1 ȵɿʍɲɶʘɶɼ ʍʏɿʎ ɀɸʌɿʃɹʎ ȴɿɲʔʉʌɿʃɹʎ ȵʇɿʍʙʍɸɿʎ Ʌɸʌɿɸʖʊʅɸʆɲ ͻ ϭϭ ȳɸʆɿʃɹʎ ȶʆʆʉɿɸʎ ϭϭϭ Ʌʌʉɴʄɼʅɲʏɲ ͻ ϭϮ ɀɹɽʉɷʉɿ ȵʋʀʄʐʍɻʎ ɀɸʌɿʃʙʆ ȴɿɲʔʉʌɿʃʙʆ ȵʇɿʍʙʍɸʘʆ ͻ ϭϯ Ⱦɲʄɳ Ɉʉʋʉɽɸʏɻʅɹʆɲ Ʌʌʉɴʄɼʅɲʏɲ ȵʐʍʏɳɽɸɿɲ ͻ ϭϰ Ɉɲʇɿʆʊʅɻʍɻ ɀɸʌɿʃʙʆ ȴɿɲʔʉʌɿʃʙʆ ȵʇɿʍʙʍɸʘʆ ȴɸʑʏɸʌɻʎ Ɉɳʇɻʎ ϭϰϭ ȴʑʉ ȴɿɲʍʏɳʍɸɿʎ ϭϰϮ Ͳ ȴɿɲʍʏɳʍɸɿʎ > ϭϰϯ Ʌʌʉɴʄɼʅɲʏɲ ͻ ϭϱ ȳʌɲʅʅɿʃɹʎ Ϯɻʎ Ɉɳʇɻʎ ʅɸ ɇʏɲɽɸʌʉʑʎ ɇʐʆʏɸʄɸʍʏɹʎ ϭϱϭ ȳɸʆɿʃɼ Ⱥɸʘʌʀɲ ϭϱϮ ȸ ȿʑʍɻ ůĞŵďĞƌƚ ʏɻʎ Ⱦʐʅɲʏɿʃɼʎ ȵʇʀʍʘʍɻʎ ϭϱϯ Ʌʌʉɴʄɼʅɲʏɲ ͻ ϭϲ ʋʀʄʐʍɻ ɀɸʌɿʃʙʆ ȴɿɲʔʉʌɿʃʙʆ ȵʇɿʍʙʍɸʘʆ ʅɸ ɀɸɽʊɷʉʐʎ ɇʐʆɼɽʘʆ ȴɿɲͲ ʔʉʌɿʃʙʆ ȵʇɿʍʙʍɸʘʆ ϭϲϭ Ʌʌʉɴʄɼʅɲʏɲ ͻ ϭϳ Ȳɿɴʄɿʉɶʌɲʔɿʃɹʎ Ȱʆɲʔʉʌɹʎ ϯ
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1
>
( ) = ∇ ( ) = , ∈ ⊆ R , ≥ ,
∇ ( ) + ( ) = , ∈ ⊆ R , ≥( ) = − ( ), ∈ ⊆ R , ≥ ,
( , ) = − ( , ), > , ∈ ⊆ R, >
( , , ) = − ( ( , , ) + ( , , )), > , ( , ) ∈ ⊆ R , > ,
+ + + + = , ( , , ) ∈ ( , ) × ( , ) ×(− , ),
R , ≥ ∂
( ) ( , ) + ( , ) = , ( , ) ∈ R ,
( ) ( , ) ( , ) + ( , ) = , ( , ) ∈ R ,
( ) ( , )− ( , ) ( , ) = , ( , ) ∈ R ,
( ) ( , ) + ( , )− ( , ) = ( , ), ( , ) ∈ R ,
( ) ( , ) + ( , ) + ( , ) = sin , ( , ) ∈ R ,
( ) ( , ) + ( , ) + log = , ( , ) ∈ R ,
( ) ( , ) + ( , ) + cosh = , ( , ) ∈ R ,
( ) ( , ) = sin sinh , ( , ) + ( , ) = , ( , ) ∈ R , ,
( ) ( , ) = sin − , ( , )− ( , ) = , ( , ) ∈ R× R+, ,
( ) ( , ) = ( ) + ( ), ( , ) = , ( , ) ∈ R ,
, ,
( ) ( , ) = ( + ) + ( − ), ( , )− ( , ) = , ( , ) ∈ R ,
, ,
( ) ( , ) = − cos( − ), ( , )− ( , ) = , ( , ) ∈ R× R+.
( − )
( , ) ( , ) < < ∞=
( , ) = ( , ), ∈ , > ,
( , ) = ( ), ( , ) = ( ), ∈ ,
× ( ,∞) ( , ) ( ), ( )
( , ) = , ( , ) ∈ ,
( , ) = ( ), ( , ) = ( ), ( , ) = ( ), ( , ) = ( ),
( , ) ∈ =: {( , ) ∈ R : < < , < < } ( ), ( ) ( )( ) ,
( , ) = ( , ) =( , ) = ( , )
′ ( ) = ′ ( ) ( ), ( )
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( , ) + ( , ) = , ( , ) ∈ ,
( , ) = , ( , ) =sin
,
= {( , ) ∈ R : ∈ R, > }
( , ) =(sinh )(sin )
→ ∞ ( , )→ ∞
sinh
( , ) ∈ R , , , , , ,
( , ) + ( , ) + ( , ) + ( , ) = ( , ), ,
( , ) + ( , ) + ( , ) + ( , ) = ( , ), ,
( , ) + ( , ) + ( , ) + ( , ) + ( , ) = ( , ), .
( , ) = + ( , ),
, �=, =
( , ) = , ∈ R, ∈ ( , / ), ( , ) = , ( , / ) = .
/ = = .
= + ( ) ( , ) = − + ( ) + ( ),
(R) ( , ) =
− + ( ) = ( ) = ( , / ) = ( ) +
( ) = ( ) = −
( , ) = − − + .
2
( , ) + ( , ) = , ( , ) ∈ ⊂ R ,
( , , ) + ( , , ) + ( , , ) = , ( , , ) ∈ ⊂ R ,
R =,
( ) = ∇ ( ) = , ∈ ⊂ R ,
,
,
( ), = , , , ... , ∈⊂ R , ≥
( ) = , ∈ ⊂ R , ≥
( ) =∑∞
= ( ), , , ...
( ), = , , , ..., , ∈⊂ R , ≥
= , = , , , ... , =∑
=
=∑
=
( ) = ∇ ( ) =
( , ) =∞∑=
sin[( − ) ] sinh[( − ) ]
( − ) sinh[( − ) ], < < , < < .
( , ) + ( , ) = , < < , < < ,
( , ) = , ( , ) = , < < , ( )
( , ) = , ( , ) = sin , < < .
( , ) = ( ) ( )( )
′′( ) + ( ) = , < < , ( ) = , ( ) = , ( )′′( )− ( ) = , < < , ( ) = , ( )
( )
( ) = cos + sin , = .
( )
= , ( ) = sin( ), < < , = , , , ... . ( )
= ( )( )
( ) = sinh( ), < < , = , , , ... . ( )
( ) ( )( )
( , ) =∞∑=
sin( ) sinh( ), < < , < < ,
( , ) = sin =
∞∑=
sin( ) sinh( ), < < .
= sinh( ) = =/ sinh( ) = , , , ... ,
=sinh
∫sin sin( ) =
∫{cos( − ) − cos( + ) } = .
( )
( , ) =sinh
sinh( ), < < , < < .
=, = = , =
( , ) + ( , ) = , < < , < < ,
( , ) = , ( , ) = , < < ,
( , ) = cos , ( , ) = cos , < < ,
′′( ) + ( ) = , < < , ′( ) = , ′( ) = ,′′( )− ( ) = , < < .
= , ( ) = , < < ,
= , ( ) = cos , = , , , ... , < < .
= , = , , , ... ,
( ) =
{+ , = , < < ,cosh + sinh , = , , , ... , < < .
( , ) = + +
∞∑=
[ cosh + sinh ] cos ,
< < , < < =
( , ) = cos = +∞∑=
cos( ), < < .
= = , = = , = , , , ... .
( , ) = + cosh cos +
∞∑=
sinh cos ,
< < , < <=
cos = + cosh cos +∞∑=
sinh cos( ),
+ cos − cosh cos = +
∞∑=
sinh cos( ).
=
= , = − cosh
sinh, =
sinh, = , = , , , ... .
< < , < <
( , ) = +
[cosh − cosh sinh
sinh
]cos +
sinh
sinhcos
=
[+
sinh( − )
sinhcos +
sinh
sinhcos
].
=
( , ) + ( , ) = , < < , < < ,
( , ) = , ( , ) = , < < , ( )
( , ) = , ( , ) = , < < .
( , ) = ( ) ( )
′′( ) + ( ) = , < < , ′( ) = , ′( ) = , ( )′′( )− ( ) = , < < , ′( ) = , ( )
( )
( ) = cos + sin , < < , = .
( )
= , ( ) = cos( ), < < , = , , , ... . ( )
( ) =( )
( ) =
{+ , = , < < ,cosh + sinh , = , , , ... , < < . ( )
( ) ( )( )
( , ) = + +
∞∑=
( )[ cosh( ) + sinh( )],
< < , < < ,
( , ) = = +∞∑=
cos( ), < < .
= , =
∫cos( ) =
(− ), = , , , ... .
( , ) = = +
∞∑=
cos( )[(− )
sinh( ) + cosh( )], < < ,
= = − (− ) sinh( )
cosh( ).
3
( , ) = ( , ) + ( , ), ∈ ⊆ R , > ,
( , )
( , ) = ( , )− ( )∇ ( , ), ∈ ⊂ R , > ,
( , )( )∇ ( , )
∂ ( , , , ) ( , , )
q( , , , )( , , )
n( , , )
( , ) + ( , ) = , − < < , < < ,
(− , ) = , ( , ) = , > ,
( , ) = sin
[( + )
]+ + sin( ( + )), − < < ,
−1 −0.5 0 0.5 1−1
0
1
2
3
4
5
x−axis
u−a
xis
diffusion
−1−0.5
00.5
1
0
0.5
1
x 10−3
0
1
2
3
4
5
x t
u
( , ) + ( , ) = , − < < , < < ,
(− , ) = , ( , ) = , > ,
( , ) = cos
[( + )
]+ + cos( ( + )), − < < .
−1 −0.5 0 0.5 1−1
0
1
2
3
4
5
x−axis
u−a
xis
diffusion
R
∂ = × [ , ] ∂∂ = ∂ × ( , )∪ ×{ }∪ ×{ }
( , )
( , )− ( , ) < , ( , ) ∈ ,
( , )∂ × ( , ) ∪ × { }
( , )( , ) ≤
( , ) =( , ) ×{ }
( , ) ≤ ( , ) ≥♦
( , )
( , )− ( , ) ≤ , ( , ) ∈ .
( , ) ≤ , ( , ) ∈ ∂ × ( , )∪ ×{ } ( , ) ≤ ( , ) ∈
( , )− ( , ) = ( , ), ( , ) ∈ ,
( , ) = ( ), ∈ , ( , ) = ( , ), ∈ ∂ , ∈ ( , ).
{ , , }, { , , }≤ , ≤ , ≤
{ , , } { , , }
{ , , }{ , , }
( , ) ≤ ( , ), ( , ) ∈( , ) = ( , )− ( , )
( , ) − ( , ) ≥ ( , ) ∈( , ) ≥ , ( , ) ∈ ∂ × ( , ) ∪ × { }
( , ) ≥ ( , ) ∈♦
( , ), ( , ){ , , }
{ , , }( , ) ≤ ( , ), ( , ) ≥ ( , ), ( , ) ∈
♦
( , ), ( , ){ , , }, { , , }
| ( , )− ( , )| ≤ , ( , ) ∈ ,
| ( , )− ( , )| ≤ , ( , ) ∈ ∂ × ( , ) ∪ × { }.
| ( , )− ( , )| ≤ + , ( , ) ∈ .
( , ) = ( , ) − ( , ){− ,− ,− }
{ , , } ( , ) = ( , ) + ( , ) ( , )( , )
( , )− ( , ) + ( ) ( , ) = , ( , ) ∈ , ( ) ≥R = × [ , ]
( , )− ( , ) + ( ) ( , ) = , ( , ) ∈ ,
( , ) = ( ), ∈ , ( , ) = , ∈ ