∑
k
x
−
1
φ
(
k
+
z
)
=
∑
ν
=
0
∞
B
ν
−
A
ν
ν
!
φ
(
ν
−
1
)
(
x
+
z
)
−
{\displaystyle \sum _{k}^{x-1}\varphi (k+z)=\sum _{\nu =0}^{\infty }{\frac {B_{\nu }-A_{\nu }}{\nu !}}\varphi ^{(\nu -1)}(x+z)-}
−
∫
0
x
A
N
′
(
x
−
ξ
)
N
!
φ
(
N
−
1
)
(
z
+
ξ
)
d
ξ
−
∫
0
x
∑
m
=
0
∞
φ
(
N
+
m
)
(
z
+
ξ
)
k
N
+
1
+
m
A
N
+
m
(
x
−
ξ
)
k
N
+
m
+
1
d
ξ
+
H
(
x
,
z
)
=
{\displaystyle -\int \limits _{0}^{x}{\frac {A_{N}'(x-\xi )}{N!}}\varphi ^{(N-1)}(z+\xi )d\xi -\int \limits _{0}^{x}\sum _{m=0}^{\infty }\varphi ^{(N+m)}(z+\xi ){\begin{array}{l}k_{N+1+m}\\A_{N+m}(x-\xi )\\k_{N+m}+1\end{array}}d\xi +H(x,z)=}
=
F
(
x
,
z
,
N
)
−
f
(
x
,
z
,
N
)
−
f
ε
(
x
,
z
,
N
)
,
lim
N
→
∞
f
ε
(
x
,
z
,
N
)
=
0
{\displaystyle =F(x,z,N)-f(x,z,N)-f\varepsilon (x,z,N),\,\lim _{N\to \infty }f\varepsilon (x,z,N)=0\,}
where
A
ν
=
0
,
ν
=
0
,
1
,
2
,
.
.
.
N
−
1
{\displaystyle A_{\nu }=0,\nu =0,1,2,...N-1\,}
and periodical function with the period one
H
(
x
,
z
)
=
{\displaystyle \,\,\,\ \ H(x,z)=}
=
∫
ζ
=
0
∫
0
x
(
A
N
″
(
ξ
+
ζ
)
N
!
φ
(
N
−
1
)
(
z
−
ζ
)
∑
m
=
0
∞
φ
(
N
+
m
)
(
z
−
ζ
)
k
N
+
1
+
m
A
N
+
m
′
(
ξ
+
ζ
)
k
N
+
m
+
1
)
d
ξ
d
ζ
=
{\displaystyle =\int \limits ^{\zeta =0}\int \limits _{0}^{x}\left({\frac {A''_{N}(\xi +\zeta )}{N!}}\varphi ^{(N-1)}(z-\zeta )\sum _{m=0}^{\infty }\varphi ^{(N+m)}(z-\zeta ){\begin{array}{l}k_{N+1+m}\\A_{N+m}'(\xi +\zeta )\\k_{N+m}+1\end{array}}\right)d\xi d\zeta =}
=
h
N
(
x
,
z
)
+
h
ε
N
(
x
,
z
)
,
and
lim
N
→
∞
h
ε
N
(
x
,
z
)
=
0
,
{\displaystyle =h_{N}(x,z)+h\varepsilon _{N}(x,z),{\text{ and }}\lim _{N\to \infty }h\varepsilon _{N}(x,z)=0,}
where
z
{\displaystyle z\,}
is a parameter,
B
ν
{\displaystyle B_{\nu }\,}
are Bernoulli numbers and
A
N
′
(
α
)
=
{
2
(
−
1
)
⌊
N
2
⌋
+
1
N
!
∑
k
=
1
k
N
−
sin
2
π
k
α
(
2
π
k
)
N
−
1
,
when N is even
.
2
(
−
1
)
⌊
N
2
⌋
+
1
N
!
