Summary of Super Calculus

Summary of Super Calculus

01 Gamma Function & Digamma Function

Although the factorial

n!

and the harmonic number

H

_n

( = 1+1/2++1/n )

are usuallydefined for a natural number, if a gamma function and a digamma function are used, these can be defined for the real number

p

. That is,

p! =  ⁽ 1+ p ⁾ , H

_p

=  ⁽ 1+ p + ⁾  (  ^=0.57721)

The former is convenient to express the coefficient of the higher order primitive or derivative of a power function, and the latter is indispensable to non-integer order calculus of the logarithmic function.

Although some formulas about these functions are described here, the following two formulas proved in Section 3 are especially important. That is, when

m , n =0, 1, 2, 3, 

,

 ( -m )

 ⁽ -n ⁾

= ( -1 )

^m-n

n !

m ! ,  ( -n )

 ⁽ -n ⁾

= ( -1 )

ⁿ⁺¹

n!

These show that the ratios of the singular points of

 ^{( )} z

or

 ^{( )} z

reduce to integers or its reciprocals. The former is necessary to express the higher order derivative of fractional functions, and the latter is required for the super calculus of a logarithmic function.

02 Multifactorial

The relational expression of multifactorial and the gamma function is shown here.

For instance, in case of double factorial,

( 2n -1 !! = 2 )

ⁿ

  ^{n +} 2 

1 /    2

1 = 2

ⁿ

  ^{n +} 2 

1 / 

( 2n !! ) = 2

ⁿ

  ^{n +} 2 

2 /    2

2 = 2

ⁿ

 ( n +1 = 2 )

ⁿ

n !

These are used to express the half-integration of a integer-power function later.

Moreover, we obtain the following Maclaurin expansions as by-products.

f   2 x =

0!!

f ( ) 0 x

⁰

+

2!!

f

^'

( ) 0 x

¹

+

4!!

f

^"

( ) 0 x

²

+

6!!

f

^{( )3}

( ) 0

x

³

+  f   3

x = 0!!!

f ( ) 0 x

⁰

+

3!!!

f

^'

( ) 0 x

¹

+

6!!!

f

^"

( ) 0 x

²

+

9!!!

f

^{( )3}

( ) 0

x

³

+ 

03 Generalized Multinomial Theorem

First, the binomial theorem and a generalized binomial theorem are mentioned. The Leibniz rule and the Leibniz rule about super-differentiation are expressed just like these later.

Next, multinomial thorem and generalized multinomial thorem are shown as follows.

Theorem 3.3.1

For real numbers

x

₁

, x

₂

,  , x

_m and a natural number

n

, the following expressions hold.

 x

1

+x

2

+ +x

_m



ⁿ

= Σ

r1=0

n

Σ

r2=0 r1

 Σ

r_m-1=0 r_m-2

  ^{r n}

1

  ^{r r}

¹2

   ^{r r}

^m-2m-1

x

1^{n -r1}

x

2^r1-r2

 x

_m-1^rm-2-rm-1

x

_m^rm-1

(1.1)

= Σ

r1=0

n

Σ

r2=0

n

 Σ

r_m-1=0

n

 r

₁

+ r

₂

+ ⁿ  + r

_m-1

  ^r r

¹

^{+ r}

₂²

+ ⁺  ^ + ⁺ r ^r

_m-1^m-1

   ^r

^m-2

⁺ r ^r

^m-1_m-1



 x

1^n-

^r

¹

^{-  - r}

^m-1

x

2^r1

x

3^r2

 x

_m^rm-1

Theorem 3.4.1

The following expressions hold for real numbers



and

x

₁

, x

₂

,  , x

_m

s.t.

 

x

₁

 _ x

₂

+ x

₃

+ +x

_m

_

.

.

 x

₁

+x

₂

+ +x

_m



^

= Σ

r1=0



Σ

r2=0 r1

 Σ

r_m-1=0 r_m-2

  ^ r

₁

  ^r r

¹₂

   ^r r

^{m -2}_{m -1}

x

₁^-r1

x

₂^r1-r2

 x

_m-1^rm-2-rm-1

x

_m^rm-1

(1.1)

= Σ

r1=0



Σ

r2=0



 Σ

r_m-1=0



 r

1

+ r

2

+ ^  + r

_m-1

  ^r r

¹

^{+ r}

2²

+ ⁺  ^ + ⁺ r ^r

_m-1^m-1

   ^r r

^m-2

^{+ r}

^m-1_m-1



 x

₁^-

^r

¹

^{-  - r}

^m-1

x

2^r1

x

3^r2

 x

_m^rm-1

(1.2) Where,  

x

₁

 _ x

₂

+ x

₃

++x

_m

_

is allowed at

 > 0

.

