-e x ( 0!x+1! ) -e x 0!x 2 +1!x+2! e t dt, the following expressions hold. t

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1 4 Higher and Super Calculus of Logarihmic Inegral ec. 4. Higher Inegral of Eponenial Inegral Eponenial Inegral is defined as follows. Ei( ) - e d (.0) Inegraing boh sides of (.0) wih respec o repeaedly by ONLINE INTEGRATOR (Wolfram Mahemaica) and arranging he resuls, we obain he following higher indefinie inegrals. Ei( ) d! Ei( ) d2 2! Ei( ) d3 3! Ei( ) dn n! Ei( ) -e 0! 2 Ei( ) -e ( 0!+! ) 3 Ei( ) -e 0! 2 +!+2! n Ei( ) -e n - Σ r! n-- r Alhough hese righ sides are he lineal primiive funcions of Ei(), since boh zeros of Ei() and e are -, zeros of he righ sides are all -. Therfore, he lineal higher primiive funcion of Ei() can be epressed by he higher inegral wih a fied lower limi -. Formula 4.. When Eponenial Inegral is Ei( ) - - Ei( ) d n - n! e d, he following epressions hold. n Ei( ) -e n - Σ r! n-- r (.n) Eample : 3rd order inegral of Ei () - -

2 - 2 -

3 4.2 Higher Inegral of Cosine Inegral Cosine Inegral is defined as follows. Ci( ) cos d Inegraing boh sides of his wih respec o repeaedly by ONLINE INTEGRATOR and arranging he resuls, we obain he following higher indefinie inegrals. Ci( ) d! Ci( ) Ci( )- 0!sin d2 2 Ci( )- 0!sin +!cos 2! Ci( ) d3 3 Ci( )- 0! 2-2! sin +!cos 3! Ci( ) d4 4 Ci( )- 0! 3-2!sin +! 2-3! cos 4! Ci( ) dn n! Ci( ) ( n -)/2 n - sin Σ (-) r ( 2r )! n--2r ( n -2)/2 + cos Σ (-) r ( 2r + )! n-2-2r Alhough hese righ sides are he lineal primiive funcions of Ci(), hese zeros are all. (See he above figure.) Therfore, he lineal higher primiive funcion of Ci() can be epressed by he higher inegral wih a fied lower limi. Formula 4.2. When is floor funcion and Ci( ) Ci( ) d! Ci( )- 0!sin Ci( ) d n n! cos d,is Cosine Inegral, he following epressions hold. Ci( ) ( n- )/2 n - sin Σ (-) r ( 2r )! n--2r - 3 -

4 ( n-2 )/2 + cos Σ (-) r ( 2r+ )! n-2-2r n 2 Eample : 4h order inegral of Ci () If boh sides are illusraed, i is as follows. Since boh sides overlap eacly, he lef side (blue) is no visible

5 4.3 Collaeral Higher Inegral of Sine Inegral Sine Inegral is defined as follows. Si( ) 0 sin d Inegraing boh sides of his wih respec o repeaedly by ONLINE INTEGRATOR and arranging he resuls, we obain he following higher indefinie inegrals. Si(! Si( ) Si( ) + 0!cos d2 2 Si( ) + 0!cos +!sin 2! Si( ) d3 3 Si( ) + 0! 2-2! cos +! sin 3! Si( ) d4 4 Si( ) + 0! 3-2!cos +! 2-3! sin 4! Si( ) dn n! Si( ) ( n -)/2 n + cos Σ (-) r ( 2r )! n--2r ( n -2)/2 + sin Σ (-) r ( 2r + )! n-2-2r Alhough hese righ sides are he lineal primiive funcions of Si(), hose zeros are all 0 a he ime of even order and are no 0 a he ime of odd order. Tha is, he lineal higher primiive funcion of Si() can no be epressed by he higher inegral wih a fied lower limi.(see he above figure.) Therfore, he higher inegra of Si() wih a fied lower limi 0 is no lineal bu collaeral. However, he idea which makes 0 a common lower limi is naural. I is because he Si() iself is defined by he inegral wih a lower limi 0. Collaeral Higher Inegral of Sine Inegral Collaeral Higher Inegrals of Si() are obained by compensaing he above lineal higher primiive funcions wih Consan-of-inegraion Polynomials. Formula 4.3. When is floor funcion and Si( ) 0 sin d,is Sine Inegral, he following epressions hold

