f
∗
(n)
class="bi x0 y2 w1 h5"
class="bi x0 y3 w2 h6"
class="bi x0 y4 w2 h7"
class="bi x0 y5 w2 h8"
h = 0 y = f(x)
x
0
, x
1
= x
0
+ h, x
2
= x
0
+ 2h, . . . , x
n
= x
0
+ nh, . . . (n ∈ N)
y
0
, y
1
, . . . ., y
n
, . . .
y

n
;
. . .
∆y = f(x + h) − f(x) = ∆f(x)
f(x) x.
∆y
0
= ∆f(x
0
) = y
1
− y
0
; ∆y
1
= ∆f(x
1
) = f (x
1
+ h) − f(x
1
) = y
2
− y
1
; . . .
∆y
n−1
= ∆f(x
n−1

(x + 1) − f(x) = x ∀x.
x = 0, 1, 2 . . . n . . . f(x) = f(0)
f(x) f(x) = f(0) ∀x.
∆f(x) = 0 ∀x
f(x)
g(x) g(x + 1) − g(x) = x ∀x.
g(x) = ax
2
+ bx
a(x + 1)
2
+ b(x + 1) − ax
2
− bx = x ∀x ⇔ 2ax + a + b = x.
a =
1
2
, b =
−1
2
g(x) =
1
2
x
2
−
1
2
x.
(f(x + 1) − g(x + 1)) − (f(x) − g(x)) = 0 ∀x

k
)) = ∆
n−1
f(x
k+1
) − ∆
n−1
f(x
k
).
∆
2
y
k
= ∆
2
f
k
= ∆f(x
k+1
) − ∆f(x
k
) = f (x
k+2
) − 2f(x
k+1
) + f(x
k
)
y = f(x) x

∆y
0
∆y
1
∆y
2
∆y
3
. . .
∆
2
y
k
∆
2
y
0
∆
2
y
1
∆
2
y
3
. . .
. . . . . .
f(n)
f(n)
n 0 1 2 3 4 5 . . .

− 2n
2
+ 3n − 1.
∆C = 0,
∆
k
y
n
=
k

i=0
(−1)
i
C
i
k
y
n+k−i
∆
k
(αf + βg) = α∆
k
f + β∆
k
g.
α, β f, g x
∀k, m ∈ N, m < k
y
k

n

k=1
∆y
k
= y
n+1
− y
1
.
F (n, y
n
, ∆y
n
, . . . , ∆
k
y
n
) = 0
F (n, y
n
, y
n+1
, . . . , y
n+k
)
y(n)
∆
k
y

n
∆
3
y
n
= y
n+3
− C
1
3
y
n+2
+ C
2
3
y
n+1
− y
n
. . .
y
k
= f(x
k
) =
k−m

i=0
C
i

) + ∆
2
f(x
n
) = y
n
+ 2∆y
n
+ ∆
2
y
n
y
n+3
= y
n
+ C
1
3
∆y
n
+ C
2
3
∆
2
y
n
+ ∆
3

f(n) = f(n + 3) − 3f(n + 2) + 3f(n + 1) −f(n)
f(n + 3) + f (n + 2) = 0
b
0
y
n+k
+ b
1
y
n+k−1
+ ··· + b
n
y
n
= F (n)
b
0
, b
1
, . . . , b
n
b
0
, b
k
= 0 y(n)
F (n)
b
0
f(n + k) + b

k−1
C
1
, C
2
, C
3
, . . . ., C
k
˜y(n)
˜y(i) = y
i
, i =
0, k −1
• y
∗
(n)
(a
n
)
F (z)
F (z) = a
1
+
a
1
z
+ . . . +
a
n

F (z)z
n−1
dz (2)
L
F (z)
(a
n
) f(n) a
n
= f(n)
f(n) |f(n)| ≤ Me
αn
f(n) = 0 n < 0.
z = e
p
, p = s + iθ
F (z) = F (e
p
) = F
∗
(p)
F
∗
(p) = f
0
+ f
1
.e
−p
+ f

−π < Imp ≤ π Rep > α.
L
D
f(n) =
1
2πi
γ+πi

γ−πi
F
∗
(p)e
np
dp.
f(n)
F (z) =
1
(z −a)(z −b)
.
a = b F (z) =
1
(z −a)(z −b)
z
1
= a, z
2
= b.
f(n) =
1
2πi

