuctio d p p - We call etc. Costat-of-differetiatio Fuctio of f( d p ). Sice p is a real uber, r geerally it is difficult to obtai this. However, it becoes easy eceptioally at the tie of f( ) e. That is, Costat-of-itegratio Fuctio i 7..3 was as follows. ( -a) r e a ( ) ( -a) r+ p r r p - e a ( +r) Differetiatig both sides with respect to p ties, d p p - e a ( -a) r e a d p r ( +r) r -a r - ( +r) ( +r+p) d p d p ( -a) r ( -a) r+ p - ( +r) ( +r+p) Fro the.3. etioed later, the followig epressios hold. d p d p ( -a) r ( +r) ( +r-p) ( -a) r-p, Substitutig this for the above, we obtai the followig epressio. d p p - e a ( -a) r d p e a r ( +r) r..4 Lieal ad Collateral d p d p ( -a) r+p ( -a) r-p ( -a) r - ( +r-p) ( +r) d r ( +r+p ) ( -a) r ( +r) I the case of the higher differetiatio, sice the costat-of-itegratio polyoial was degree -, the costat-of-differetiatio fuctio which differetiated this ties becae. However, i the case of the super differetiatio, sice the costat-of-itegratio fuctio is epressed by a series i geeral, the costat-of-differetiatio fuctio which differetiated this p ties does ot becoe. This shows that there are lieal ad collateral i the super differetiatio. Defiitio..4 d p d p f() f () p d p p - () () + d p f r I this epressio, whe Costat-of-differetiatio Fucti is, r a( p -r) a( d r (.) a() p p-r) - -
d p we call d p f() Lieal Super Differetiatio ad we call the fuctio equal to this Lieal Super Derivaive Fuctio. whe Costat-of-differetiatio Fucti is ot, d p we call d p f() Collateral Super Differetiatio ad we call the fuctio equal to this Collateral Super Derivaive Fuctio. These are the sae also i (.). I short, Lieal Super Derivaive Fuctio is what differetiated f( ) with respect to cotiuously without cosiderig the costat-of-differetiatio fuctio. Eaple: lieal derivative ad collateral derivative of e I the case of easier fied lower liit, fro (.) i the theore d p d p e e d p p - + d p e a ( -a) r r ( +r) Here, usig the forer epressio i.e. we obtai d p p - e a ( -a) r d p e a r ( +r) r d p d p e e + e a r ( -a) r-p ( -a) r - ( +r-p) ( +r) ( -a) r-p ( ) - ( +r-p) ( ) -a r +r e a ( -a) r-p r ( +r-p) Whe a -, sice the costat-of-differetiatio fucti ca ot be, this is a collateral differetiatio. Whe a -, sice e a, we obtai the followig lieal differetiatio. d p d p e e Whe p /, a, if the differetial quotiets o.3 are copared with the calculatio result by Riea-Liouville differitegral (later..), it is as follows. - 3 -
Ad if the lieal super derivative ad the collateral super derivative are illustrated side by side, it is as follows. Reark It is thought that this collateral super derivative is a asyptotic epasio. Ad this collateral super derivative is correspodig with the terwise super differetiatio. I geeral, a terwise super differetiatio sees to becoe a collateral super differetiatio...5 The basic forulas of the Super Differetiatio The followig forulas hold like the higher differetiatio. () p c f( ) c f () p () c : costat ultiple rule f( ) + g( ) () p f () p () + g () p () : su rule - 4 -