∑
k
=
1
k
N
cos
2
π
k
α
(
2
π
k
)
N
−
1
,
when N is odd
{\displaystyle A_{N}'(\alpha )={\begin{cases}\displaystyle {2(-1)^{\lfloor {\frac {N}{2}}\rfloor +1}N!\sum _{k=1}^{k_{N}}{\frac {-\sin 2\pi k\alpha }{(2\pi k)^{N-1}}}},&{\text{when N is even}}\\.\\\displaystyle {2(-1)^{\lfloor {\frac {N}{2}}\rfloor +1}N!\sum _{k=1}^{k_{N}}{\frac {\cos 2\pi k\alpha }{(2\pi k)^{N-1}}}},&{\text{when N is odd}}\end{cases}}}
A
ν
(
0
)
=
A
ν
and
{\displaystyle A_{\nu }(0)=A_{\nu }\ \ {\text{and}}}
k
N
+
1
+
m
A
N
+
m
(
x
)
k
N
+
m
+
1
=
{
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
{\displaystyle \left.{\begin{array}{l}k_{N+1+m}\\A_{N+m}(x)\\k_{N+m}+1\end{array}}\right.={\begin{cases}\displaystyle {2(-1)^{\lfloor {\frac {N+m}{2}}\rfloor +1}}\sum _{k=k_{N+m}+1}^{k_{N+1+m}}{\frac {\cos 2\pi kx}{(2\pi k)^{N+m}}},&{\text{when N+m even}}\\.\\\displaystyle {2(-1)^{\lfloor {\frac {N+m}{2}}\rfloor +1}}\sum _{k=k_{N+m}+1}^{k_{N+1+m}}{\frac {\sin 2\pi kx}{(2\pi k)^{N+m}}},&{\text{when N+m odd}}\end{cases}}}
The Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle {\emph {floor}}}
of
x
{\displaystyle x}
(
x
{\displaystyle x}
is real)
⌊
x
⌋
{\displaystyle \lfloor x\rfloor }
is the largest integer less then
x
{\displaystyle x}
.
The boundaries of summation
k
ν
{\displaystyle k_{\nu }}
are determined for example from the folloving condition
|
B
ν
(
x
)
−
A
ν
(
x
)
|
≤
1
ν
!
ν
ν
,
0
≤
x
≤
1
{\displaystyle |B_{\nu }(x)-A_{\nu }(x)|\leq {\frac {1}{\nu !\nu ^{\nu }}},\ 0\leq x\leq 1\ \,\ \ }
or
|
B
ν
(
x
)
−
A
ν
(
x
)
|
=
r
{\displaystyle \quad \,\ |B_{\nu }(x)-A_{\nu }(x)|=r}
where
r
{\displaystyle r}
is a constant.
k
ν
{\displaystyle k_{\nu }\ }
are chosen the least that satisfy the inequality.
Definite sum is defined as:
∑
k
=
a
x
−
1
φ
(
k
+
z
)
=
∑
k
x
−
1
φ
(
k
+
z
)
−
∑
k
a
φ
(
k
+
z
)
{\displaystyle \sum _{k=a}^{x-1}\varphi (k+z)=\sum _{k}^{x-1}\varphi (k+z)-\sum _{k}^{a}\varphi (k+z)}
More details one can see at www.oddmaths.info/indefinitesum .
Summation of non-analytical functions
edit
Let there be a set
S
{\displaystyle S}
of functions
φ
(
x
)
{\displaystyle \varphi (x)}
such that a
φ
(
x
)
∈
S
{\displaystyle \varphi (x)\in S}
streams to zero when
|
x
|
{\displaystyle |x|}
streams to infinity faster then any power of the inverse of
x
{\displaystyle x}
, i.e.
lim
|
x
|
→
∞
φ
(
x
)
x
n
=
0
{\displaystyle \displaystyle {\lim _{|x|\to \infty }\varphi (x)x^{n}}=0}
for any
n
=
1
,
2
,
…
{\displaystyle n=1,2,\dots }
. The set
S
{\displaystyle S}
is the space of basic functions. Let there on the space of basic functions be defined a functional
(
f
,
φ
)
=
∑
k
=
−
∞
∞
f
(
k
)
φ
(
k
)
{\displaystyle (f,\varphi )=\sum _{k=-\infty }^{\infty }f(k)\varphi (k)}
The functional of finite difference of a function
f
(
x
)
{\displaystyle f(x)}
is defined as follows:
(
∇
f
,
φ
)
=
−
(
f
,
Δ
φ
)
{\displaystyle (\nabla f,\varphi )=-(f,\Delta \varphi )}
where
∇
f
(
x
)
=
Δ
f
(
x
−
1
)
.