Higher order calculus of the product of many functions and super order calculus of the product of many functions are expressed just like these later.

What should be paid attention here are the following property of generalized binomial coefficients.

Σ

r=0

n n

Ｃ

r

= Σ

r=0



  ⁿ r

That is, once generalized binomial coefficients was used, the upper limit of



should be



. Therefore, when

n

is a natural number and

p

is a real number, the following holds in most cases.

n  p  　 Σ

r=0

n n

Ｃ

r

f( n,x  ) Σ

r=0



  ^p r f( p,x )

When the original coefficient is not binomial coefficient e.g. 1 ,

n  p  　 Σ

r=0

n n

Ｃ

r

f( n,x  ) Σ

r=0



  ^p r f( p,x )

Although

Σ

r=0 p

1 f( ) p,x

is difficult,

Σ

r=0



  ^p r f( p,x )

is satisfactory. What enables super calculus in this text is just this property of the generalized binomial coefficient. Newton is great !

04 Higher Integral

(1) Definitions and Notations

The 1st order primitive function of

f (x )

is usually denoted

F (x )

. However, such a notation is unsuitable for the description of the 2nd or more order primitive functions. Then,

f

^{< >1}

( ) x , f

^{< >2}

( ) x ,  , f

^{< >n}

( ) x

denote the each order primitive functions of

f (x )

in this text. Here, for example, when

f (x )= sin x

,

f

^{< >1}

( ) x

might mean

-cos x

or might mean

-cos x + c

. Which it means follows the definition at that time.

Furthermore, each order integrals of

f (x )

are denoted as follows.



_a

1

x

f( ) x dx

¹

, 

_a

2



x_a

1

x

f( ) x dx

²

,  , 

_a

n

x

 

_a

1

x

f( ) x dx

ⁿ

And these are called higher integral with variable lower limits . On the other hand,



_a^x

^{f( ) x dx}

¹

^, 

_a



^x_a^x

^{f( ) x dx}

²

^{,  ,} 

_a^x

^ 

_a^x

^{f( ) x dx}

ⁿ

are called higher integral with a fixed lower limit .

(2) Fundamental Theorem of Higher Integral

There is Fundamental Theorem of Calculus for the 1st order integral. The same theorem holds for the higher order integral.

Theorem 4.1.3

Let

f

^{< >r}

r=0, 1,  , n

be continuous functions defined on a closed interval

I

^and

f

^<r+1> be the arbitrary primitive function of

f

^{< >r} . Then the following expression holds for

a

_r

, x  I

^.



_a

n



x



_a

1 x

f( ) x dx

ⁿ

= f

^{< >n}

( ) x - Σ

r=0 n -1

f

^{<n -r>}

_ a

_n-r

_ 

_a

n



x



_a

n- r+ 1 x

dx

^r

Especially, when

a

_r

= a for r=1, 2,  ,n

^,



_a

^

^x



_a^x

^{f( ) x dx}

ⁿ

^{= f}

^{< >n}

^{( ) x - Σ}

^{n -1}_r=0

^f

^{<n -r>}

^{( ) a ( x-a} _r! ⁾

^r

(3) Lineal and Collateral

We call Constant-of-integration Polynomial the 2nd term of the right sides of these. And when Constant-of-integration Polynomial is 0, we call the left side Lineal Higher Integral and we call

f

^{< >n}

( ) x

Lineal Higher Primitive Function.

Oppositely, when Constant-of-integration Polynomial is not 0, we call the left side Collateral Higher Integral and we call the right side Collateral Higher Primitive Function.

For example,



2 3

x



2 2

x



2 1

x

sin xdx

³

= cosx

Left: Lineal 3rd order integral Right: Lineal 3rd order primitive



₀



^x₀



^x₀^x

^{sin xdx}

³

^{= cosx+} 2 x

²

-1

Left: Collateral 3rd order integral Right: Collateral 3rd order primitive

Furthermore, from Theorem 4.1.3, we see that

a

_r must be all zeros of

f

^{< >r}

for r =1, 2, , n

in order for the higher integral of

f (x )

to be lineal.

(4) Higher Integral and Reimann-Liouville Integral

The higher integral with a fixed lower limit reduce to the 1st order integral which called Reimann-Liouville Integral.

Theorem 4.2.1 ( Cauchy formula for repeated integration )

When

f (x )

is continuously integrable function and

 ^{( )} z

denotes a gamma function ,



_a

^

^x



_a^x

^{f( ) x dx}

ⁿ

⁼ _{ ( )} ¹ _n 

_a^x

^{( x-t )}

^n-1

^{f( ) t dt}

Reimann-Liouville Integral of the right side is important. However, in the higher integral, since the left side itself has an operation functions, the right side is not indispensable.

By the way, replacing the left side in Theorem 4.1.3 for Reimann-Liouville Integral and shifting the index by

-n

, we obtain the following.

f

^{< >0}

( ) x = Σ

r=0 n -1

f

^{( )r}

( ) a r!

( x-a )

^r

+  ^{( )} n

1 

_a^x

^{( x-t )}

^n-1

^f

^{( )n}

^{( ) t dt}

This is the Taylor expansion of

f (x )

around

a

. And Reimann-Liouville Integral of

f

^{( )n} is the remainder term called Bernoulli form

(5) Higher Integrals of Elementary Functions.

When

m

is a natural number, the 2nd order integral of

x

^m becomes as follows.



₀



^x₀^x

^x

^m

^dx

²

⁼ _{( m +1 ( )} ¹ _{m +2 )} ^x

^m+2

⁼ _{( m +2 !} ^{m !} ₎ ^x

^m+2

Then, when



is a positive number, it is as follows.



₀



^x₀^x

^x

^

^dx

²

⁼ _{( } ^ _{+2 !} ^! ₎ ^x

^+2

Here,

 ^!

can be expressed by gamma function

 ( 1+  )

. Thus



₀



^x₀^x

^x

^

^dx

²

⁼ _{ (} ^ ₁₊ ^{( 1+} _ ^ ₊₂ ⁾ ₎ ^x

^+2

By such an easy calculation, we obtain the following expressions for elementary functions. Where,

,

denote the ceiling function and the floor function respectively.

Higher Integrals of Power Function etc.



₀

^

^x



₀^x

^x

^

^dx

ⁿ

⁼ _

_

1+  + n

_

 ( 1+  )

x

^+n

(   0 )



_^x

^ 

_^x

^x

^

^dx

ⁿ

^{= ( -1 )}

ⁿ

^{ (} _ ^- ₍ ^ _{- } ^{- n} ₎ ^{) x}

^+n

^{(  < -n )}



_^x

^ 

_^x

^e

^{ x}

^dx

ⁿ

^{= ( ) 1}

ⁿ

^e

^{ x}



₀

^

^x



₀^x

^{log x dx}

ⁿ

⁼ _{n !} ^x

ⁿ

_ ^{log x - Σ}

_{k =1}ⁿ

_k ¹ _ x > 0



2 n

x

 

2 2

x



2 1

x

sin xdx

ⁿ

= sin  ^x- 2 

n 



2 (n -1)

x

 

2 1

x



2 0

x

cosxdx

ⁿ

= cos  ^x- 2 

n 



2 n i

x

 

2 2 i

x



2 1 i

x

sinh xdx

ⁿ

=

2 e

^x

- ( -1 )

ⁿ

e

^-x



2 (n -1) i

x

 

2 1 i

x



2 0 i

x

cosh xdx

ⁿ

=

2 e

^x

+ ( -1 )

ⁿ

e

^-x

Higher Integrals of Inverse Trigonometric Functions



₀

^

^x



₀^x

^tan

^-1

^{x dx}

ⁿ

^{= tan} _{n !}

^-1

^{x Σ}

^{n /2}_{k =0}

^{( -1 )}

^kn

^Ｃ

^n-2k

^x

^n-2k

+ 2n ! log  1+ x

²



Σ

k =1 n /2

( -1 )

^kn

Ｃ

n+1-2k

x

^n+1-2k

- n!

1 Σ

r=1 n /2

(-1)

^r n

Ｃ

n+1-2r 

 ⁽ 1+n - ⁾  ⁽ 2r ⁾



x

^n+1-2r



₀

^

^x



₀^x

^cot

^-1

^{x dx}

ⁿ

⁼ _{n !} ^x

ⁿ

^cot

^-1

^{x - tan} _{n !}

^-1

^{x Σ}

^{n /2}_{k =1}

^{( -1 )}

^kn

^Ｃ

^n-2k

^x

^n-2k

- 2 n ! log  x

²

+1 

Σ

k =1 n /2

( -1 )

^kn

Ｃ

n+1-2k

x

^n+1-2k

+ n!