. Fractioal Derivative.. Riea-Liouville differitegral Aog the super itegrals of fuctio f(), the super itegral whose lower liit fuctio a( k) is a costat a was calculable by Riea-Liouville itegral. The super derivativeof such a fuctio f() is calculable by Riea-Liouville itegral ad iteger ties differetiatio. It is as follows. Let p ceil( p ). First, itegrate with f () -p ties. Net, differetiate it ties. Ad, sice the result is - (-p), it eas that f() was differetiated p ties. f () p () ( -p) d d ( -t) -p- f() t dt a p (.) This epressio is called Riea-Liouville differitegral. "differitegral" is a coied word which cobied "differetial" ad "itegral". Although the uerical itegratio ad the uerical differetiatio are possible for (.) with this, the accuracy of uerical differetiatio is bad ad the desired result ay ot be obtaied. I this case, the followig forula which replaced the calculatio order of itegratio ad differetiatio is effective. f () p () ( -p) ( -t) a -p- d dt f() t dt p Although this forula has a possibility of cuttig off a costat of itegratio as a result of differetiatig previously, i ay cases, it is correctly calculable. This (.') is ofte used i the followig chapters... Riea-Liouville differitegral epressios of super derivatives of eleetary fuctios Riea-Liouville differitegral epressios of super derivatives of soe eleetary fuctios are as follows. I the right side, super derivatives obtaied by super differetiatio are show i advace. Needless to say, Riea-Liouville differitegral holds oly if the lower liit fuctio a( k) is a costat a. I additio, p is a positive o-iteger ad p ceil( p ) i all the epressios. Note () p ( -p) e () p ( -p) ( -p) ( log ) () p ( -p) d ( -t) -p- d t ( +) dt + -p d ( ) t dt (-) -p ( - +p) -p ( -) -t -p- dt d d ( -t) -p- e t dt ( ) -p e d d ( -t) -p- log t dt Whe p, -( -p) < <, the followig epressio does ot hold. () p ( -p) (.') -p ( ) log -( -p) - -p ( -p) ( <) d d ( -t) -p- t dt (.) I this case, -p ties itegral of + -p ( <) is carried out, ad super priitive fuctio + -p > is obtaied. Accordig to this, the zero of the super priitive fuctio chages fro to. The itegral with a fied lower liit is iapplicable to such a itegral. I this case, the followig forula which replaced the calculatio order of itegratio ad differetiatio is effective. - 5 -
() p ( -p) d ( -t) -p- dt t dt [ p ] If this forula is used, -p ties itegral of - ( - <) is carried out, ad super priitive fuctio -p ( -p <) is obtaied. Sice the zero of the super priitive fuctio does ot chage, the itegral with a fied lower liit is applicable. (See Eaple 5 i the followig chapter). Super Derivative & Fractioal Derivative I Super Derivative which I developed, first, we obtai the higher derivative, et, etedig the ide of the operator to real uber, we obtai the super derivative. O the cotrary, i traditioal Fractioal Derivative, the super derivative is directly draw fro Riea- Liouville differitegral. However, the calculatio is very difficult. Three eaples are show below. I each eaple, the st is Super Derivative, ad the d is Fractioal Derivative. (.') Eaple ( +) +-/ ( -/) ( /) d d - ( ) 3/ d - - d ( -t) t dt d - d ( -t) t dt 3 4 3 d d - -t ( +t) 3 Eaple e - e - - - ( ) ( -/) e - i e - d - - d ( -t) e -t dt + d - d ( -t) e -t d dt -e erfi -t + d d e - d erfi - e - erfi - d i d d e - d erf- -e - e - i d i d i Eaple 3 ( log ) log - -/ - - -/ log - / - - / - 6 -