1
(2 − 1)!
lim
z→a
d
dz
[(z −a)
2
z
n−1
(z −a)
2
)]
= (n − 1)a
n−2
.
f(n) = e
−n
.
F
∗
(p) =

n≥0
f(n)e
−np
=

n≥0
e

p
− e
−1
.
f(n), g(n) f
j
(n), g
j
(n)
F
∗
(p), G
∗
(p), F
j
∗
(p), G
j
∗
(p)
n

j=1
α
j
f
j
(n) 
n


)
•
d[F
∗
(p)]
dp
 −n.f(n).
d
k
[F
∗
(p)]
(dp)
k
 (−1)
k
.n
k
.f(n)
f(0) = 0,
f(t)
t
|
t=0
= lim
t→0
+
f(t)
t
= 0

n−1

k=0
f(k) 
F
∗
(p)
e
p
− 1
f(n)
F
∗
(p)
A
e
p
− e
p
0
 Ae
p
0
(n−1)
,
A
(e
p
− e
p

∗
(p).e
(n−1)p
; p
m
].
p
m
[F
∗
(p).e
(n−1)p
; p
m
] = lim
p→p
m
[F
∗
(p)(e
p
− e
p
m
).e
(n−1)p
].
p
m
[F

F
∗
(p) =
G(p)
H(p)
=
c
0
e
mp
+ c
1
e
(m−1)p
+ ··· + c
m
b
0
e
rp
+ b
1
e
(r−1)p
+ . . . ··· + b
r
.
z = e
p
F (z) =


k
G(z
k
)
H

(z
k
)
z
n−1
k
F (z) z
k
m
k
f(n) =

k
1
(m
k
− 1)!
lim
z→z
k
d
m
k

p
=2
e
p
=1
⇔ [
p
2
= ln 2
p
1
= 0
p
1
= 0 p
2
= ln 2
F
∗
(p)
[F
∗
(p).e
(n−1)p
; 0] = lim
p→0
[F
∗
(p)(e
p

n
f(n) = [F
∗
(p).e
(n−1)p
; 0] + [F
∗
(p).e
(n−1)p
; ln 2] = −1 + 2
n
.
F
∗
(p) =
e
p
e
2p
− 1
.
p
1
= 0 p
2
= πi
−π ≤ imp ≤ π
[F
∗
(p).e

(n−1)p
] = lim
p→πi
e
pn
e
p
− 1
e
πin
e
πi
− 1
=
(−1)
n−1
2
f(n) =
1
2
+
(−1)
n−1
2
.
F
∗
(p) =
e
p

k=1
z
n
k
3z
2
k
− 12z
k
+ 11
=
1
2
− 2
n
+
1
2
.3
n
.
F
∗
(p) =
e
p
(e
p
− 1)
2

2−1
dz
2−1
[(z −1)
2
z
(z −2)(z −1)
2
z
n−1
] = lim
z→1
d
dz
[
z
n
z −2
]
= lim
z→1
nz
n−1
(z −2) −z
n
(z −2)
2
= −n − 1.
f(n) = 2
n

+ 2
−
e
p
e
2p
+ 2e
p
+ 2
)
e
an
sin αn 
e
p
.e
a
. sin α
e
2p
− 2e
p
.e
a
. cos α + e
2a
.
a α e
a
. cos α = 1 e

√
2. cos
π
4
+ 2
e
p
e
2p
− 2e
p
+ 2

√
2

n
sin
3π
4
n 
e
p
.
√
2. sin
3π
4
e
2p

2p
+ 2e
p
+ 2
) 
1
4

√
2

n

sin
π
4
n − sin
3π
4
n

F
∗
(p) = e
p
e
2p
e
4p
+ 4

2p
e
4p
+ 4
=
e
p
2
(
1
e
2p
− 2e
p
+ 2
−
1
e
2p
+ 2e
p
+ 2
).
1
e
2p
− 2e
p
+ 2
= e

+ 2e
p
+ 2


√
2

n−1
sin
3π
4
(n − 1).
F
∗
(p) 