1 Σ

r=1 n /2

(-1)

^r n

Ｃ

n+1-2r 

 ( 1+n - )  ( 2r )



x

^n+1-2r



_a

n



x



_a

1

x

sin

^-1

x dx

ⁿ

= Σ

r=0 n/2

( 2r !! )

²

( n -2r ! ) x

^n-2r

sin

^-1

x

+ Σ

r=1 n /2

Σ

s=0 n -2r+1

( -1 )

^sn -2r+1

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r+1 ! ) x

^n-2r+1

1-x

²

Where,

a

₁

= i 1.508879 , a

₂

= 0 , a

₃

= -i 0.475883 , a

₄

= 0 , 



_a

n



x



_a

1

x

cos

^-1

x dx

ⁿ

= n !

x

ⁿ

cos

^-1

x - Σ

r=1 n/2

( 2r !! )

²

( n -2r ! ) x

^n-2r

sin

^-1

x - Σ

r=1 n /2

Σ

s=0 n -2r+1

( -1 )

^sn -2r+1

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r+1 ! ) x

^n-2r+1

1-x

²

Where,

a

₁

= 1 , a

₂

= 0 , a

₃

= ? , a

₄

= 0 , 



₁

^

^x



₁^x

^sec

^-1

^xdx

ⁿ

⁼ _{n !} ^x

ⁿ

^sec

^-1

^{x -}

^{(n -1 /2}

^Σ

_r=0⁾

( 2r !!( ) 2r+1 ! ) ( 2r-1 !! )

( n -2r-1 ! ) x

^n-2r-1

log  x+ x

²

-1  + Σ

r=1 n /2

Σ

s=0 n -2r

( -1 )

^s

2r+s

Ｃ

n -2r s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r ! ) x

^n-2r

x

²

-1



_a

n



x



_a

1

x

csc

^-1

x dx

ⁿ

= n !

x

ⁿ

csc

^-1

x + Σ

r=0 (n -1 /2)

( 2r !!( ) 2r+1 ! ) ( 2r-1 !! )

( n -2r-1 ! ) x

^n-2r-1

log  x+ x

²

-1 

- Σ

r=1 n /2

Σ

s=0 n -2r

( -1 )

^s

2r+s

Ｃ

n -2r s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r ! ) x

^n-2r

x

²

-1

Where,

a

₁

, a

₂

,  , a

_n are all complex numbers.

Higher Integrals of Inverse Hyperbolic Functions



₀

^

^x



₀^x

^tanh

^-1

^{x dx}

ⁿ

^{= tanh} _{n !}

^-1

^{x Σ}

^{n /2}_{k =0}ⁿ

^Ｃ

^n-2k

^x

^n-2k

^{+ log} ₂ ^ _{n !} ^{1- x}

²

^{ Σ}

^{n /2}_{k =1}ⁿ

^Ｃ

^n+1-2k

^x

^n+1-2k

- n!

1 Σ

r=1 n /2

n

Ｃ

n+1-2r 

 ⁽ 1+n - ⁾  ⁽ 2r ⁾



x

^n+1-2r



₀

^

^x



₀^x

^coth

^-1

^{x dx}

ⁿ

⁼ _{n !} ^x

ⁿ

^coth

^-1

^{x + tanh} _{n !}

^-1

^{x Σ}

^{n /2}_{k =1}ⁿ

^Ｃ

^n-2k

^x

^n-2k

+ 2 n ! log  1-x

²



Σ

k =1 n /2

n

Ｃ

n+1-2k

x

^n+1-2k

- n!

1 Σ

r=1 n /2

Ｃ

n n+1-2r 

 ( 1+n - )  ( 2r )



x

^n+1-2r

| | x <1



_a

n



x



_a

1

x

sinh

^-1

x dx

ⁿ

= Σ

r=0 n/2

( 2r !! )

²

( n -2r ! ) ( -1 )

^r

x

^n-2r

sinh

^-1

x + Σ

r=1 n /2

Σ

s=0 n -2r+1

( -1 )

^r+sn -2r+1

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r+1 ! ) x

^n-2r+1

1+x

²

Where,

a

₁

= 1.508879 , a

₂

= 0 , a

₃

= -0.475883 , a

₄

= 0 , 



₁

^

^x



₁^x

^cosh

^-1

^{x dx}

ⁿ

^{= Σ}

^n/2_r=0

_{( 2r !! )} ^x

^2n-2r

_{( n -2r ! )} ^cosh

^-1

^x

- Σ

r=1 n /2

Σ

s=0 n -2r+1

( -1 )

^sn -2r+1

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r+1 ! ) x

^n-2r+1

x

²

-1



_a

n



x



_a

1

x

sech

^-1

x dx

ⁿ

= n !

x

ⁿ

sech

^-1

x + Σ

r=0 (n -1 /2)

( 2r !!( ) 2r+1 ! ) ( 2r-1 !! )

( n -2r-1 ! ) x

^n-2r-1

sin

^-1

x + Σ

r=1 n /2

Σ

s=0 n -2r

( -1 )

^s

2r+s

n -2r

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r ! ) x

^n-2r

1- x

²

Where,

a

₁

= a

₃

= a

₅

=  = 0

,

a

₂

, a

₄

, a

₆

, 

are complex numbers.