6 Si( ) d 0 Si( ) + 0!cos - 0! 0! 0 Si( ) 2 d Si( ) 2 Si( ) + 0!cos +!sin - 2!! d 3 3 Si( ) + 0! 2 2-2! cos +! sin - + 3! 2! 30! 0 Si( ) d 4 4 Si( ) + 0! 3-2!cos +! 2-3! sin 0 4! ! 3! 0 0 Si( ) d n 0 n! Si( ) ( n- )/2 n + cos Σ (-) r ( 2r )! n--2r ( n-2 )/2 + sin Σ (-) r ( 2r+ )! n-2-2r ( n- )/2 - Σ (-) r n--2r ( 2r+ )( n --2r)! Eample : Collaeral he 4h order inegral of Si () If boh sides are illusraed, i is as follows. Since boh sides overlap eacly, he lef side (blue) is no visible

7 4.4 Higher Inegral of Logarihmic Inegral Logarihmic Inegral is defined as follows. li( ) 0 d log 6 (.0) Firs, we prepare wo Lemmas. Lemma 4.4. When Eponenial Inegral is Ei( ) - e d, he following epressions hold. Ei( 2log ) d Ei( 2log ) -Ei( 3log ) Ei( 3log ) d Ei( 3log ) -Ei( 4log ) Ei( nlog ) d Ei( nlog) -Ei( n +) log (.n) Proof Le 2log. Then e 2 Ei( 2log ) d 2 Ei() e 2, d d e 2 d. Hence 2 2 Calculaing he inegral of he righ side by ONLINE INTEGRATOR, we obain Using his, Ei() e 2 d 2Ei() e 2 d 3-2Ei 2 Ei( 2log ) d Ei() e -Ei 2 3 Ei( 2log ) -Ei( 3log ) 2 Ne, le 3log. Then we obain he following epresssion by he same calculaion. Ei( 3log ) d Ei() 3 e 3 d 3 Ei() e 3 d Ei() e 3 4 -Ei 3-7 -

8 Ei( 3log ) -Ei( 4log ) Hereafer, by inducion, we obain he desired epresson. Noe Since log - a he ime +0, 0 is clearly a zero of hese funcions. Then, (.n) can be wrien as follows. Ei( nlog) d Ei( nlog) -Ei ( n +) log 0 (.n') Lemma When Eponenial Inegral is Ei( ) - e d, he following epressions hold. n Ei( log ) d n+ Ei( log ) - n + n + Ei ( n +2) log (2.n) Calculaion Calculaing by ONLINE INTEGRATOR, we obain (2.n) immedialy. Noe Since log - a he ime +0, 0 is crealy a zero of hese funcions. Then, (2.n) can be wrien as follows. n Ei( log ) d n+ Ei( log ) - 0 n + n + Ei ( n +2) log (2.n') Formula When li( ) 0 d, Ei() log - e d he following epression holds for 0. 0 li( ) d n n 0 n! Σ( -) r nc r n- r Ei( r+) log (3.n) Proof Le log. Then [ 0, ] [ -, ], d e d. Hence 0 Ne, le 0 0 d log - e d [ Ei() ] - d 2 log 0 li( ) d Ei(log ) li( ) Calculaing he inegral of he righ side by ONLINE INTEGRATOR, we obain li( ) d li()-ei(2log ) - 8 -

9 Since he zero of his righ side is 0 obviously, li( ) d Ei( log ) -Ei( 2log ) 0 Ne, inegraing boh sides of his wih respec o and applying Lemma 4.4., o he resul, we obain he following 0 li( ) d 2 0 Ei( log ) d - 0 Ei( 2log ) d 0 2 Ei( log ) - Ei( 3log ) Ei( 2log ) -Ei( 3log ) 2 Ei( log ) -2Ei( 2log ) +Ei( 3log ) 2 Ne, inegraing boh sides of his wih respec o and applying Lemma 4.4., o he resul, we obain he following 0 0 li( ) d Ei( log ) d Ei( 2log ) d + 2 Ei( 3log ) d 0 3 Ei( log ) - Ei( 4log ) - 2 Ei( 2log ) + Ei( 4log ) 3! 3! Ei( 3log ) - Ei( 4log ) 2 2 3! 3 Ei( log ) -3 2 Ei( 2log ) +3Ei( 3log ) -Ei( 4log ) Hereafer, by inducion, we obain he desired epresson. Eample : 2nd order inegral of li( ) - 9 -