Here ( ) log log - (--log ) - - ( -/) d - - d ( -t) log t dt d - d ( -t) log t dt d d tah - + log - -4 tah - -t log + log - log( + ) - log( ) - -t ( log t-) - < The Fro this Therefore Usig this tah - -t 4 tah - -t log + -t - log - -t log + -t - log - -t 4 log + -t - log + -t + log - -t 4 log + -t - log-( -t) 4 log + -t - log t 4 tah - -t ( ) + -t ( log t-) 4 log + -t - log t + -t log t - 4 -t 4 log + -t - - -t log t - 4 -t log d d -4 tah - -t - -t ( log t-) d -4 log + -t + - -t log t +4 -t d d -4 log + log d d + 4 log + - - log -4 d d 4 log - 4 log log d - 7 -
Furtherore Thus 4 log - 4 4 log + 4 log - 4 ( ) Referece log 4 log + log - 4 4 ( log - ) + log d log +log - 4 ( log - ) + log d ( -t) - t dt - -t ( +t) 3 ( -t) - e t dt -e erf -t erf() z z e -t dt ( -t) - e -t dt -e - erf i -t erf i() z erf( iz )/i ( -t) - log t dt -4 tah - -t..3 The deerit ad the strog poit of Fractioal Derivative - -t ( log t-) As see i three upper eaples, Fractioal Derivative based o Riea-Liouville differitegral is difficult like this. Although Eaple 3 was the / ties derivative of a logarithic fuctio, for this calculatio, the delicate techique was used abudatly, ad oe day was required. Whe p is a real uber, the p ties derivative of this fuctio is hopelessly difficult. Thus, i Fractioal Derivative, it is very difficult to obtai the derivative fro Riea-Liouville differitegral. Moreover, how to take a lower liit is ot clear, ad it caot treat trigooetric fuctios etc. These are the sae as that of Fractioal Itegral. A peculiar proble to Riea-Liouville differitegral is ot to be able to use it whe p is a iteger. It is because of becoig ( ) of p at this tie. I this case, it will rely o the Higher Derivative. However, (.) ad (.') are the powerful tools of uerical coputatio ad ca obtai super derivative of the arbitrary poits of arbitrary fuctios easily. Ad super derivatrive ca be verified uerically with these. - 8 -
.3 Super Derivative of Power Fuctio.3. Forula of Super Derivative of Power Fuctio Aalytically cotiuig the ide of the differetiatio operator i Forula 9.. ( 9. ) to [,p ] fro [,] we obtai the followig forula. I additio, Riea-Liouville differitegrals are also epressed together Forula.3. Whe ( z) is gaa fuctio ad p is ceilig fuctio, the followig epressios hold () Basic for () p ( +) ( + -p) () Liear for ( a+b) () p -p ( -p) (-) -p ( - +p) -p ( -) ( -p) ( ) a d ( -t) -p- d t dt ( ) -t -p- dt -p ( +) ( + -p) d d ( -t) -p- ( at+b) dt ( -p) - a b d t dt ( <) ( a+b) -p ( ) ( a+b) () -p ( - +p ) - ( ) -p a+b a ( -) ( -p) ( -t) -p- d dt ( at+b ) dt p ( <) Cautio! Do ot describe -p to be a p i the upper forula. Because, sice the law of epoets a p q a pq a does ot hold for the arbitrary real ubers p,q at the tie a <, a -p (a - ) -p a (-)( -p) a p i.e. a If it is described as a istake result is caused. For eaple, whe a -3, p /, -p -3 a a p (-3) -p a p -p a p, the rotatio directio o the cople plae becoes reverse ad the - - ( -) 3 ( -) 3 (-) - 3 (-) 3 -i 3 i 3-9 -
Eaple > p 9 ( +) +-/ ( +) +-9/ - - 9 7 9/ / 9 9.54739.955579 Whe /, / < / >, / < 9/ >, are draw o a figure side by side, it is as follows. Eaple p () ( +) - +- Eaple 3 < p () + + - ( +) +-/ ( +) - +- - - ( ) 3 ( ) ( /) ( ) - () - - - Eaple 4 < - (-) - - () ( ) - () ( ) -(-) +/ -- -( -) - - -(-) + -- -(-) 3 - -i ( ) - - () 5 - - -(-/) + - - ( 5/) - -(-/) ( /) - - 4 3-5 - -