_a

n



x



_a

1

x

csch

^-1

x dx

ⁿ

= n !

x

ⁿ

csch

^-1

x + Σ

r=0 (n -1 /2)

( 2r !!( ) 2r+1 ! ) ( -1 )

^r

( 2r-1 !! )

( n -2r-1 ! ) x

^n-2r-1

sinh

^-1

x

+ Σ

r=1 n /2

Σ

s=0 n -2r

( -1 )

^r+s

2r+s

n -2r

Ｃ

s

( s+2r-1 !! )

²

( s-1 !! )

²

( n -2r ! ) x

^n-2r

x

²

+1

Where,

a

₁

= 0 , a

₂

= 0.6079 , a

₃

= 0 , a

₄

= 1.5539 , 

(6) Termwise Higher Integral and Taylor series of higher primitive function

Theorem 4.6.1

Let

f

^{< >r}

r =0, 1,  , n

be continuous functions defined on

[a , b ]

and

f

^{<r +1>}be the arbitrary primitive function of

f

^{< >r} . At this time, if

f (x )

can be expanded to the Taylor series around

a

, the following expressions hold for

x [ a , b ]

.

f

^{< >n}

( ) x = Σ

r=0 n -1

f

^{<n -r>}

( ) a r!

( x-a )

^r

+ Σ

r=0



f

^{( )r}

( ) a

( n +r ! ) ( x-a )

^{n+ r}

= Σ

r=0



f

^{<n -r>}

( ) a r!

( x-a )

^r

This expression shows that the Taylor series of

f

^{< >n} consists of the constant-of-integration polynomial and the termwise higher integral of

f (x )

. The following can be said from this.

(1)

A termwise higher integral with a fixed lower limit is collateral generally.

(2)

It is the following case that a termwise higher integral with a fixed lower limit is lineal.

f

^{< >r}

( ) a =0 for r =1, 2,  ,n & f

^{( )s}

( ) a  0,  for at least one s  0

For example,



_a

^

^x



_a^x

^e

^x

^dx

ⁿ

^{= Σ}

_r=0^

^e

^a

( n +r ! ) ( x-a )

^{n+ r}

a -

is collateral higher integral.



₀

^

^x



₀^x

^tan

^-1

^{x dx}

ⁿ

^{= Σ}

_r=0^

^{( -1 )}

^r

_{( n +2r+1 !} ^{( 2r ! )} ₎ ^x

^{n+2r+ 1} is lineal higher integral.

Next, the following is obtained from the Taylor series of the higher integral of

log x

.

Σ

r=1



r( r+1 ( ) r+n ) 1 =

n ! ( -1 )

^n-1

Σ

r=0 n -1

( -1 )

^rn

Ｃ

r

H

_n-r



^{Hn =}

^Σ

k =1



n

k 1

Σ

r=1



r( r+1 ( ) r+n ) ( -1 )

^r-1

= n !

2

ⁿ

 log 2 -H

_n

 + n !

1 Σ

r=0

n -1n

Ｃ

r

H

_n-r

Σ

r=0 n -1

( -1 )

^rn

Ｃ

r

H

_n-r

=

n

( -1 )

^n-1



₀¹

^{( 1- x} _{1+ x} ⁾

^p

^{dx = 2}

^p

^{log 2-  ( 1+p - ) }

^

^{+ Σ}

_r=1^

  ^p _r

^

^{ ( 1+r + ) }

^

Example

1234

1 +

2345

1 +

3456

1 +

4567

1 +  = 18

1

1234

1 -

2345

1 +

3456

1 -

4567

1 +- = 3

4 log 2 - 9 8

05 Termwise Higher Integral (Trigonometric, Hyperbolic)

In this chapter, for the function which second or more order integral cannot be expressed with the elementary functions among trigonometric functions and hyperbolic functions, we integrate the series expansion of these function termwise and obtain the following expressions. Where,

,

denote the ceiling function and the floor function respectively. And Bernoulli Numbers and Euler Numbers are as follows.

B

₀

=1, B

₂

= 6

1 , B

₄

=- 30

1 , B

₆

= 42

1 , B

₈

=- 30

1 , B

₁₀

= 66

5 ,  E

₀

=1, E

₂

=-1, E

₄

=5, E

₆

=-61, E

₈

=1385, E

10

=-50521, 

(1) Taylor Series



₀^x

^ 

₀^x

^{tan x dx}

ⁿ

^{= Σ}

_{k =1}^

² _{2k (}

^2k

^{ 2} _{2k+n -1 !}

^2k

^{-1   B} ₎

^2k

^{ x}

^{2k + n-1}

^{| | x <} 2





₀^x

^ 

₀^x

^{tanh x dx}

ⁿ

^{= Σ}

_{k =1}^

_{2k (} 2k+n -1 ! ) 2

^2k

 2

^2k

-1 B 

_2k

x

^{2k + n-1}

| | x <

2





₀^x

^ 

₀^x

^{secx dx}

ⁿ

^{= Σ}

_{k =0}^

_{( 2k+n !} ^{ E}

^2k

^ ₎ ^x

^{2k + n}

^{| | x <} 2



(2) Fourier Series

When

 ^{( )} x = Σ

k =1



k

^x

( -1 )

