Vortrag - des Instituts für Mathematik

( )

( )

( )

≤ , , ≤
+ + = .

= + .
+ + =
− − = ,

−
−
= , = .
≥
≤ ≥ ≤
≤
≤ ≤ ≥

¤
() = − .
() = − . ¤

×
¤
()
() ¤

()
() = ( − .) = − .
¤ ¤
= − + . = ( − + .).
() = ( − .)( − + .(( − .))
= −. + . − .

≤ () ≤ ,
≤ − . ≤
≤ ≤ .

¤ ¤

130
120
110
100
90
80
g(p)
70
10 15 20 25 30 35 40

¤
′ () = − · . + . =
= .
= .,
· .

¤
′ () = − · . + . =
= .
= .,
· .

¤
′ () = − · . + . =
= .
= .,
· .

( )
( )
( )
( )

( ( ), ( ), ( ), ( ))
( )
+
( + ) = . ( ).
+
( + ) = . ( )
( + ) = . ( ).

+
( )
( )
( )
( )
( )
( )
( + ) = ( ) + ( ) + ( )

( + ) = ( ) + ( ) + ( )
( + ) = . ( )
( + ) = . ( )
( + ) = . ( ),
+

x 104
2.5
2
1.5
1
0.5
0
22 464
4 320
1 728
1 2 3 4
1 555

x 104
2.5
2
1.5
1
0.5
0
22 464
4 320
1 728
1 2 3 4
1 555

()
∆
( + ∆) + ∆
∆ = ( + ∆) − ().
∆ = α()∆
∆
= α().
∆

∆
+ ∆
∆
∆
∆
∆ →

∆
∆→ ∆ = ˙ ().
˙() = α()
() = α(−)

( ) =
=
() = .

( ) =
=
() = .

α <
α >

α <
α >

α
β δ
α = β − δ,
˙() = β() − δ(),

δ δ
˙() = β() − δ ()
˙() = λ()( − ()).
λ = δ = β
λ

˙() > () <
˙() < () >
˙() = () =
() =

≡

=
( ) =
() =
+ (
.
− )−λ (−)

20
18
16
14
12
10
8
6
4
2
P 0 > K
P 0 = K
P 0 < K
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 107
16
14
12
10
8
6
4
2
0
1780 1800 1820 1840 1860 1880 1900 1920 1940 1960

α = . δ = . · −
= . ·

α = . δ = . · −
= . ·

˙() = λ() − λ ()
λ()
λ ()

= ().
˙() = λ() − λ () − ().

˙ () ≡
λ() − λ () − () = ()(λ − − λ) = ,
() ≡
∗ () ≡ −
λ .
∗ < ∗ >
< λ .

()
() < λ() = β().
= ∗
( ) = ∗ = ( −
).
λ
′ ( ) =
∗ =
λ =
β.

∗
∗
( ∗ ) = ∗ ∗ =
λ =
β ,
∗ ( ∗ ) =
.
= .

β = λ = .
( ∗ ) =
· . · . =
∗ =
= .

β = λ = .
( ∗ ) =
· . · . =
∗ =
= .

∈ /∈

= {, , , }.
= {, , , } = {, , , } = {, , , , , , , , , }.

= { ∈ : ( )}.
( ) <
= { ∈ : < }
, , ,
= .

∅
= {, , , , . . . } . . .
= {, ±, ±, ±, . . . } . . .
= {
: , ∈ , = } . . .
. . .
= { + : , ∈ } . . .

⊂
P()
P() = { : ⊂ }.

∩ = { ∈ : ∈ ∈ } . . .
∪ = { ∈ : ∈ ∈ } . . .
\ = { ∈ : ∈ /∈ } . . .
∩ = ∅
= \ .

∩ = { ∈ : ∈ ∈ } . . .
∪ = { ∈ : ∈ ∈ } . . .
\ = { ∈ : ∈ /∈ } . . .
∩ = ∅
= \ .

, . . . ,
( , . . . , ) ∈ ∈
× · · · × = {( , . . . , ): ∈ , . . . , ∈ }
, . . . ,
= = , . . . , = × · · · ×
= = =
(, ) = (, ) = , {, } = {, }
(, ) = , {, } = {}

= { , . . . , }
# = .
#∅ = .

#
#
#( ∪ ) = # + # − #( ∩ )
# × = # · #
#P() = #

= { ∈ : ≤ ≤ }
= { ∈ : < ≤ }
× = {( , ): ≤ ≤ , < ≤ }
5
4
3 B
2
1
A x B
A
0
0 0.5 1 1.5 2 2.5 3

, . . . , , . . . ,
· · . . . ·

= {, , }
.

! = · ( − ) · . . . · · , ≥
! =
!

!
≤
· ( − ) · . . . · ( − + ) = !
(−)!

−
− ( − )
!

−
− ( − )
!

!
! ! !
! = ! ! !
= !
!!!

= , . . . ,
!
! ! . . . ! , + + · · · + = .

!
· ( − ) · · · · · ( − + ) =
( − )!
≤

· · =

− = =
= −
!
( − )! !
≤

!
!!
= · · · · ·
· · · · ·
=

, ∈
=
!
(−)! !
≤
>

, ∈
=
!
(−)! !
≤
>

, ∈
=
!
(−)! !
≤
>

=

{, , . . . , }
{, , . . . , , + , . . . , + − }
−

+ −
+ −
=
−

= =
=
= +−

!
(−)!
−+

≥ ≤
= =
=
+
−
=
+ , ≤ ≤
−

+
∗
\ { ∗ } ( − )
\ { ∗ } ∗

∈ ∈
(+)
=
+
− +
− +· · ·+
−
− +
( + ) = + +
( + ) = + + + .
.

∈ ∈
(+)
=
+
− +
− +· · ·+
−
− +
( + ) = + +
( + ) = + + + .
.

−2√2
1/2√2
−1 −1/2 0 1/2 1 2
√2 2√2
√
/∈

=
< =
≤ = <
> =
≥ = >

⎧
⎪⎨ + ≤ + ∈
≤ λ ≤ λ
⎪⎩
λ ≥ λ
λ >
λ < ,
>

[, ] = { ∈ : ≤ ≤ }
(, ) = { ∈ : < < }
(, ] = { ∈ : < ≤ }
[, ) = { ∈ : ≤ < }
[, ]
(, )

≥
| | =
−

| | = =
|λ | = |λ| | | λ ∈
| + | ≤ | | + || ∈

−∞
∞
−∞ < < ∞.
±∞
(−∞, ) = { ∈ : < }
(−∞, ] (, ∞) [, ∞)

| + | > −

+ ≥ ≥ −
| + | = + .
+ > − ,

≥ −
− ≤ < .
L = [−, ).
+ < < −
| + | = −( + ),

− − > −
<
.
< −
L = (−∞, −).
L = L ∪ L = (−∞, ).

| − |
| + | >
= −
L = (−∞, −) ∪ (, ∞)

ε >
ε
( − ε, + ε) = { ∈ : | − | < ε}

, ∈
+ + =
= −
±
− ,
− ≥
+ =

+ = −
= .
= (−
+
− )(−
−
− ) = ((−
)
−
−
)
=
− ( − ) =

= ± √ − + =
= √ −(− √ −) = − √ −
= −(−) = .