Eaple 5 - - 3-3 p -p < < (-) - - - - - 3-3 + - ( -t) - - 3 5 ( 5/6) - -i 6 ( /3) 5.87 - - i 6.6789 d dt t - 3 dt - 3 -.435i - 5 6 - ( -t) t - 4 3 dt This last itegral caot be epressed with a eleetary fuctio. The, if the value o is uerically itegrated by atheatical software, it is as follows. This result is cosistet with the previous value -.435i eactly. Riea-Liouville differitegral a : -/3: p : /: df : diff(t^a,t) 4 3 3 t fl : -> /gaa(ceil(p)-p) *it((-t)^(ceil(p)-p-)*df, tifiity..) ( t) p p df d t (p p) float(fl()).4356763 i.3. Half Derivative of a power fuctio Especially, Super Derivative of order / is called Half Derivative. Forula.3. Let be a o-egative iteger, -!!, ( -)!! 35( -),!!,!! 46, the followig epressios hold. () Basic for + ( )!! ( -)!! - ( + )( + )!! ( ) +!! - -
() Liear for ( a+b) ( a+b) + ( )!! ( -)!! +!! a - ( a+b) ( + )( + )!! a ( a+b) ( ) Proof Upper rows ( a+b) of the liear for of Forula.3. were as follows. ( a+b) + Here, whe is a o-egative iteger, The + ( +) + - + + ( +) - a - a ( +) + - + + ( +) ( a+b) - ( a+b) ( -)!!, ( )!!! ( +) + ( + ) + ( +)! ( -)!! ( + )( ) + ( + )!! ( )!! ( -)!! ( +) -!! + +!! ( + )( + )!! ( + )!! Substitutig these for the previous forula, we obtai the liear for, adgivig a, b to the, we obtai the basic for. Eaple!! (-)!!!!!! 4!! 3!! 3 - - 8 3 3 ( a - ) ( a ) ( a 3 ) 3 6!! 5!! 5 5 6 5 ( a 3 5 ) - -
Eaple 3 5 7!!!! 3!! 4!! 35!! 6!! 47!! 3 8!! 3 4 5 6 35 3 3 ( ) a ( ) a 3 ( ) a3 4 ( 3 ) a4.3.3 Half Derivative of a iteger power fuctio (Fractioal Derivative) Net, usig Riea-Liouville differitegral, we obtai the Half Derivative of a iteger power fuctio. Forula.3.3 Whe deots a atural uber, the followig epressio holds. Proof + k - k k ( ) k+ - Let be a atural uber,, p/. The, sice >p, fro Forula.3. we obtai the followig epressio. ( /) d - d ( -t) t dt Here, accordig to 岩波数学公式 Ⅰ p96, the followig epressios hold -r The t -t dt -t ( ) - + ( ) r r - r ( -t) -r+ t dt -t r -r ( -t) (- ) -t (-) + r r -r+ (-) ( ) r r - r Differetiate both sides of this with respect to as follows. d d t -t d dt d (-) (-) r( ) + -r+ r - r r( ) - r r d d -r+ -r+ t -t + dt - - 3 -
Here, we devise further, r (-) r- -r+ r Usig this, we obtai By-product ( +) r + r + k r ( ) k+ (-) r- -r+ r (-) r- -r+ r ( ) ( -r) + - - -r -r - k k - - k Coparig Forula.3. ad Forula.3.3, we obtai the followig forula. k (-) k k+ k ( )!! ( + )!! This is the sae as the by-product i 7.3.3..3.4 Fractioal Derivative of a iteger power fuctio (-) k k+ k Geeralizig Forula.3.3, we calculate a fractioal derivative of a iteger power fuctio. First, we prepare the followig lea. Lea Whe, are atural ubers, the followig epressio holds. ( -t) - t dt - ( -t) (-) + ( ) r - r -r ( -t) r ( -r) +( -) (4.) Proof Let F() t - ( -t) (-) + ( ) r Differetiate this with respect to t. The - d d F() t dt dt + ( -t) - + ( ) - ( -t) (-) + ( - )( -t) - (-) + - r - -r ( -t) r ( -r) +( -) ( ) r - r d dt r( ) ( ) r -r ( -t) r ( -r) +( -) - r - r -r ( -t) r ( -r) +( -) -r ( -t) r ( -r) +( -) - 4 -