^{k -1}

,

 ^{( )} n = Σ

k =0



( 2k+1 )

ⁿ

( -1 )

^k

,



₀^x

^ 

₀^x

^{tan x dx}

ⁿ

⁼ ₂

^n-1

^{1 Σ}

_{k =1}^

^{( -1} _k ⁾

^{nk -1}

^sin _ ^{2k x- n} ₂ ^ _

+ Σ

k =1 n/2

( -1 )

^{k -1}

2

^{2k -2}

 ⁽ 2k-1 ⁾

( n +1-2k ! ) x

^n+1-2k

| | x <

2





₀^x

^ 

₀^x

^{tanh x dx}

ⁿ

^{= ( -1} ₂

^n-1

⁾

^n-1

^Σ

_{k =1}^

^{( -1 )}

^{k -1}

^e _k

^{-2k xn}

⁺ _{n !} ^x

ⁿ

- Σ

k =0 n -1

( -1 )

^k

2

^k

 ⁽ k+1 ⁾

( n -1-k ! ) x

^{n-1- k}

x>0



₀^x

^ 

₀^x

^{sec x dx}

ⁿ

^{= 2 Σ}

_{k =0}^

_{( 2k+1} ^{( -1 )} ₎

^kn

^cos _ ^{( 2k+1 x- ) n} ₂ ^ _

+ Σ

k =1 n /2

( -1 )

^{k -1}

( n -2k ! ) 2  ⁽ 2k ⁾

x

^n-2k

| | x <

2





_^x

^ 

_^x

^{sech x dx}

ⁿ

^{= ( -1 )}

ⁿ

^{2 Σ}

_{k =0}^

^{( -1 )}

^k

₍ ^e _2k+1

^{-(2k +1 x}

₎

⁾ⁿ

^x>0

(3) Riemann Odd Zeta & Dirichlet Odd Eta

Comparing Taylor series and Fourier series, we obtain Riemann Odd Zeta and Dirichlet Odd Eta. For example,

 ( 2n +1 = ) ^{( -1} _ ⁾

n

2

²ⁿ

-1 2

²ⁿ

 ^Σ

k =1





2k ( 2k +2n +1 ! )

 2

^2k

-1 

_

B

_2k



^2k+2n+1

-

( 2n +1 ! )



²ⁿ⁺¹

log 2

-  ( -1 )

ⁿ

2

²ⁿ

-1 2

²ⁿ

Σ

k =1 n -1

( -1 )

^k

( 2n +1-2k ! )



^2n+1-2k

2

^2k

2

^2k

-1

 ( 2k +1 )

 ⁽ 2n +1 = ( ⁾ -1 )

ⁿ



1  ( 2n ! ) 



²ⁿ⁺¹

 ^{log  - Σ}

k =1



2n +1

k

1 - Σ

k =1



2k ( 2k +2n +1 ! )



B

_2k



^2k+2n+1

+ ( -1 )

ⁿ

 1 Σ

k =1 n -1

( -1 )

^k-1

( 2n -2k +1 ! )



^2n-2k+1

 ( 2k +1 )

 ( ) n =

2

^n-1

-1 2

^n-1

 ^Σ

^{k =1}

^{( -1 )}

^{k -1}

k

ⁿ



e

^-k

+ Σ

j=1 n -2

( -1 )

^j-1

2

^{n-1- j}

2

^{n-1- j}

-1

j!

1  ( n -j )

-

2

^n-1

-1

( -2 )

^n-1

 ^Σ

k =1





2k ( 2k+n -1 ! )

 2

^2k

-1 B 

_2k

- 2 1

n ! 1 +

( n -1 ! ) log 2

 ⁽ 2n = ⁾ 2 ( -1 )

^n-1

Σ

k =0



( 2k +2n ! )



E

_2k

  2



^2k+2n

+ Σ

k =1 n

( -1 )

^k-1

( 2k ! )

 ⁽ 2n -2k ⁾

  2



^2k

06 Termwise Higher Integral (Inv-Trigonometric, Inv-Hyperbolic)

Here, we integrate the series of an inverse trigonometric function or an inverse hyperbolic function term by term.

Then, we obtain formulas simpler than what were obtained in "04 HIgher Integral". Moreover, both are compared and we obtain various by-products.