10 4.5 Higher Inegral of Double Logarihmic Funcion Double Logarihmic Funcion is defined as follows. f( ) log log (.0) Inegraing boh sides of his wih respec o repeaedly and arranging he resuls, we obain he following Ei( log ) is Logarihmic Inegral menioned in he previous. higher indefinie inegrals. Where, li( ) log log log -li( )! log 2! 2 log log -2li() +Ei( 2log ) log log d 3 3! log log d n n! 3 log log -3 2 li( ) +3Ei( 2log ) -Ei( 3log ) n n log log +Σ (-) r nc r n-r Ei( rlog) r Alhough hese righ sides are he lineal primiive funcions of log log, since boh zeros of n log log and Ei( nlog ) are 0, zeros of he righ sides are all 0. Therfore, he lineal higher primiive funcion of log log can be epressed by he higher inegral wih a fied lower limi 0. Formula 4.5. When Ei( ) log log d n n! e d, he following epressions hold for 0. Eample : 2nd order inegral of log log n n log log +Σ (-) r nc r n-r Ei( ) r When he one arbirary poin.6 is given), he values of he boh sides are as follows. rlog (.n) - 0 -

11 4.6 Super Calculus of Logarihmic Inegral Among he higher inegrals menioned in previous secions, Higher Inegral of Logarihmic Inegral is eensible even o Super Calculus. I is because his higher inegral is epressed wih binomial coefficiens Super Inegral of Logarihmic Inegral Formula 4.6. When li( ) 0 d, Ei( ) - log he following epression holds for p 0 and 0. 0 li( ) d p 0 ( +p) e d Σ (-) r p r p- r Ei( r+) log Proof Firs, replace n!, nc r wih ( +n ), n r respecively in Formula Ne, analyically coninuing he inde of he inegraion operaor o [ 0,p ] from[,n ], we obain he desired formula. Eample : 3/2h order inegral of li( ) We calculaed he funcion values on arbirary one poin 4 according o he formula and Riemann- Liouville inegral. As he resul, wo values were almos corresponding. Compared wih he figure of he 2nd order inegral in 4.4, we can find ha his curvaure is loose Super Derivaive of Logarihmic Inegral Formula When - -

12 li( ) 0 d, Ei( ) - log e d he following epression holds for p >0, p,2,3, and 0. () p -p li( ) ( -p) Σ ( ) -p- r Ei( r+) log r Proof - r In fac, Formula 4.6. holds for p -,-2,-3,, 0. Therefore, in Formula 4.6., replacing he inegraion operaor <> p wih he differeniaion operaor () < > Eample : /2h order derivaive of li( ) p -p, we obain he desired epression. We calculaed he he super differenial coefficiens on arbirary one poin 2 according o he formula and Riemann-Liouville differinegral. As he resul, wo values were almos corresponding

13 4.7 Super Calculus of Double Logarihmic Funcion Among he higher inegrals menioned in previous secions, Higher Inegral of Double Logarihmic Funcion is eensible even o Super Calculus. I is because his higher inegral is epressed wih binomial coefficiens Super Inegral of Double Logarihmic Funcion Formula 4.7. When Ei( ) - e d, he following epressions hold for p >0, 0. 0 log log d p 0 ( +p) p log log +Σ ( ) r p r - r p-r Ei( rlog) Proof Firs, replace n!, nc r wih ( +n ), n r respecively in Formula Ne, analyically coninuing he inde of he inegraion operaor o [ 0,p ] from[,n ], we obain he desired formula. Eample : 5/3h order inegral of log log We calculaed he funcion values on arbirary one poin.5 according o he formula and Riemann- Liouville inegral. As he resul, wo values were almos corresponding Super Derivaive of Double Logarihmic Funcion Formula When Ei( ) - e d, he following epression holds for p >0, p,2,3,, 0, - 3 -

14 he following epression holds. log log () p p ( -p) log log +Σ ( ) r -p r - r Ei( rlog) r Proof In fac, Formula 4.7. holds for p -,-2,-3,, 0. Therefore, in Formula 4.7., replacing he inegraion operaor < p> wih he differeniaion operaor () Eample : 0.3h order derivaive of log log p < > -p, we obain he desired epressions. We calculaed he he super differenial coefficiens on arbirary one poin 0.5 according o he formula and Riemann-Liouville differinegral. As he resul, wo values were almos corresponding. Since he number of he order of he differeniaion is small, i resembles he figure of log log well Alien's Mahemaics K. Kono - 4 -

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