- ( -t) - + ( ) - ( -t) (-) - ( -t) (-) - ( -t) t r r - ( -r)( ) r ( - ) +( ) ( -) r ( -t) r - r -r ( ) -r -r- ( -t) r ( -r) +( -) -r - r ( -t) r ( -r) +( -) - ( -t) (-) (-t ) Usig this Lea, we obtai the followig forula. Forula.3.4 Whe, are atural ubers such that,, the followig epressios hold. Proof k ( +-) (-) k - ( -/) k k+( -) k +( ) ( -/) (-) k k+( -) k () p ( +) + -p - k (-) k k+( -) k ( +, -/) - ( +-) -p ( k-p) Give, p/ to this. The sice k /, ( -/) - ( -/) Usig the above Lea, we calculate as follows. - ( -t) t dt d - d ( -t) t dt d - d ( -t) t dt - ( -t) (-) + - (-) ( ) r ( ) r - r - - (4.) (4.') (): beta fuctio (4.) d k d ( -t) k-p- k t dt ( p) - r -r ( -t) r ( -r) +( -) -r r ( -r) +( -) - 5 -
r Furtherore, sice r, are itegers, r Usig this, (-) r- ( -r) +( -) r - ( -t) t dt k Differetiatig this with respect to, Thus (-) r- ( -r) +( -) - + r r k - -r -r ( ) ( -r) +( -) (-) k k+( -) k (-) k k+( -) k d - d ( -t) t dt ( +-) k - + (-) k k+( -) k ( +-) (-) k - ( -/) k k+( -) k Moreover, (4.') follows iediately fro this. Net, ( +) ( + -/) Sice this have to be equal to (4.), Fro this -( +-) k k Reark (-) k k+( -) k - (-) k k+( -) k ( +) ( -/) ( +, -/) - ( +-) - - ( -/) ( + -/) ( +) ( -/) + -/ - (4.) suggests that (4.) ca be epressed with a beta fuctio ad, ca be real ubers. Actuality, ( +) ( +) ( ) The, ( + -p) ( -p) () p ( +) ( + -p) -p ( + -p) -p ( +, -p) ( -p).3.5 Super Derivative of a iteger power fuctio +, -p ( -p) Replacig / with p i Forula.3.4, we obtai the followig forula. Forula.3.5 Whe is a atural uber, the followig epressios hold for <p. -p ( ) (4.) (4.) - 6 -
() p +-p ( -p) k (, -p) - -p - p k (-) k k+-p k - k k ( ) k+-p - -p (5.) ()deots beta fuctio. (5.) Eaple 3 ( +) +- 3 3-3 3-3 - 3 3-3 - 3 - - 3-3.5-3 + 3-3 - 3.5-3 Eaple - 3, - 3-3 - 3 3-e ( 3, -e) - e -e - - 3 - -e + 3-e.788675 -.59637.3.6 Super Derivative of a positive power fuctio Sice Forula.3.5 are bioial fors, the further geeralizatio is possible. Forula.3.6 The followig epressios hold for p,q such that <p q. q () p q +-p ( -p) r ( q, -p) - q -p p r (-) r q r+-p r - r r ( ) r+-p.3.7 Super Derivative of a polyoial q - q-p (6.) ()deots beta fuctio. (6.) I the case of a polyoial f( ) c-k -k, as for the zero of the super derivative, it is good to k perfor it as follows. It is based o eperiece of a writer. () Whe f( ) is factored by priary forula a+b, let -b /a be the zero. () Whe f( ) is ot factored by priary forula a+b, let be a the zero. For eaple, i the case of the / ties derivative of f( ) - +, the followig calculatio is right i ay case. -+ ( -) 3 8 ( -) 3 If this is calculated terwise as follows, the result is differet fro the forer. - 7 -
-+ - 3 8-4 3 Needless to say, this cause is the differece betwee + + - -+ d - ad -d - + d - That is, it is because the latter regarded it as ad although the forer regarded the zero of the super derivative as pi. The latter is right whe there is a special reaso why the zero of the super derivative should be. However, such a case is rare, ad i alost all cases the forer is right. - 8 -
.4 Super Derivative of Epoetial Fuctio Aalytically cotiuig the ide of the differetiatio operator i Forula 9.. ( 9. ) to [,p ] fro [,] we obtai the followig forula. I additio, Riea-Liouville diffeitegrals are also epressed together Forula.4. Whe p is ceilig fuctio, the followig epressios hold () Basic for e () p () Liear for e a+b () p (3) Geeral for a+b () ( ) -p e ( -p) a Proof of the geeral for -p e a+b ( -p) p alog -p Let calog, d blog, the a+b ( -p) d d ( -t) -p- e t dt d d ( -t) -p- e at+b dt a >:-, a <:+ d d ( -t) -p- at+b dt a >:-, a <:+ e c+d e alog+blog e alog e blog e log a e log b a b a+b Applyig this to () Liear for, we obtai (3) Geeral for iediately. Eaple e - - - ( ) e - i e - -ie - e 3-4 () ( ) 3 i log -3 3 - e 3-4 4.7884 e 3-4 -i (.93358 -.35836 i ) log( -3) - (-3) (.48774 +.558 i ) (-3) -/ 3 log 3.4847 3 Whe 3 (), 3 ( / ), 3 are draw o a figure side by side, it is as follows. - 9 -
y 8 6 4...4.6.8...4.6.8. 3^*l(3) 3^/(/l(3))^(/) 3^ - -
.5 Super Derivative of Logarithic Fuctio Reversig the sig of the ide of the itegratio operator i Forula 7.5. ( 7.5 ), we obtai the followig forula. I additio, Riea-Liouville differitegrals are also epressed together Forula.5. Whe ( z ), ( z ), p deote zeta fuctio, psi fuctio, ceilig fuctio respectively, the followig epressios hold. () Basic for ( log ) () p () Liear for Proof a+b () p log( ) Where, ( -p) ( -p) log -( -p) - -p ( -p) ( -p) d d ( -t) -p- log t dt log( a+b) -( -p) - b ( -p) + a d d ( -t) -p- log( at+b) dt ( -p) Forula 7.5. ( 7.5 ) was as follows. log d p - a b - b a - a b (-) p ( p -)! for p,,3, log( a+b) d p log -( +p) - p ( +p) -p log( a+b) -( +p ) - b ( +p) + a Sice differetiatio is the reverse operatio of itegratio, replacig the ide p of the itegratio operator with -p, we obtai the desired epressios. Ad Forula.3. (.3 ) was ( -) ( -) (-) +!,,,,3, The, replacig with p -, we obtai ( -p) (-) p ( p -)!, ( -p) Eaple ( log ) () log( ) 3+4 p,,3, log - - - - - log( ) - 3+4 - -.564( log +.386 )/ -(-)! - - - 4 + 3 - p +.3333 - -
( log ) ( log ) 9 log - - - 9 log - - 9 - - - - - 9.9357 - ( log +.777).5-9 ( log +9.8465) Whe log, ( log ) ( / ), ( log ) ( 9/ ), ( log ) () are draw o a figure side by side, it is as follows. y 3 4 5 6 7 - -4-6 l() -/^(/)/gaa(9/)*(euler - l() + psi(9/) -/^(9/)/gaa(/)*(EULER - l() + psi(/) / - -
.6 Super Derivative of Trigooetric Fuctio.6. Super Derivatives of si, cos Aalytically cotiuig the ide of the differetiatio operator i Forula 9..4 ( 9. ) to [,p ] fro [,] fwe obtai the followig forula. Forula.6. () Basic for ( si ) () p p si + ( cos ) () p p cos+ () Liear for a+b () p si( ) a+b () p cos( ) a -p p sia+b + a -p p cosa+b + Eaple ( si ) ((si ) ) si- ( cos ) ( ) cos () si+ (si (+ )) 4 - si+ 4 si+ + 4 - si - + cos+ cos+ cos+ 4 cos( +) si+ si cos Whe cos, ( cos ) ( / ), -si are draw o a figure side by side, it is as follows. Red shows / order super derivative. It is clear also i the figure that super derivative which is the easiest to uderstad is super derivative of trigooetric fuctios. - 3 -
.6. Terwise Super Derivative of si, cos Reversig the sig of the ide of the operator of the collateral super itegrals of si, cos i 7.6. (7.6 ), we obtai the followig terwise super derivatives. These are collateral super derivatives as uderstood fro the costat-of-differetiatio fuctio i the right side. ( si ) () p k ( cos ) () p k (-) k k+- p p ( k+-p) si+ p C( p, ) k (-) k k- p p ( k+-p) cos+ p C( p, ) k + C( ) p, -p-k ( -p -k) - C( ) p, -p-k ( -p -k) si k cos k Whe the /th order collateral super derivative of cos is draw as cos ad -si side by side, it is as follows. Red shows the /th order collateral super derivative. - 4 -
Copared with the upper figure, the collateral super derivative is curvig uaturally ear the coordiate origi i this figure. Sice this is siilar also i si, it is thought that the terwie super derivatives of si ad cos are asyptotic epasios of the lieal super derivatives. - 5 -