(1) Taylor Series



₀

^

^x



₀^x

^tan

^-1

^{x dx}

ⁿ

^{= Σ}

_{k =0}^

^{( -1 )}

^k

( 2k+n +1 ! ) ( 2k ! )

x

^{2k + n+1}

| | x < 1



₀

^

^x



₀^x

^cot

^-1

^{x dx}

ⁿ

⁼ 2

 n !

x

ⁿ

- Σ

k =0



( -1 )

^k

( 2k+n +1 ! ) ( 2k ! )

x

^{2k + n+1}

0< x 1



₀

^

^x



₀^x

^sin

^-1

^{x dx}

ⁿ

^{= Σ}

_{k =0}^

( 2k+n +1 ! )



( 2k-1 !! )

²

x

^{2k + n+1}

| | x < 1

^collateral



₀

^

^x



₀^x

^cos

^-1

^{x dx}

ⁿ

⁼ 2

 n !

x

ⁿ

- Σ

k =0



( 2k+n +1 ! )



( 2k-1 !! )

²

x

^{2k + n+1}

| | x < 1

collateral



₀

^

^x



₀^x

^tanh

^-1

^{x dx}

ⁿ

^{= Σ}

_{k =0}^

( 2k+n +1 ! ) ( 2k ! )

x

^{2k + n+1}

| | x < 1



₀

^

^x



₀^x

^sinh

^-1

^{x dx}

ⁿ

^{= Σ}

_{k =0}^

^{( -1 )}

^k

( 2k+n +1 ! )



( 2k-1 !! )

²

x

^{2k + n+1}

| | x < 1

collateral



₀

^

^x



₀^x

^sech

^-1

^{x dx}

ⁿ

⁼ _{n !} ^x

ⁿ

 ^log x 

2 + Σ

j=1 n

j

1 - Σ

k =1



2k ( 2k+n ! )



( 2k-1 !! )

²

x

^{2k +n}

0< x<1

collateral



₀

^

^x



₀^x

^csch

^-1

^{x dx}

ⁿ

⁼ _{n !} ^x

ⁿ

 ^log x 

2 + Σ

j=1 n

j

1 - Σ

k =1



( -1 )

^k

2k ( 2k+n ! )



( 2k-1 !! )

²

x

^{2k +n}

0< x<1

^collateral

(2) By-products



₀¹

⁽ _{1- x} ^{1- x )}

²ⁿ

^{dx = 2}

^n-1

^{log 2 - Σ}

^{n /2}_r=1ⁿ

^Ｃ

^{n+1-2r }

^{ ( 1+n - )  ( 2r )}

^



₀¹

⁽ _{1+ x} ^{1- x )}

²ⁿ

^{dx =  4 Σ}

^{n /2}_{k =0}

^{( -1 )}

^kn

^Ｃ

^n-2k

⁺ 2 log 2

Σ

k =1 n /2

( -1 )

^kn

Ｃ

n+1-2k

- Σ

r=1 n /2

(-1)

^r n

Ｃ

n+1-2r 

 ( 1+n - )  ( 2r )



Σ

k =0



( 2k+n +1 ! ) ( 2k ! )

= n!

2

^n-1

log 2 - n!

1 Σ

r=1 n /2

n

Ｃ

n+1-2r 

 ⁽ 1+n - ⁾  ⁽ 2r ⁾



Σ

k =0



( -1 )

^k

( 2k+n +1 ! ) ( 2k ! )

= 4 n!

 Σ

k =0 n /2

( -1 )

^kn

Ｃ

n-2k

+ 2 n!

log 2

Σ

k =1 n /2

( -1 )

^k n

Ｃ

n+1-2k

- n!

1 Σ

r=1 n /2

(-1)

^r n

Ｃ

n+1-2r 

 ( 1+n - )  ( 2r )



Σ

k =0



( 2k+n +1 ! )



( 2k-1 !! )

²

= ( 2n !! )

2n -1

Ｃ

n-1

 - Σ

k =1 n /2

( 2k-1 !! )

²

( n -2k+1 ! ) 1

Σ

k =1



2k ( 2k+n ! )



( 2k-1 !! )

²

= n !

1  ^{log 2+ Σ}

j=1



n

j

1 + Σ

r=1 n /2

2r( 2r-1 !! )

²

1

( n -2r ! ) 1

- 2

 Σ

r=0 (n -1 /2)

( 2r !!( ) 2r+1 ! ) ( 2r-1 !! )

( n -2r-1 ! ) 1

Example

1234

1 +

3456

1 +

5678

1 +

78910

1 +  = 3

2 log 2 - 12

5

1234

1 -

3456

1 +

5678

1 -

78910

1 +-  = 12

5 - 12

 - 6 log 2

2!

( -1 !! )

²

+ 4!

1!!

²

+ 6!

3!!

²

+ 8!

5!!

²

+  = 2

 - 1

23!

1!!

²

+ 45!

3!!

²

+ 67!

5!!

²

+ 89!

7!!

²

+  = 1!

1 ( log 2+1 - ) 2



07 Super Integral (Non-integer order Integral)

It is

Σ

j=0 p

a = ap

which extended the domain of index

j

of

Σ

j=1 m

a

to the real number interval

[0 , p ]

from the natural

number interval

[1 , m ]

. And it is

Π

k =0 q

b = b

^q which extended the domain of index

k

of

Π

k =1 n

b

to the real number interval

[0 , q ]

from the natural number interval

[1 , n ]

. It is called analytic continuation to extend the domain generaly. Although usually analytic continuation is used for extending the domain of a function, it can be used also for extending the domain of the index of a operator. Here, extending the domain of the index of the integration operator, we obtain Super Integral (Non-integer order Integral ).

(1) Definitions and Notations

f

^{< >p}

( ) x

denotes the non-integer order primitive function of

f (x )

. And we call this Super Primitive Function of

f(x)

. Since there is a super primitive function innumerably, which

f

^{< >p}

( ) x

means follows the definition at that time.

We call it Super Integral to integrate a function

f

with respect to an independent variable

x

from

a (0)

^to

a (p )

continuously. And it is described as follows.



_{a( )}^x_p

^ 

_{a( )}^x₀

^{f( ) x dx}

^p

 ⁼ 

_{a( )}^x_p

^ 

_{a( )}^x₀

^{f( ) x dx  dx} 

And

when

a( ) k = a for all k [ 0 , p ]

, we call it super integral with a fixed lower limit , when

a( ) k  a for some k [ 0 , p ]

, we call it super integral with variable lower limits .

(2) Fundamental Theorem of Super Integral

Let

f

^{< >r}

r [ 0 , p ]

be an continuous function on the closed interval

I

and be arbitrary the

r

-th order primitive function of

f

. And let

a (r )

be a continuous function on the closed interval

[0 , p ]

.

Then the following expression holds for

a( ) r , x  I

.



_{a( )}^x_p

^ 

_{a( )}^x₀

^{f( ) x dx}

^p

^{= f}

^{< >p}

^{( ) x - Σ}

^{p -1}_r=0

^f

^{<p -r>}

^{a( p -r )}

^



_{a( )}^x_p

^ 

_a(^x_{p -r+1)}

^dx

^r

Especially, when

a( ) r = a for all k [ 0 , p ]

,



_a

^

^x



_a^x

^{f( ) x dx}

ⁿ

^{= f}

^{< >p}

^{( ) x - Σ}

^{p -1}_r=0

^f

^{<p -r>}

^{( ) a} _ ⁽ ₍ ^x-a _{1+ r} ⁾ ₎

^r

(3) Lineal and Collateral

We call Constant-of-integration Function the 2nd term of the right sides of these.

And when Constant-of-integration Function is 0, we call the left side Lineal Super Integral . and we call

f

^{< >p}

( ) x

Lineal Super Primitive Function.

Oppositely, when Constant-of-integration Function is not 0, we call the left side Collateral Super Integral and we call the right side Collateral Super Primitive Function.

For example,



2 p

x

 

2 0

x

sin xdx

^p

= sin  ^x- 2 

p 

Left: Lineal

p

th order integral Right: Lineal

p

th order primitive



₀^x

^ 

₀^x

^{sin xdx}

^p

^{= Σ}

_r=0^

_{ ( 2r+2+p} ^{( -1 )}

^r

₎ ^x

^{2r+1+ p} Left: Collateral

p

th order integral Right: Collateral

p

th order primitive

(4) Super Integral and Reimann-Liouville Integral

The super integral with a fixed lower limit reduce to the 1st order integral which called Reimann-Liouville Integral.



_a^x

^ 

_a^x

^{f( ) x dx}

^p

⁼ _{ ( )} ¹ _p 

_a^x

^{( x-t )}

^p-1

^{f( ) t dt}

This is what extended the parameter

n

to the real number in Cauchy formula for repeated integration. Since the left side has lost the operating function, Reimann-Liouville Integral of the right side is very important. All the super integral with a fixed lower limit can be verified numerically by this. On the other hand, since the super integral with variable lower limits cannot apply Reimann-Liouville Integral, the verification is vary difficult.

(5) Fractional Integral & Super Integral

In traditional Fractional Integral , the super primitive function is drawn from Riemann-Liouville Integral.

For example, in the case of

f( ) x = x

^, it is as follws.