√
2

n−1
2
[sin
π
4
(n − 1) − sin
3π
4
(n − 1)].
F

(e
p
+ e)
(e
p
− e)
3
n
[2]
e
n

2.e
p
.e
2
(e
p
− e)
3
1
2
n
[2]
e
n−1

e
p
.e

e
n−1
−
1
2
n
[2]
e
n−1
= e
n−1

n
2
−
n(n − 1)
2

F
∗
(p) =
e
p
(e
p
− e)
3
= e
−p
e

3
F (z) =
z
(z −e)
3
z = e
f(n) =
1
2!
lim
z→e
d
2
dz
2
[(z −e)
3
.F (z).z
n−1
]
=
1
2!
lim
z→e
d
2
dz
2
(z

p
+ 1)
(e
p
− 1)
3
n
3
e
p
(e
2
p + 4e
p
+ 1)
(e
p
− 1)
4
n
4
e
p
(e
3
p + 11e
2
p + 11e
p
+ 1)

ne
an
e
p
.e
a
(e
p
− e
a
)
2
n
2
e
an
e
p
.e
a
(e
p
+ e
a
)
(e
p
− e
a
)

e
p
. sin α
e
2
p − 2e
p
cos α + 1
cos αn
(e
p
− cos α)e
p
e
2
p − 2e
p
cos α + 1
sinh(αn)
e
p
. sinh α
e
2
p − 2e
p
chα + 1
cosh(αn)
(e
p

a
cosα)e
p
e
2
p − 2e
p
cosα + e
2
a
n
[
k]
k!
e
an
e
p
.e
ka
(e
p
− e
α
)
k+1
n
[
k]
k!

−np
=
e
p
e
p
− 1
, p
f(n) = n
F
∗
(p) =

≥
f e
−np
=

≥
e
−np
n = −
d
dp
[

≥
e
−np
]

p
(e
p
− 1)
2
)
=
e
p
(e
p
+ 1)
(e
p
− 1)
3
n
2

e
p
(e
p
+ 1)
(e
p
− 1)
3
.
f(n) = n

)
e
p
(e
2p
+ 4e
p
+ 1)
(e
p
− 1)
4
n
3

e
p
(e
2p
+ 4e
p
+ 1)
(e
p
− 1)
4
.
n
4
= n.n

.
n
4

e
p
(e
3p
+ 11e
2p
+ 11e
p
+ 1)
(e
p
− 1)
5
f(n) = n
[2]
= n(n−1) = n
2
−n
n
2
−n 
e
p
(e
p
+ 1)

= n(n − 1) . . . .(n − k + 1)
k
f(n) 
k!.e
p
(e
p
− 1)
k+1
k
k > 2, k ∈ N k
n
[k+1]

(k + 1)!.e
p
(e
p
− 1)
k+2
n
[k+1]
= n(n − 1) . . . .(n − k + 1).(n − k) = n
[ ]
.n − n
[k]
.k
 −
d
dp

n
[k]

k!.e
p
(e
p
− 1)
k+1
.
f(n) = e
a.n
f(n)  F
∗
(p) =

≥
e
−np
e
a.n
=

≥
e
−n(p−a)
=
1
1 − e
−(p−a)

p
− e
−iα
=
e
p
e
p
− (cos α − i sin α)
e
p
[(e
p
− cos α) − i. sin α]
(e
p
− cos α)
2
+ sin
2
α
e
p
(e
p
− cos α)
e
2p
− 2e
p

(
e
p
e
p
− e
α
−
e
p
e
p
− e
−α
)
=
1
2
(e
α
− e
−α
)e
p
e
2p
− e
p
(e
α

e
p
. sinh α
e
2p
− 2e
p
. cosh α + 1
.
cos α =
1
2
(e
iα
+e
−iα
) cos(iαn) =
1
2
(e
αn
+e
−αn
) = cosh(αn)
cos iα = cosh α
cosh αn = cos iαn 
e
p
(e
p

f(n)  F
∗
(p −a) F
∗
(p)
f(n)
e
an
sin αn 
e
p−a
. sin α
e
2(p−a)
− 2
p−a
. cos α + 1
e
p
.e
a
. sin α
e
2p
− 2e
p
.e
a
. cos α + e
2a

(e
p
− e
a
cosα)e
p
e
2p
− 2e
p
.e
a
. cos α + e
2a
n
[k]
k!
e
α.n
= C
k
n
e
αn
n
[k]
k!
e
α.n


= n
[k]
.e
(ln a).n
 k!
e
p
.e
k lna
(e
p
− e
ln a
)
k+1
= k
e
p
.a
k
(e
p
− a)
k+1
n
[k]
k!
a
n
= C

(p) =
e
p
e
2p
− 7e
p
+ 10
5
n
− 2
n
3
F
∗
(p) =
e
p
e
2p
+ 1
sin
nπ
2
F
∗
(p) =
e
p
e

F
∗
(p) =
e
p
(e
p
− e)
3
n(n − 1)
2
e
n−2
F
∗
(p) =
1
(e
p
− 3)(e
p
− 4)
4
n−1
− 3
n−1