.7 Super Derivative of Hyperbolic Fuctio.7. Super Derivatives of sih, cosh Aalytically cotiuig the ide of the differetiatio operator i Forula 9..5 ( 9. ) to [,p ] fro [,] we obtai the followig forula. Forula.7. () Basic for ( sih ) () ( cosh ) () () Liear for Eaple p i p i -p sih + p i p i -p cosh + a+b () p sih( ) a+b () p cosh( ) ( sih ) ((sih ) ) -p i a a -p i a a e - (-) -p e - e + (-) -p e - p i sih a+b + - e a+b - (-) -p e -( a+b) p i cosh a+b + - e a+b + (-) -p e -( a+b) i - sih+ (i - i - i i sih (+ )) 4 i - i i sih+ + 4 i sih - i + i - i sih+ 4 cosh i cosh+ ( cosh ) ( cosh ) i 9 i () - i i - i i - 9 i i i cosh + + -icosh( + i ) icosh cosh - 9 cosh -.788 cosh - 9 4.7 cosh - - 6 -
Super derivative of hyperbolic fuctio is the ost icoprehesible i super derivatives. The reaso is that the super derivative turs ito a cople fuctio ecept the order p is a iteger or a purely iagiary uber. The, whe cosh, ( cosh ) ( i/ ), ( cosh ) ( 9 i/ ), sih which ca be displayed o a real uber doai are draw o a figure side by side, it is as follows. y 4 3-4 -3 - - 3 4 5 6 cosh() /(I)^(/*I)*cosh( - /*PI) /(I)^(9/*I)*cosh( - 9/*PI) sih() All of four curves have overlapped i the positive area. It is atural that ( cosh ) ( i/) is ear cosh i a egative area. But ( cosh ) ( 9 i/) is far apart fro sih i why. Eaple ( sih ) ((sih ) ) i cosh+ () - - e - ( ) e + i e - e - e i + -i e - i e + i e - e i + e - e + e - cosh - e + + (-) - e - + i e - i e - e - i e i e - e - i i e + e - icosh ( ) i sih - - i e - ( ) e - e - e i i - e - e - e - - 7 -
.7. Terwise Super Derivative of sih, cosh Reversig the sig of the ide of the operator of the collateral super itegrals of sih, cosh i 7.7. we obtai the followig terwise super derivatives. These are collateral super derivatives as uderstood fro the costat-of-itegratio fuctio i the right side. ( sih ) () p k ( cosh ) () p k k+- p i -p p i ( k+-p) sih+ p C( p, ) k k- p i -p p i ( k+-p) cohs+ p C( p, ) k - C( p,) -p-k i -k k i sih ( -p -k) - C( p,) -p-k i -k k i cosh ( -p -k) Whe the values of the lieal ad the collatera l.7th order derivatives of cosh o, 6 are calculated respectively, it is as follows. Though the differece of both is large where is sall, both are alost correspodig ig where is large. Sice this is siilar also i sih, it is thought that the terwise super derivatives of sih ad cosh are asyptotic epasios of the lieal super derivatives. - 8 -
.8 Super Derivative of Iverse Trigooetric Fuctio.8. Super Derivatives of ta -, cot - Reversig the sig of the ide of the itegratio operator i Forula 7.8. ( 7.8 ), we obtai the followig forula. Forula.8. Whe ( ),( ) hold for. ta - () p ta - ( -p) k( ) deote gaa fuctio ad digaa fuctio respectively, the followig epressios + -p - k -p -k log+ ( -p) ( ) k - ( -p) (-) r r -p -p-k - k -p +-k -p -p +-r -p+-k ( -p) -( r) cot - () p -p cot - ta - - ( -p) ( -p) k( - ) k -p -p -k log+ - ( -p) ( - ) k -p -p+-k k -p +-k -p + ( -p) (-) r ( -p) -( r) r -p +-r Eaple: /th order derivative of cot - -p+-r -p-k -p+-r Riea-Liouville differitegral g : -> /gaa(-p)*it((-t)^(-p-)*arccot(t), t..) - 9 -
( t) p arccot(t) d t ( p).7.7 Alie's Matheatics K. Koo - 3 -