= −
= { : = + , , ∈ }

= +
= ℜ
= ℑ
| | = √ +
¯ = −
{ : ∈ }

(ℜ , ℑ )
2
1.5
1
0.5
0
−0.5
−1
−a+ib
−a−ib
Imaginäre Achse
i
1
a+ib
reelle Achse
a−ib

= −

= + = α + β
± = ( + ) ± (α + β) = ± α + ( ± β),
· = ( + ) · (α + β) = α + α + β + β
= α − β + (α + β).

= β =
ℑ =
ℑ =
(, ) ∈

= +
ℜ =
( + ¯ ), ℑ = ( − ¯ )
¯ = | | ,
= ¯
| |

=
¯
=
||
.

+
−
= + = −
= ¯
||
= +
+
=
( + ),
= ( + )( + ) = ( − + ( + )) = (− + ).

+ = ¯ + ¯
· = ¯ · ¯
| | = |¯ |

| | = =
|| = | | · ||
| + | ≤ | | + ||

| | = =
|| = | | · ||
| + | ≤ | | + ||

= {( , . . . , ): ∈ , = , . . . , }
= {( , . . . , ): ∈ , = , . . . , }
= ( , . . . , )

=
⎛
⎜
⎝
⎞
⎟
⎠ ,

=
⎛
⎜
⎝
⎞
⎟
⎠ ,

= ( , . . . , ) ∈ = ( , . . . , ) ∈
+ = ( + , . . . , + ).
= ( , . . . , ) ∈ λ ∈
λ
λ = (λ , . . . , λ).

= ( , . . . , ) ∈ = ( , . . . , ) ∈
+ = ( + , . . . , + ).
= ( , . . . , ) ∈ λ ∈
λ
λ = (λ , . . . , λ).

∈ λ µ ∈
+ = +
+ ( + ) = ( + ) +
+ =
λ( + ) = λ + λ
(λ + µ) = λ + µ
(λµ) = λ(µ )
= , =

= (, . . . , ) ∈
= || + . . . ||

·
λ ∈
≥
= =
λ = |λ|
+ ≤ +

−
ε >
( , ε) = { ∈ : − < ε}
ε = =
( , ε)
ε

−
ε >
( , ε) = { ∈ : − < ε}
ε = =
( , ε)
ε

= ∅
∈ ∈
: → = ( )
() = { ∈ : = ( ) ∈ }
( )
= { , ( ) ∈ × : ∈ }

=
⊂ ⊂
≤

f(x)
x

f(x 0 )
f(x)
x 0
(x 0 ,f(x 0 ))
x

2
1.5
1
0.5
0
1
0.5
0
−0.5
−1
−1
−0.5
: →
0
0.5
1

10
9
8
7
6
5
4
3
2
1
0
0 5 10 15 20 25 30

→
:
↦→ ( ).
= ( ) ∈
( ) = ( ) ∈

∈
∈ = ( ) ∈
˜ =
∈
∈ ˜ ∈
( )
( )

↦→ − +
→ ( ) = ( − )
∈
: ↦→ = −
+
= −
= \ {−}

:
→
↦→ .
= ∅
→
:
↦→ .
| · |:
→
↦→ | |.

:
→
↦→ ()
()

c
f(x)
x
id
x

|x|
x

: →
∈
∈ = ( )
, ˜ ∈
= ˜ ( ) = (˜ )

∈
( ) = (˜ ) = ˜
∈
| − | = ||
( ) =
∈
∈

↦→
+
\ {−}
( ) = (),
+
=
+
+ = + , = .

=
+

, ⊂

⊂ : →
∈ < ( ) ≤ ()
( ) < ()
∈ < ( ) ≥ () ( ) > ()

−

−

, : → ⊂ λ ∈
λ +
: →
(λ )( ) = λ ( )
( + )( ) = ( ) + ( )
()( ) = ( )( )
( )
( ) = ( )
( ) =

, : → ⊂ λ ∈
λ +
: →
(λ )( ) = λ ( )
( + )( ) = ( ) + ( )
()( ) = ( )( )
( )
( ) = ( )
( ) =

: → : → ⊃
→
◦ =
↦→ ( ())

x
A
f g
C
f(x)
g ◦ f
B
g(f(x)) =
(g ◦ f)(x)
D

⊃
( ) ∈ ◦
◦ = ◦
◦ ◦
() = −
() = + ∈
( ◦ )() = (()) = () − = ( + ) − = + −
( ◦ )() = (()) = () + = − + = − .

= ◦
→ ( ) =
+
+
( ) = + ( ) = () = \ {}
() = [, ∞) = ◦

↦→
+
\ {−} →
\ {−}

∈
= ( ) ∈
∈
=
+ .

+ =
= −
∗ =
− .
∗
=
− = −
= ( )
=
\ {−} → \ {}

( ) =
=
: \ {} → \ {−} ↦→
( ◦ )() =
() + =
−
(
− ) + = , ◦ = ,
( ◦ )( ) =
− = ( + ) − = , ◦ = .
( )

= ∅ : →
= − ()
= ( )
− : →

: →
◦ − = − ◦ =
, ⊂

( ) = + ∈ [, ]

≤ < ≤ <
≤ + < + [, ]
−
([, ]) = [, ]
−
( − ()) = = − ()
( ) = , + =

= − () = ± − .
− () ∈ [, ]
− () = − , ∈ [, ].

: →
( ) = {( , ): ∈ , = ( )}
( − ) = {(, ): ∈ , = − ()}
= {(, ): ∈ , = ( )}
= {(, ): ∈ , = ( )}
−
=

f(x)
(f(x),x)
f −1 (x)
(x,f(x))
f(x)
x

f(x)
(f(x),x)
f −1 (x)
(x,f(x))
f(x)
x

f(x)
L
f(x)
f(x)
x
c
x
x
f(x)
L
f(x)
f(x)
x
c x

( )
−
+

∈
( ) = ( ) = ( )
( ) = √ \ {} (, ∞) [, ∞)
=

L −
L +
f(x)
c
x
f(x)
x

⊂
: → ∈
ε (, ε)
δ ( , δ)
( ( , δ) \ { }) ∩ ⊂ (, ε).
= ( )
→

ε >
δ >
| ( ) − | < ε ∈ < | − | < δ
→( − ) = −

ε >
δ >
| ( )−| = |(− )−(−)| < ε < | −| < δ
|( − ) − (−)| | − |
|( − ) − (−)| = | − | = | − |.

| − | < δ
|( − ) − (−)| = | − | < δ.
|( − ) − (−)| < ε | − | < δ
δ ≤ ε,
δ ≤ ε δ
δ

δ | − | < ε ∈
|( − ) − (−)| = | − | < ε
= ε

∈ : →
→ ( ) = → ( ) =
± λ λ ∈
→( ± )( ) = ±
→(λ )( ) = λ
→()( ) =
=
(
→
)( ) =
∈ : ( ) ≤ ( ) ≤

= =
→
→
: → ( ) = + + · · · +
( ) = (). →

=
( )
( ) =
→
→ ( ) = () () , () = .
(
→−
+ − ) = (−) + (−) − = ,
→
−
+
−
=
=
+ .