{\displaystyle \nabla f(x)=\Delta f(x-1).}
Definition of the functional of the sum of a function
f
(
x
)
{\displaystyle f(x)}
.
A function
Ψ
(
x
)
=
Δ
φ
(
x
)
{\displaystyle \Psi (x)=\Delta \varphi (x)\,}
belongs to the space of basic functions
S
{\displaystyle S}
. First I define the functional of sum on the functions
Ψ
(
x
)
∈
S
1
⊂
S
{\displaystyle \Psi (x)\in S_{1}\subset S}
. From the previous result
(
f
(
x
)
,
Δ
φ
(
x
)
)
=
−
(
Δ
f
(
x
−
1
)
,
φ
(
x
)
)
{\displaystyle (f(x),\Delta \varphi (x))=-(\Delta f(x-1),\varphi (x))\,}
therefore
(
∑
k
x
f
(
k
)
,
Ψ
)
=
−
(
f
(
x
)
,
{\displaystyle \left(\sum _{k}^{x}f(k),\Psi \right)=-{\big (}f(x),}
∑
{\displaystyle \sum }
Ψ
)
{\displaystyle \Psi {\big )}}
where
∑
Ψ
{\displaystyle \sum \Psi }
is an indefinite sum of
Ψ
(
x
)
{\displaystyle \Psi (x)\,}
. For the rest functions
φ
(
x
)
∉
S
1
{\displaystyle \varphi (x)\notin S_{1}}
I choose
(
∑
k
x
f
(
k
)
,
φ
)
=
0.
{\displaystyle \left(\sum _{k}^{x}f(k),\varphi \right)=0.}
Therefore the functional is defined on the entire space
S
.
{\displaystyle S.}
Heaviside function of the second type
U
(
x
)
=
Θ
(
x
)
{\displaystyle U(x)=\Theta (x)\,}
and Dirac delta function of the second type
δ
(
x
)
{\displaystyle \delta (x)\,}
(
Θ
,
φ
)
=
∑
k
=
−
∞
∞
Θ
(
k
)
φ
(
k
)
=
∑
k
=
0
∞
φ
(
k
)
,
{\displaystyle (\Theta ,\varphi )=\sum _{k=-\infty }^{\infty }\Theta (k)\varphi (k)=\sum _{k=0}^{\infty }\varphi (k)\,,}
and
(
δ
,
φ
)
=
∑
k
=
−
∞
∞
δ
(
k
)
φ
(
k
)
=
φ
(
0
)
,
{\displaystyle (\delta ,\varphi )=\sum _{k=-\infty }^{\infty }\delta (k)\varphi (k)=\varphi (0),}
or their shifted forms
(
Θ
,
φ
)
=
∑
k
=
−
∞
∞
Θ
(
k
−
x
0
)
φ
(
k
)
=
∑
k
=
x
0
∞
φ
(
k
)
,
{\displaystyle (\Theta ,\varphi )=\sum _{k=-\infty }^{\infty }\Theta (k-x_{0})\varphi (k)=\sum _{k=x_{0}}^{\infty }\varphi (k),}
and
(
δ
,
φ
)
=
∑
k
=
−
∞
∞
δ
(
k
−
x
0
)
φ
(
k
)
=
φ
(
x
0
)
.
{\displaystyle (\delta ,\varphi )=\sum _{k=-\infty }^{\infty }\delta (k-x_{0})\varphi (k)=\varphi (x_{0}).}
Summation with non-number boundaries
edit
Let
A
,
B
,
{\displaystyle A,\ B,\ }
zero matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle \ \,{\emph {0}}\ ,}
and identity matrix
I
{\displaystyle I\ \ }
are {
n
×
n
{\displaystyle n\times n}
} matrices.
