indefinite and definite sums

indefinite and definite sums

An

indefinite sum

, like an indefinite integral, is an operator which acts on a

function

. In other words, it transforms a given function to another

via

a certain law. This article presents the so called

Caves summation formula

. The advantages of the formula in comparison with other summation methods are that it gives the indefinite sum for any analytical function, and that it also completely reduces

summation

to integration. One can do with the Caves summation formula everything that one can do with an

integral

. For example, one can take a sum along a path either in the

complex plane

or along a

contour

with a singular point inside the contour, and so on.

∑

k

x

-

1

φ

⁢

(

k

+

z

)

=

∫

0

x

∑

ν

=

0

∞

B

ν

-

A

ν

ν

!

⁢

φ

(

ν

)

⁢

(

ξ

+

z

)

⁢

d

⁢

ξ

-

∫

0

x

A

N

′

⁢

(

z

-

ξ

)

N

!

⁢

φ

(

N

-

1

)

⁢

(

z

+

ξ

)

⁢

𝑑

ξ

-

-

∫

0

x

∑

m

=

0

∞

φ

(

N

+

m

)

⁢

(

z

+

ξ

)

⁢

A

N

+

m

⁢

(

x

-

ξ

)

⁢

k

N

+

m

+

1

k

N

+

1

+

m

d

⁢

ξ

+

H

⁢

(

x

,

z

)

=

F

⁢

(

x

,

z

,

N

)

-

f

⁢

(

x

,

z

,

N

)

-

f

⁢

ε

⁢

(

x

,

z

,

N

)

+

H

⁢

(

x

,

z

,

N

)

F

⁢

(

x

,

z

,

N

)

=

F

N

⁢

(

x

,

z

,

N

)

+

F

N

⁢

ε

⁢

(

x

,

z

,

N

)

.

I choose that

⁢

|

B

ν

-

A

ν

|

≤

(

r

⁢

(

ν

)

)

ν

,

where

⁢

B

ν

⁢

are Bernoulli numbers,

|

F

N

⁢

ε

⁢

(

x

,

z

,

N

)

|

=

|

∫

0

x

∑

ν

=

N

∞

B

ν

-

A

ν

ν

!

⁢

φ

(

ν

)

⁢

(

z

+

z

1

)

|

⩽

|

x

|

⁢

(

r

⁢

(

N

)

)

N

N

!

⁢

sup

z

+

z

1

∈

G

⁡

|

φ

(

n

)

⁢

(

z

+

z

1

)

|

|

f

ε

(

x

,

z

,

N

)

|

=

|

∫

0

x

∑

m

=

0

∞

φ

(

N

+

m

)

(

z

+

ξ

)

A

N

+

m

(

x

+

α

-

ξ

)

k

N

+

m

+

1

k

N

+

1

+

m

|

⩽

|

x

|

⁢

(

r

⁢

(

N

)

)

N

N

!

sup

ζ

∈

G

|

φ

(

N

)

(

ζ

)

|

where

G

is the region of summation. In case of summation in complex plain

r

⁢

(

ν

)

must

be a

positive

constant,

r

⁢

(

ν

)

=

r

z

=

max

⁡

(

r

,

e

ln

⁡

r

+

1

e

⁢

r

)

where

r

is a positive value less or equal to the minimal

radius of convergence

of Tailor series of the function

φ

⁢

(

z

)

on the

intersection

of the area of summation

G

with the

x

-axis. In case of summation exclusively on a

segment

of the

x

-axis it is more convenient to choose

r

⁢

(

ν

)

=

1

ln

⁡

ν

or

r

⁢

(

ν

)

=

1

ln

⁡

(

ln

⁡

ν

)

, especially in a case when there is a singular point on the path of summation.The same for a path parallel to the

x

-axis when

φ

⁢

(

z

)

is regarded as a function of real valued argument. The more close

r

z

is to zero the more close the possible area of summation is to the hole area where

φ

⁢

(

z

)

is analytical.

A

ν

=

0

,

ν

=

0

,

1

,

2

,

…

,

N

-

1

,

(

N

⩾

2

)

,

A

2

⁢

ν

=

0

,

ν

=

0

,

1

,

2

,

…

Periodical function with the period

1

H

⁢

(

α

,

z

)

=

=

∫

x

=

0

∫

0

α

(

A

N

′′

⁢

(

ξ

+

x

)

N

!

φ

(

N

-

1

)

(

z

-

x

)

+

∑

m

=

0

∞

φ

(

N

+

m

)

(

z

-

x

)

A

N

+

m

′

(

ξ

+

x

)

k

N

+

m

+

1

k

N

+

1

+

m

)

d

ξ

d

x

=

=

h

N

⁢

(

α

,

z

)

+

h

⁢

ε

N

⁢

(

α

,

z

)

,

lim

N

→

∞

|

h

ε

N

(

α

,

z

)

|

=

lim

N

→

∞

|

∫

x

=

0

∫

0

α

(

∑

m

=

0

∞

φ

(

N

+

m

)

(

z

-

x

)

A

N

+

m

′

(

ξ

+

x

)

k

N

+

m

+

1

k

N

+

1

+

m

)

d

ξ

d

x

|

⩽

⩽

lim

N

→

∞

⁡

|

D

⁢

α

|

⁢

r

N

+

1

(

N

+

1

)

!

⁢

sup

ζ

∈

G

⁡

|

φ

(

N

+

1

)

⁢

(

ζ

)

|

=

0

,

where

D

is the diameter of the area of summation and

z

is a

parameter

.

A

N

⁢

(

α

)

=

{

2

⁢

(

-

1

)

⌊

N

2

⌋

+

1

⁢

N

!

⁢

∑

k

=

1

k

N

cos

⁡

2

⁢

π

⁢

k

⁢

α

(

2

⁢

π

⁢

k

)

N

-

1

,

when

N

even

2

⁢

(

-

1

)

⌊

N

2

⌋

+

1

⁢

N

!

⁢

∑

k

=

1

k

N

sin

⁡

2

⁢

π

⁢

k

⁢

α

(

2

⁢

π

⁢

k

)

N

-

1

,

when

N

odd

,

A

N

⁢

(

0

)

=

A

N

, and

A

N

+

m

⁢

(

x

)

⁢

k

N

+

m

+

1

k

N

+

1

+

m

=

{

2

⁢

(

-

1

)

⌊

N

+

m

2

⌋

+

1

⁢

∑

k

=

k

N

+

m

+

1

k

N

+

1

+

m

cos

⁡

2

⁢

π

⁢

k

⁢

x

(

2

⁢

π

⁢

k

)

N

+

m

,

when

N

+

m

even

2

⁢

(

-

1

)

⌊

N

+

m

2

⌋

+

1

⁢

∑

k

=

k

N

+

m

+

1

k

N

+

1

+

m

sin

⁡

2

⁢

π

⁢

k

⁢

x

(

2

⁢

π

⁢

k

)

N

+

m

,

when

N

+

m

odd

The

floor

of

x

(

x

is real)

⌊

x

⌋

is the largest integer less then

x

.

From the condition

|

B

ν

(

x

)

-

A

ν

(

x

)

|

⩽

(

r

(

ν

)

)

ν

=

r

ν

,

(

0

⩽

x

⩽

1

)

(

B

ν

(

x

)

are

Bernoulli polynomials

) I find out that

k

ν

=

⌊

ν

2

⁢

π

⁢

r

⁢

e

⌋

⁢

(

1

-

δ

ν

,

1

)

,

ν

=

1

,

2

,

…

where

⁢

δ

ν

,

1

⁢

is the Kronecker delta,

⁢

δ

ν

,

1

=

1

⁢

when

⁢

ν

=

1

⁢

and

⁢

0

⁢

otherwise.

The

definite sum

is defined as:

∑

k

=

a

x

-

1

φ

⁢

(

k

+

z

)

=

∑

k

x

-

1

φ

⁢

(

k

+

z

)

-

∑

k

a

-

1

φ

⁢

(

k

+

z

)

In the case of integer summation boundaries the summation formula can be simplified.

∑

k

=

n

1

n

2

-

1

=

∫

n

1

n

2

(

∑

ν

=

0

∞

B

ν

-

A

ν

ν

!

⁢

φ

(

ν

-

1

)

⁢

(

ξ

+

z

)

-

A

N

′

⁢

(

z

-

ξ

)

N

!

⁢

φ

(

N

-

1

)

⁢

(

ξ

+

z

)

)

⁢

𝑑

ξ

+

ε

N

,

where

|

ε

N

|

⩽

|

∫

n

1

n

2

(

∑

m

=

0

∞

φ

(

N

+

m

)

⁢

(

z

+

ξ

)

⁢

A

N

+

m

⁢

(

-

ξ

)

⁢

k

N

+

m

+

1

k

N

+

1

+

m

d

)

⁢

ξ

|

⩽

|

n

2

-

n

1

|

⁢

(

r

⁢

(

N

)

)

N

N

!

⁢

sup

n

1

⩽

ζ

⩽

n

2

⁡

|

φ

(

N

)

⁢

(

ζ

)

|

.

r

⁢

(

ν

)

=

r

,

r

⁢

(

ν

)

=

1

l

⁢

n

⁢

ν

⁢

or

⁢

r

⁢

(

ν

)

=

1

l

⁢

n

⁢

(

l

⁢

n

⁢

ν

)

.

Notes:

1.

Complete

details are provided through the link to the following

http://www.oddmaths.info/indefinitesum

web site: http://www.oddmaths.info/indefinitesum.

2. The complete pdf of the entire article can be downloaded here from the

http://planetmath.org/files/papers/554/Summation.pdf

complete article on “Summation” uploaded to the Papers section

.

Title

indefinite and definite sums

Canonical

name

IndefiniteAndDefiniteSums

Date of creation

2013-03-22 19:22:22

Last modified on

2013-03-22 19:22:22

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

80

Author

bci1 (20947)

Entry type

Topic

Classification

msc 34A36

Classification

msc 39B72

Classification

msc 33E30

Classification

msc 39A99

Related topic

IndefiniteSum

Defines

non-analytical function

Defines

definite sum

Defines

Caves summation formula

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