∈ : →
→ ( ) = → ( ) = =
→(
)( )

→(
)( ) → ( ) =
→ ( ) =

→
→−
− −
−
+
( + +)

=
=
− − = ( + )( − ),
− −
−
→
− −
−
= +
= (
+ ) = .
→

= −
+ + +
→−
+ + = ( + )( + ),
+
( + + )
=
→−
( + ) ( + ) .

( − δ, ] [, + δ)
− +
( ±
( −
) =
→ ±
) =
→,
<
( ),
( ), ( +
) =
→,
>
( )
+ −
− ( ) = → − + ( ) = →
+
= ∞ = −∞ (∞, δ) = (δ, ∞)

(−∞, δ) = (−∞, δ)
= ±∞

→ ( ) =
→, < ( ) = →, > ( ) =

→ ( ) =
→, < ( ) = →, > ( ) =

→ ( ) =
→, < ( ) = →, > ( ) =

( ) =
=
+ ≤ ,
− >

≤ ( ) = +
→, < ( ) = ≥ ( ) = −
→, > ( ) =
=

⎧
⎪⎨ +
( ) =
⎪⎩
≤ ,
−
=
= ,
>
( ) = ( +
→, <
→, <
) =
( ) = ( − ) = .
→, >
→, >

=
( ) = .
→
() =

=
( ) = .
→
() =

f(x)
c
x

f(x)
c x

∞
−∞ ¡¡
±∞
→
+ ( ) = ∞
→
− ( ) = −∞.

(ξ, ∞) ∞ (−∞, ξ)
−∞ ε ξ
ξ
( ) = ∞
→
ξ (∞, ξ) ∞
δ (, δ)
(, δ) \ {} ⊂ (∞, ξ),

ξ
δ >
( ) > ξ < | − | < δ

±∞
∞ ∞ ∞
· ∞ ∞ > · ∞ −∞ <
= ∈
∞
∞ · ∞∞
−∞ < < ∞ ∈
· ∞ ∞ − ∞ ∞
∞

→ ( ) → ( ) =
( )
( ) > = ( )
( )
→ = ∞ <
( ) ( )
( )
= → = −∞
( )
→ ( ) = → ( ) = ∞
( )
→ =
( )

→
−
−
+
→∞ − →( − ) = − →( − ) =
→ ( )
− > > − < <
→ +
−
= −∞,
− →− −
= ∞.
−

∞
∞
+
−
+
=
− ,
→∞
=
→∞( +
) = →∞( −
) =
+
→
)
− = →∞( +
→∞( − = .
)

: → ∈
→ ( ) = ()
∈

: →
ε ( ( ) − ε, ( ) + ε) ( ) δ
( − δ, + δ)
(( − δ, + δ)) ⊂ ( ( ) − ε, ( ) + ε)

f(x 0 )+ε
f(x 0 )
f(x 0 )−ε
f(x)
x 0 −δ
x 0 x 0 +δ
x

ε
δ

: →
ε ( ( ) − ε, ( ) + ε) ( ) δ
( − δ, + δ)
(( − δ, + δ)) ⊂ ( ( ) − ε, ( ) + ε)

= →
→ ( ) = ( )
( ) = ( ).
→
→

: →
λ ∈
± λ ( ) =

: →
: → ⊃ ( )
( )
◦

→ ( ) = ( +
+ )
+ >
→ +
→ + + = ∈
: → +
+
: → → = ◦

−
( ) − ( )
[, ∞) → [, ∞)
:
→ √
[, ∞) → [, ∞)
:
→

= → ( )
( ) =

: → ( )

[α, β] λ (α) (β)
ξ α ≤ ξ ≤ β (ξ) = λ
αβ < λ =
[, ]
(, )

[, ]
() < () >
(, )
+
+
( )
( +
) =

[, ] () () <
[ , ] ( ) ( ) ≤
= ( + )/ ( )
[+, +] = [ , ] [ ,
(+) (+) ≤
(+ − +)

ξ
−
| − ξ| ≤ ,

: →
∈
( ) ≤ ( ) ( ) ≥ ( )
∈
∈
δ > ( ) ≤ ( )
( ) ≥ ( ) ∈ ( − δ, + δ) ∩

3
2
1
0
−1
−2
−3
lokales
Minimum
globales
Maximum
globales Minimum
−4
−0.5 0 0.5 1
lokales
Maximum

3
2
1
0
−1
−2
−3
lokales
Minimum
globales
Maximum
globales Minimum
−4
−0.5 0 0.5 1
lokales
Maximum

[, ]

: →
()
() ⊂ : →

() ⊂ ∈
ε > (ε)
≥ (ε) ε
=
→∞ .
() ()
() →∞ =

ε ε >
( − ε, + ε)

()
ε
ε
() = ((
,
)) = ( , )
ε >
|| − || = |
− | + |
− | / < ε.
|
− | < ε, |
− | < ε

() = ((
, . . . , ))
= ( , . . . , )
(
) ⊂ ()
= , . . . ,

() ⊂ >
|| ≤ ∈
() ⊂
≤ + ≥ + ∈

() ⊂ >
|| ≤ ∈
() ⊂
≤ + ≥ + ∈

() ⊂ () ⊂
( )

→∞
α = α >
( ) ∈
|| < =
→∞ =
α >
→∞ =
⇔ || <
⇔ = .
→∞ α = .
→∞( +
) = = . . . .

() () ≤
∈ →∞ ≤ →∞
() ()
→∞ = →∞
()
≤ ≤ ,
()
→∞ = →∞

() () λ ∈
(λ) ( ) ( ± )
→∞ λ = λ →∞
→∞ = →∞ →∞
→∞ ± = →∞ ± →∞
→∞ =
(
)
→∞
= →∞
.
→∞

= = (−) ∈

= + +
+ +
= ( +
+
)
( +
+
)
= ( +
+
)
( +
+
)
=
.

→∞ = +
→∞
→∞ = +
→∞
+
→∞
+
→∞
= + + = .
→∞ =
(
)
→∞
=
→∞
= →∞
→∞
= .
=
.