U
{\displaystyle U\ \ }
is {
n
×
n
{\displaystyle n\times n}
} orthonormal matrix with
n
{\displaystyle n\,}
orthonormal vectors and
|
U
|
=
1
{\displaystyle |U|=1\,}
. Let
A
=
U
D
a
U
∗
,
B
=
U
D
b
U
∗
{\displaystyle A=UD_{a}U^{*},\ B=UD_{b}U^{*}}
, where
U
∗
{\displaystyle U^{*}\,}
is Hermitian conjugate of matrix
U
{\displaystyle U\,}
and
D
a
=
(
d
a
,
1
0
⋯
0
0
d
a
,
2
⋯
0
.
.
.
.
.
.
.
.
.
.
.
.
0
0
⋯
d
a
,
n
)
,
D
b
=
(
d
b
,
1
0
⋯
0
0
d
b
,
2
⋯
0
.
.
.
.
.
.
.
.
.
.
.
.
0
0
⋯
d
b
,
n
)
,
{\displaystyle D_{a}={\begin{pmatrix}d_{a,1}&0&\cdots &0\\0&d_{a,2}&\cdots &0\\...&...&...&...\\0&0&\cdots &d_{a,n}\end{pmatrix}},\ \ D_{b}={\begin{pmatrix}d_{b,1}&0&\cdots &0\\0&d_{b,2}&\cdots &0\\...&...&...&...\\0&0&\cdots &d_{b,n}\\\end{pmatrix}},}
then by definition
∑
k
=
A
B
φ
(
k
)
=
∑
k
=
U
D
a
U
∗
U
D
b
U
∗
φ
(
k
)
=
U
(
∑
k
=
D
a
D
b
φ
(
k
)
)
U
∗
=
{\displaystyle \sum _{k=A}^{B}\varphi (k)=\sum _{k=UD_{a}U^{*}}^{UD_{b}U^{*}}\varphi (k)=U\left(\sum _{k=D_{a}}^{D_{b}}\varphi (k)\right)U^{*}=}
=
U
(
∑
k
=
d
a
,
1
d
b
,
1
φ
(
k
)
0
0
⋯
0
0
∑
k
=
d
a
,
2
d
b
,
2
φ
(
k
)
0
⋯
0
0
0
∑
k
=
d
a
,
3
d
b
,
3
φ
(
k
)
⋯
0
.
.
.
.
.
.
.
.
.
⋯
.
.
.
0
0
0
⋯
∑
k
=
d
a
,
n
d
b
,
n
φ
(
k
)
)
U
∗
{\displaystyle =U{\begin{pmatrix}\displaystyle {\sum _{k=d_{a,1}}^{d_{b,1}}\varphi (k)}&0&0&\cdots &0\\0&\displaystyle {\sum _{k=d_{a,2}}^{d_{b,2}}\varphi (k)}&0&\cdots &0\\0&0&\displaystyle {\sum _{k=d_{a,3}}^{d_{b,3}}\varphi (k)}&\cdots &0\\...&...&...&\cdots &...\\0&0&0&\cdots &\displaystyle {\sum _{k=d_{a,n}}^{d_{b,n}}\varphi (k)}\end{pmatrix}}U^{*}}
If
A
=
U
a
D
a
U
a
∗
,
B
=
U
b
D
b
U
b
∗
,
{\displaystyle A=U_{a}D_{a}U_{a}^{*},\ B=U_{b}D_{b}U_{b}^{*},}
then
∑
k
=
A
B
φ
(
k
)
=
∑
k
=
A
=
U
a
D
a
U
a
∗
U
a
0
U
a
∗
φ
(
k
)
+
∑
k
=
U
b
I
U
b
∗
U
b
D
b
U
b
∗
φ
(
k
)
{\displaystyle \sum _{k=A}^{B}\varphi (k)=\sum _{k=A=U_{a}D_{a}U_{a}^{*}}^{U_{a}0U_{a}^{*}}\varphi (k)+\sum _{k=U_{b}IU_{b}^{*}}^{U_{b}D_{b}U_{b}^{*}}\varphi (k)}
--Ascoldcaves (talk ) 00:37, 3 November 2011 (UTC)