→∞ −
− − =
= −
− −
= −
−
=
( − (
) )
→∞ =
→∞ (−(
) ) =
(−
→∞ (
) ) =
(−) =
.

() = ()
±∞
() ⊂
∞ −∞ ξ ∈ (ξ)
≥ ξ ( ≤ ξ)
≥ (ξ) ±∞
()
→∞ = ∞ →∞ = −∞

(), () ⊂ →∞ = ∈
→∞ = ∞ →∞
→∞ =
→∞
= ∞
→∞ =
→∞
= −∞
=
> ∈
< ∈

() = ( − +
−
− √ + ), () = ( √ )
− +

∞
∞
√
= −
−
√
+
−
, =
√ −
√
√
− √ +
√

√ −
=
√ ( −
√ ) > ,
→∞ = −∞
√
→∞ = .

∈ ∈
=
=
·
· · · ·
,
= .

= +
( ) =
() =
< < < <
< < > < <
> > >

→
∈
( ) =

: →
( ) = (− )
∈
( ) = −(− )
∈
→∞ ( ) = ∞ ∈
→−∞ ( ) =
∞ ,
−∞ .

8
7
6
5
4
3
2
1
x 4
x 2
0
−2 −1 0 1 2
4
3
2
1
0
−1
−2
−3
x 4
−4
−2 −1 0 1 2
x 5

[, ∞)

: [, ∞) → [, ∞)
: →
√ = ⇔ =
√ ≡ /

→∞ ( ) = ∞
< < >
>
> >

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
√x
x 1/4

∈ [, ∞) ∈ √
( √ )
= , =
> .

∈ \ {} ∈ − =
−
> ∈ =

→
→∞
→
→ −
→ +
=
→−∞
= ,
= ∞, ,
= −∞, ,
= ∞, ,

100
90
80
70
60
50
40
30
20
10
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
1/x n
n gerade
−
10
8
6
4
2
0
−2
−4
−6
−8
−10
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
1/x n
n ungerade
−+

∈ () ⊂
→∞ = .

∈ >
→∞
()

∈ (, ∞) ∈ () ⊂
→∞ =
:=
→∞

∈
+ =
() =
( ) = = ( )
= − =
(
) = −
> < < <
> < <
∈ (, ) <

∈ =
(, ∞) →
=
↦→
(, ∞) (, ∞)
>

3
2.5
2
1.5
1
0.5
a = 0
a < 1
a 1
0
0 0.5 1 1.5 2

∆()
= λ()
∆
− ∆()
∆
= λ()
()
∆()
= λ()
∆
()

∆
∆()
= λ().
∆

()
= () =
( + ) = ().

∆ =
( + ) − () = λ(),
( + ) = ()
= λ +
( + ) = ( + ) = ( + ) = () ! = ()
= , = √ .

() = ,
() = () = ,
() = () = ,
() = =
.

( + ∆) − () ≈ λ()∆.
∆
=
() = () >

() [, ]
∆ =
=
= , . . . ,
() ≈ (−) + λ∆(−)
= ( + λ
)(−)
≈ ( + λ
)((−) + λ∆(−))
= ( + λ
)(−) = · · ·
≈ ( + λ
) ().

[, ] ()
() =
→∞ ( + λ
)
( +
→∞ ) = , ∈ ,
= , ...
() = λ .

=
→
↦→ →∞( +
) .
( )

( )

=
:=
→
↦→
(, ∞)
>
<
∈ ( ) > () =
→∞ ( ) = ∞ →−∞ ( ) =
>

= =
8
7
6
5
4
3
2
1
a = 1/2
a = 2
0
−4 −3 −2 −1 0 1 2 3 4

λ λ
→∞
= , >
λ λ λ
→∞ λ = , >

(, ∞)
: → (, ∞) > =
: (, ∞) →
= () ⇔ = ( ) =
()

=
=
=
() ()

3
2
1
0
−1
−2
−3
−4
a = 2
a = e
a = 10
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

=
∈ (, ∞) =
∈ =
>
=
() = +
=
= − ∈
= > =

= = =

→∞ = ∞,
→ + = −∞.
>
α >
→∞
α
= ,
→ + α = .

= =
= , > =
= =
= (
) = −
√
/ = = −
√
=

=
= ( ) = .
= =
= .

= ,
= ,
= −,
= , − = −.

= ,
= ,
= −,
= , − = −.

() = −λ , > .
( + ) =
()
−λ(+) =
−λ ,
−λ =

−λ =
= − ,
λ =
.

= ,
=

= + , ,
= + , .

= + , ,
= + , .

= λ = λ
λ

. . −. .
. . −. .
. . .
. . . .
. . . .
. . . .

log y
5
4.5
4
3.5
3
2.5
2
1.5
1
0.2 0.4 0.6 0.8 1 1.2
x
1.4 1.6 1.8 2 2.2

5
4.5
4
3.5
3
2.5
2
1.5
1
−2 −1.5 −1 −0.5
log x
0 0.5 1

= λ
λ
= + λ

λ
λ
= (., .) = (., .)
λ = −
−
= . − .
. − .
= − = −..
.

= . − (−.). = .
= . = ..
= . −.

5
4.5
4
3.5
3
2.5
2
1.5
P 1
∆ y = −3
∆ x = 1.7
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
P 2

log y
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
P 1
∆ x = 2
∆ y = 1.4
1.8
−2.5 −2 −1.5 −1
log x
−0.5 0 0.5
P 2

= + λ
λ
= (, .) = (−, .).
λ = −
−
= . − .
= .
− (−)

= . = .
= .

log y
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
0 0.2 0.4 0.6 0.8 1 1.2 1.4
x

log y
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
−2.5 −2 −1.5 −1
log x
−0.5 0 0.5

: → >
∈
( + ) = ()

◦ ′ ”

π
π ◦
π = ◦
= ◦
π
◦ = π

π
π
π
α
(α)
(, )
α > α < |α|
π
(, )
(α) ( α, α)
, : → [−, ],
π
π
π
π
π

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
α
cos α
sinα
P(α)

3
2
1
0
−1
−2
−3
−3 −2 −1 0 1 2 3 4 5 6
sin
cos

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
α
−α
P(α)
sin(α)
cos(α)
sin(−α)
P(−α)
cos(−α)

1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
P(π−α)
sin(π−α)
cos(π−α)
π−α
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
α
cos(α)
P(α)
sin(α)

π
= π ∈
= ( + ) π
α
α + α =
| α| ≤ | α| ≤
α = (−α) α = − (−α)
α = − (π − α) α = (π − α)
α = − (π + α) α = − (π + α)
α = ( π
π
+ α) α = − ( + α)
∈

1
0.8
P(π/2+α)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
sin(π/2+α)
cos(π/2+α)
π/2+α
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
α
cos(α)
P(α)
sin(α)
8
6
4
2
0
−2
−4
−6
−8
−4 −3 −2 −1 0 1 2 3 4
tan
cot

α β ∈
(α ± β) = α β ± α β
(α ± β) = α β ∓ α β.

:
:
π
\ { + π, ∈ } →
α ↦→
α
α
\ { + π, ∈ } →
α ↦→
α =
α
α
α

π
π
(α + π)
(α + π) =
(α + π)
− α
= = α.
− α

(− π π
, )
(, π)
π
π

α
(α)
(α)
: = : α, α =
,
: = : α, α =
.

α =
, α =
.

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
−0.2
α
c
sin α
P(α)
cos α
A Q
b
C
0 0.2 0.4 0.6 0.8 1 1.2 1.4
a
B

[−
π π
, ] → [−, ]
[, π] → [−, ]
(− π π
, ) →
(, π) →

[−, ] → [− π
, π
]
= ⇔ =
[−, ] → [, π]
= ⇔ =
→ (− π
, π
)
= ⇔ =
→ (, π)
= ⇔ =

1.5
1
0.5
0
−0.5
−1
−1.5
arcsin
−1.5 −1 −0.5 0 0.5 1 1.5
sin
3
2.5
2
1.5
1
0.5
0
−0.5
arccos
−1
−1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5
cos

6
4
2
0
−2
−4
arctan
−6
−6 −4 −2 0 2 4 6
tan
6
4
2
0
−2
−4
arccot
−6
−6 −4 −2 0 2 4 6
cot

π = ◦ =

ϕ
π
ϕ ∈ [, π]
(, ϕ) ( , )

= ϕ,
= ϕ.
= + ,
⎧
⎪⎨
ϕ =
⎪⎩
π
= , > ,
, = , > ,
π
+ π < , = ,
, = , < ,
+ π, > , < .

1
0.8
0.6
0.4
0.2
0
r = (x 2 + y 2 ) 1/2
φ
x = r cos φ
y = r sin φ
(x,y)
−0.2
−0.2 0 0.2 0.4 0.6 0.8 1 1.2

() = + (ω + φ)
ω
ω π
ν =
φ
= π
ω

φ φ
ω
ω

= . =
≈ ..
ν =
π
≈ ., ω = ≈ .

() = + (ω( − )),
=

c
3
2.5
2
1.5
1
0.5
0
−0.5
φ/ω T
A
−1
−4 −3 −2 −1 0 1 2 3 4

φ = −ω
= −.
φ ≈ −(−.) = ..
= .
φ = −ω ≈ − · . = −..

π
. − (−.) ≈ π
() = . + ( + .),
() = . + ( − .).

(ω + φ) = (ω) + (ω)
= (φ), = (φ)
= +

(ω + φ) = (ω) + (ω)
= (φ), = (φ)
= +

1.5
1
0.5
0
−0.5
−1
(π)
π
(π)
π
(π) + (π)
π π
−1.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
1.5
1
0.5
0
−0.5
−1
−1.5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(π)

=

()
( ) =
(, ) = () − ( )
−
[ , ]
( )
() (, )
(, )

→
()
() = (, ) =
→
→
( ) − ( )
−
() − () .
− [, ] [ , ]
( ) − ()
→ −

( , ( ) ( , ( )
Anstieg =
(f(x)−f(x 0 ))/(x−x 0 )
(x 0 ,f(x 0 ))
x 0
x−x 0
(x,f(x))
x
f(x)−f(x 0 )

x 0
x 3
x 2
x 1

: → ⊂
∈
( ) − ()
→ −
′ ()
()
∈
′ :
→
↦→ ′ ( )

( , ( )) ( , ( )) →
( , ( ))
′ ( )
( ) = ( ) + ′ ( )( − ).

( ) − ( )
( ) − ( ) = ( ) − ( ) − ′ ( )( − )
= ( ) − ( )
−
− ′ ( ) ( − )

( ) − ( )
= .
→ −
−
( − )( ) − ( − )( )
−
→

f(x)
f(x 0 )
x 0
p(x)
x
(x,f)x))
f(x)−p(x)
(x,p(x))

′ ( ) = ( ) =
∈
( ) − ( ) = −
= ( + +
)( − )
( ) − ()
=
→ − →
( + +
) = .

′ ( ) = ( ) =
∈
( ) − ( ) = −
= ( + +
)( − )
( ) − ()
=
→ − →
( + +
) = .

( ) = | | =
| | −
→ −
>
=
→
>
| | −
=
→ −
<
−
=
→
<
= −

( ) ′ ( )
α α α−
−
( )
α α α
, >
+

, : →
±
( ± ) ′ ( ) = ′ ( ) ± ′ ( )
λ ∈ λ
(λ ) ′ ( ) = λ ′ ( )
·
() ′ ( ) = ′ ( )( ) + ( ) ′ ( )
() =
(
)′ () = ′ ()()− ′ () ()
()

, : →
±
( ± ) ′ ( ) = ′ ( ) ± ′ ( )
λ ∈ λ
(λ ) ′ ( ) = λ ′ ( )
·
() ′ ( ) = ′ ( )( ) + ( ) ′ ( )
() =
(
)′ () = ′ ()()− ′ () ()
()

( ) =
+

′ ( ) =
( )
( +
) − (+ )
( + )
( )
=
+
= + .
′ ( ) = ( + )( + ) − ( )
( + )
= ( − ) + ( + )
( + )
.
=

( ) = ( )
: → : →
⊃ ( )
◦ : →
∈
( ◦ ) ′ ( ) = ′ ( ( )) ′ ( )

′
( )
′ ( )

( ) = ( )
( ) = ( ) ◦
: ↦→ : ↦→
( ) = ′ ( ) =
′ ( ) =
′ ( ) = ( )

: →
− ∈ ′ ( ) =
= ( )
−′ ( ( )) =
′ ( ) , −′ ( ) =
′ ( − ( ))

=
( ) = − () =
′ ( ) = > ∈
() = =
.
′ ( ) = ′ ( − ()) = ′ ( ) =

: [, ] → (, )
ξ ∈ (, )
() − () = ′ (ξ)( − ).

(, )
ξ (, ())
(, ())
(ξ, (ξ))
a
(ξ, f(ξ))
ξ
b

, : (, ) → ′ ( ) =
(, )
→
′ ( )
′ = .
( )
( ) = , ( ) = ,
→ →
( ) = ∞, ( ) = ∞,
→ →
( )
= .
→ ( )
→

( )
( )
∞
∞
→
→
= ,
−
= ,
=
→ +

→
→
−
=
→
=
→
= ,
−
= .
· ∞
=
→ + → +
=
→ +
−
= ) = .
→ +(−

: [, ] →
′ ( ) ≥ ′ ( ) > ∈ [, ]
[, ] ′ ( ) ≤ ′ ( ) <
∈ [, ] [, ]
≤ < ≤
ξ ∈ ( , )
() − ( ) = ′ (ξ) ( − ) ≥ (> ).
≥
′ ( ) = ∈ [, ] [, ]
>

: [, ] →
∈ (, )
′ ( ) =

( ) =

′

′
<
′ ( ) ≥ >
′ ( ) ≤

f’ ≥ 0
f’ ≤ 0

: (, ) → ′ ( ) =
∈ (, )
′ ( ) ≥ ∈ ( − δ, )
′ ( ) ≤ ∈ ( , + δ)
δ >
′ ( ) ≤ ∈ ( − δ, )
′ ( ) ≥ ∈ ( , + δ)

( ) =
−
≥
= () =

(, ∞)
( ) ( ) ≥ ≥
( ) > > =
′ ( ) =
( −
−
) .

∈ ′ ( ) =
−
= , = .
′ ( ) −
′ ( ) > ⇔ −
′ ( ) < ⇔ −
> ⇔
< ⇔
< ⇔ < ,
> ⇔ > .

(, ∞) ′
() = =
→∞ ( ) = () = ( ) >
() = .

=
( ) = ′ ( )( − ) + ( )
ξ ξ

8
6
4
2
0
−2
(x 1 ,f(x 1 ))
−4
0 0.5 1 1.5 2 2.5 3 3.5
x 1
(x 0 ,f(x 0 ))
x 2
x 0

()
= − ′
()
(, ())
()
+ = −
′ , ∈ .
()
ξ
>
|+ − ξ| ≤ (+ − ) ,

|+ − | ≤ ε ε >
′

= ξ = √ = .
( ) = −
+ = −
−
.

()
( )
′ ( )
+−ξ
(+− )
. . . .
. . . .
| − ξ|
. . . . .
. . . . .
. . − . − . . −
. −. −. .
√

: → ∈
′ : →
′′ ( ) = ( ′ ) ′ ( )
∈
() ( ) = ( (−) ) ′ ( ).
() ∈
∞

( ) = () ( ) = ∈
( ) = ′ ( ) = ′′ ( ) = −
′′′ ( ) = − () ( ) = · · ·
( ) = ′ ( ) = ′′ ( ) = · ′′′ ( ) = !
() ( ) =

′′ ( ) ≥
′
′′ ( ) ≤ ∈ ′

˙() = λ()( − ()).

˙() > () <
˙() < () >
˙() = () = .

= () >
() ≡
=
() ≡
() =
() <
() >

¨() = λ ˙ () − () .
( ) = =
() = ∈
= ∗
( ∗ ) =

() > ∈
¨() = λ ˙
() − () > .
<
<
< <
¨ ¨ () < ≥
<
¨ = ∗ () < ∗

∗

: (, ) → ′ ( ) =
∈ (, ) ′′ ( ) < ′′ ( ) >

: (, ) → ′ ( ) =
∈ (, ) ′′ ( ) < ′′ ( ) >

+ + · · · +
=
= + + · · · + .
=
=
=
=
−
=
+
+ =
=
−.

,
=
=
=
,
=
,
=
.
=
α = α
=
=
=
= α,
=
=
−
−
, <
=

= = +− −
= =
= =
( + )
∈ \ {}
( + )( + )
= =
( + )

() =
( ) =
>

[, ]
< < . . . · · · < =
() ≈ (−), ∈ [−, ].
() ()
[−, ] ()
() − (−) = ( ∗ )( − −) ≈ (−)( − −),
∗ ∈ (−, )

(−) ( − −)
(−)( − −)
( ) − = () − ( ) =
= () − (−) + (−) − (−)+
+ (−) − (−)+
· · · + () − () = ( () − (−))
=
=
( ∗ )( − −) ≈
=
(−)( − −).
=

() = = =
= = , · · · ,
() = () ≈
−
=
() ≈
=
( − ) =
=
−
=
=
−
=
= ( + )
( − ) →
→∞ .

[ , ]
( )
τ ∈ [−, ] τ = −

: [, ] →
= < · · · < =
[, ] ξ ∈ [−, ]
R( ) =
(ξ)( − −)
=

→∞ R( )

( ) =
→∞ R( )

[, ]

, : [, ] →
λ ∈
± λ
( ± )( )) = ( ) ± ( )
(λ )( ) = λ ( )

( ) = ,
( ) = −
( ) ,
( ) = ( ) + ( ) , ∈ [, ]

( )
= =

( )

, : [, ] →
′ ( ) = ( )
∈ [, ]
( )

, , : [, ] →
( ) = ( ) + , ∈ [, ].

′ ( ) =
( ( ) + ) = ′ ( ) + = ( ), ∈ [, ].
( − )( ) = ′ ( )− ′ ( ) = ( )− ( ) = , ∈ [, ].

− [, ]
− [, ]
( ) = ( ) + , ∈ [, ].

α + β α + β
(α + β)( ) = α ′ ( ) + β ′ ( ) = α ( ) + β( ).

( ) ( )
α
α+ α+ , α = −
| |
α
α α
α −
α
α
α
α
α
+

( ) + .

( ) = − + −
=

( ) = −
− +
− + ,
() =
= −
+ , =
.
( ) = −
− +
− +
.

( ) = −
− +
− + ,
() =
= −
+ , =
.
( ) = −
− +
− +
.

: [, ] →
: [, ] →
() =
( ) , ∈ [, ]
[, ]
′ () = (), ∈ [, ]
Φ
( ) = Φ() − Φ()
Φ( )
≡ Φ() − Φ()

: [, ] →
: [, ] →
() =
( ) , ∈ [, ]
[, ]
′ () = (), ∈ [, ]
Φ
( ) = Φ() − Φ()
Φ( )
≡ Φ() − Φ()

Φ Ψ
Φ = Ψ + .
( ) = Ψ()−Ψ() = (Φ()+)−(Φ()+) = Φ()−Φ().

( ) = −
− +
− +
( − + − ) = ( ) − ()
= (−
− +
( ) − +
= −
+
( ) − +
.
( − + − )
) − (− +
)

( ) = −
− +
− +
( − + − ) = ( ) − ()
= (−
− +
( ) − +
= −
+
( ) − +
.
( − + − )
) − (− +
)

() ′ = ′ + ′ = + ,
+
( ( )( ) + ( )( )) = ( )( ) + .
( ( )( ) + ( )( )) = ( )( )
.

, : [, ] →
= ′
( )( ) = ( )( ) − ( )( ) ,
( )( ) = ( )( ) −
( )( ) .

′ = −
′ .

( ) = , ( ) =
,
( ) = , ( ) = ′ ( ) =
.

=
−
( )
=
( )
=
( ) ( )
−
=
=
−
+ .

· .
( ) = , ( ) = ,
( ) = , ( ) = ′ ( ) =
,

· =
( ) ( )
−
·
( )
( )
= − − ( − ) = .

′ = ′ =
[ (( ))] ′ = ′ (( )) ′ ( ) = (( ))( ).
◦ ( ◦ )
(( ))( ) = (( )) + .
= ( )
() = () + .

(( ))( ) = () = ( ),
()
(( ))( ) = ().
()

, : [, ] →
= ( )
(( )) ′
( ) = ().
(( )) ′ ()
( ) = ().
()

(( )) ′ ( )
√ +

( ) = ( + ) () =
√
√ +
= (( )) ′ ( ),
() =
√ =
+
= + .
√ = √ =

= + ,
=
( + ) =
√ =
+
√ ,
= ( ) = +

√
−

√
−
= , =
= ,

√ − = − = √ =
√
− =
= +
√ = +
−
√
− =
= π

√ − = − = √ =
√
− =
= +
√ = +
−
√
− =
= π

√ − = − = √ =
√
− =
= +
√ = +
−
√
− =
= π

( )

( )
= < > . . . < + < . . . = .
[−, ] = , . . . ,
| ( ) − ( )| ∈ [−, ]
( ) =
=
−
( ) ≈
( ) .
=
−

( )

( )
−
=
−
,
= +
= + , = , . . . , .

( )
[−, ]
= − +
,
( ) = (), ∈ [−, ],
= , . . . ,
−
( ) ≈
().
=

( )

( )
[−, ]
( ) = (−)+
= , . . . ,
( ) ≈
=
=
( ()− (−))( − −), ∈ [−, ],
( (−) +
=
−
( (−) + ()).
( () − (−))( − −))

( )
(−, ) (, ) (, ()) (−, (−))
−
( ) ≈
(
−
() + () + ()).
=

( )
( )

( )
( )

∈
ε ( ( ), ε) ( ) δ
( , δ) ( ) ∈ ( ( ), ε) ∈ ( , δ)

: →
( , δ)
δ

= ( , )
ε > δ > = ( , )
( − ) + ( − ) < δ
| ( ) − ( )| < ε

ϕ: → ψ : →
, : × →
( , ) = ϕ( ) + ψ() ( , ) = ϕ( )ψ()

( , ) = √ →
→ √

: → >
∗ = ( ∗ , . . . , ∗ )
∗

( ∗, . . . , ∗ , , ∗
− + , . . . , ∗ )
( ) = ( ∗ ∗ ∗
, . . . , , , − + , . . . , ∗ ), ∈ .

∗

: →
∗ ∈
→
( ∗
( ( ∗
, . . . , ∗
−
, . . . , ∗
−
, ∗
, ∗
, ∗
+ , . . . , ∗ ))
+ , ∗
+ , . . . , ∗ ) −
∗ ∂
( ∂
∗ )
( ∗ )
( ∗ )
= , . . . ,

( , , ) = ( + ) ( + )

( , , ) ∈
∂
∂
∂
∂ ( , , ) = ( + ) + ( + )
∂
∂ ( , , ) = ( + ),
∂
∂ ( , , ) = (
+ )
+
.
+
,

=
∂
( , )
∇ ( , ) = ∂
( , )
∂
∂
( , )

( , )
−∇ ( , )

∂
∂
( ∂
) =
∂
∂ ∂∂
= .

( , ) = +
∂
∂ = +
∂
∂ = .
∂
∂
= + ,
∂
∂∂ = ,
∂
∂ ∂ = ,
∂
∂
= .

= , = ,
= , = ,
= , = ,
= , =

= , = ,
= , = ,
= , = ,
= , =

: →
= ,
= = = = =

∂
( ∂
∗ )
− ∗
= , . . . ,
( ∗ + , ∗ + , . . . , ∗ +) ≈ ( ∗ , ∗ , . . . , ∗ )+
=
∂
∂
( ∗ , ∗ , . . .

=
( ∗ + , ∗ + ) ≈ ( ∗ , ∗ ) + ( ∗ , ∗ ) + ( ∗ , ∗ ).

(. . )

( , ) = ( ∗ , ∗ ) = (, )
(, )
(, ) = ,
( , ) =
( , ) =
(, ) = ,
(, ) = .
(., .) ≈ (, ) + (, )(. − ) + (, )(. − )
= − · . + · . = ..

(. . ) = . . . .
( , ) = − = =
( , ) = (, ) ≡ (, )
(, ) = (, ) =

(. . ) = . . . .
( , ) = − = =
( , ) = (, ) ≡ (, )
(, ) = (, ) =

(. . ) = . . . .
( , ) = − = =
( , ) = (, ) ≡ (, )
(, ) = (, ) =

: →
∗
∗
∈ ∗
( ) ≤ ( ∗ ) ( ) ≥ ( ∗ ).

¡¡
( ∗ ∗ ∗
, . . . , , , − + , . . . , ∗ )
∈ ( ∗ − δ, ∗ + δ) = . . . ,
δ >
( ) = ( ∗ ∗ ∗
, . . . , , , − + , . . . , ∗ ∗ ∗
), ∈ ( − δ, + δ)
∗

∗
∗ ∂
( ) = (
∂
∗ ) = , = , . . . ,

: → ∗
∗
∂
( ∗ ) = , = , . . . , .
∂

⊂ ∗

: →
∗
( ∗ ) = ( ∗ )( ∗ ) − ( ∗ ) ,
( ∗ ) > ( ∗ ) > ∗
( ∗ ) > ( ∗ ) < ∗
( ∗ ) < ∗
( ∗ ) =

∗ ∗

( , ) = − (, )
( , ) = ( , −)
( , ) = (, )
( , ) = (, )
= = − =
(, ) = −

1
0.5
0
−0.5
−1
1
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1

( , ) = −

( , ) = + − =
= −
˜ () = −
=
= ±
( , ) = (, ) ( , ) = (−, )
(±, ) =

1.5
1
0.5
0
−0.5
−1
−1.5
c = 1
c = 0.6
c = 1.4
c = 0.2
∇ g
∇ g
−1 −0.5 0 0.5 1 1.5 2 2.5 3
∇ f
∇ f

{( , ): ( , ) = − = },
∈
(, )

, : →
( ) =
∇( ) = λ
∇ ( ) + λ ∇( ) =
λ

L( , λ) = ( ) + λ ( ),
( , λ )

λ = ( , λ )

∂L
∂
∂L
∂
∂L
∂λ
∂
=
∂
∂
=
∂
+ λ ∂
∂
+ λ∂
∂
= + λ =
=
=

( ) =

L( , , λ) = − + λ( + − )
∂L
= + λ = ( + λ) =
∂
∂L
= − + λ = (− + λ) =
∂
∂L
∂λ = + − .

= =
= λ = −
= =
= ± =
λ = =
= ±
( , ) = (±, ) λ = − ( , ) = (, ±)
λ = ∇( , ) =
(±, ) = (, ±) = −
(±, ) (, ±)

= (−), ∈ .
( − )
−
−

+ = +
∈
() >
>
∈ >
→∞( + )
= + =
→∞ →∞
= ± √
= →∞
+
→∞
= +
,

= √
> >
> ∈
+
− = (
+
) − =
(
− )
≥ .

≥ , ≥
+ =
+
≤
= , ≥ .
> < ≤
≥

≥ , ≥
+ =
+
≤
= , ≥ .
> < ≤
≥

− / %
− /
= − − − + = ( −
)− + = − + ,
= ( −
).

.

= + ,
= + = ( + ) + = + + ,
= + = ( + + ) + = + + + ,
= + =
= ( + + + ) = + + + + ,
= + ( − + − + · · · + + ) = +
−
− .

→∞ =
− = ∗ .
= ( − ∗ ) + ∗ ,
() < ∗ ∗
> ∗ ∗
∗

=
= −
= .

= .− + .
∗ =
= /.
.
=
= − · . + .
∗ = /
±. / = . /
= . /

= /

= /.
= /
= , = , = ..

= − = .
= ∗ =
−
= ( − ) = · . = . /.
= .

= , =
=
.
= = .
.
=
= .

()
:=
=
()
∞ =
∞ =
= →∞

∞
=

∞ =
() (−)
− − =
=
−
−
=
=
→∞ =
→∞ ( − −) = .

∞
=
→∞ = .

∞
= →∞ =

∞ || <
∞
=
=
− .
=

=
=
=
= + −
− .
|| <
→∞ = − =
−

∞
=

= =
15
14
13
12
11
10
9
8
7
0 100 200 300 400 500 600 700 800 900 1000

= : = ,
= : = +
,
= : = +
= : = +
+
+ > + + (
+ ) = +
,
+ + + > + + ( + + +
)
= +

+
.
→∞ = ∞.

( )
()

( )
()

∞
= ∈
ρ =
→∞
||.
ρ < ∞
=
ρ > ∞
=
ρ =

∈ ∞
=

=
ρ =
→∞
| | =
| |
| | =
√ .
| | =
→∞
| |
√ =
| |
→∞ √
= | |.
| | <
| | > | | =

∞
= ∈
ρ =
→∞
|+|
.
||
ρ < ∞
=
ρ > ∞
=
ρ =

|+|
| | = | |+ | |
= | |,
( + ) +
ρ = →∞ = |+| = | |
| |

ρ

( +
)
∈ = ∞
=
!

∞
= (−)+ ≥ ∈
→∞ =
+ ≤ ∈

∞
= (−)+ ≥ ∈
→∞ =
+ ≤ ∈

∞
(−)+ =
→∞
= +

∞ = ∞ = λ ∈
∞ = λ ∞ = ±
∞
= λ λ ∞
=
∞
= ± = ∞
= ± ∞
=

∞
( = −
!
+
)

∞
= ∞
=
∞
=
!
+ !
=
|+|
| |
(+)!
→∞
=
+ →

∞
=
∞
= ||
∞
=

∞ =

∞ =

∞ = ∈
∞ =

+
() = ((−) )
(
) ( )
+

= −
− − − + − − − − + − . . .

= −
− − − + − − − − + − . . .

∞
= | | <
−
→
−
(−, ) ∞
=

( ) ⊂ ∈ ∞
= ( − )

∞
= ( − ) =

∞ =
!
∈
| | +
( + )!
! | |
· =
| | + →
→∞ .

∞
= =
| |
=
= | | →
→∞ ∞ = .

∞ = (−)− ( −)
| − | >
| − | <
| − |
= | − |
√ →
→∞ | − |.
=
=

∞
= ( − )
∈
>
| − | < | − | >
| − | =
( , ) = { ∈ : | − | < }
( − , + )

ρ = →∞ | +|
| |
ρ = →∞ | |
=
ρ
ρ >
= ∞ ρ =
= ρ = ∞

∞ = ( − )
>
(, ) →
:
↦→ ∞
= ( − ) ,

∞ = ( − )
>
(, ) →
:
↦→ ∞
= ( − ) ,

( ) =
=
=
− +
=
+
−
=
∞
=
(−) ( −
| − | < | |
=
)

( ) =
+
( ) =
∞
(−) , | | <
=
(−, )
( ) =
+ =

() →∞ =
∞
( − ) =
=
∞
( − )
=
∈
= ∈ .

() →∞ =
∞
( − ) =
=
∞
( − )
=
∈
= ∈ .

=
=
∞
, ∈
!
=
∞
(−) +
, ∈
( + )!
=
∞
= (−) , ∈
()!
=
( − ) = −
=
∞
=
, − ≤ <
∞
+
(−) , | | ≤
+
=

→
−
→
→
−
=
=
=

−
=
( + + + + · · · − ) = ( + + . . . )
! ! ! !
= +
+ + . . . →
! ! → .
= ( − + ∓ . . . )
! !
= − + ∓ . . . →
! ! →
=
( − + ∓ · · · − ) = (− + ∓ . . . )
! ! ! !
= −
+ ∓ . . . →
! ! → .
−

( − , + )
( ) =
∞
( − ) , ′ ( ) =
=
∞
( − ) − .
=

∞

=
=
∞
=
∞
=
!
−
! =
∞
=
−
( − )! =
∞
=
!
=

=
=
∞
=
(−)
∞
(−)
=
+
( + )!
( + )
( + )! =
∞
(−)
=
= .
()!

∞
=

≥
( ) = = +
+
= , . . . ,
( ) = ( ),
() ( ) = () ( ), = , . . . , .

= ( ),
= () () , = , . . . , ,
!
T( ) =
() () ( − ) !
.
=

() ( ) = ( )
T =
( ) − T( )

= (, )
+ ∈ T
=
∈ ξ
( ) = T( ) + R+( ),
R+( ) = (+) (ξ)
( + )! ( − ) + .
R

ξ
( )
T( )
[, π
]
=

1.5
1
0.5
0
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
x − x 3 /6
x
sin x
x−x 3 /6+ x 5 /120

() = = , ′ () = =
′′ () = − = , () () = − = −
() () = = , () () = =
T ( ) = , T ( ) = T ( ),
T( ) = −
! , T( ) = T( )
T( ) = − +
! !

∞

→∞ R( ) =
∈ (, )
( ) = →∞ T( ) + →∞ R( ) = →∞ T( ) =
→∞ =
() ()
( −
!
)

( , )
( ) =
∞ () () ( − ) !
=
=

( ) = λ
=

′ ( ) = λ λ , ′′ ( ) = λ λ , ′′′ ( ) = λ λ , . . .

λ
() ( ) = λ λ , () () = λ .
=
λ =
∞ () ()
=
!
=
∞
=
λ
! .

λ
() ( ) = λ λ , () () = λ .
=
λ =
∞ () ()
=
!
=
∞
=
λ
! .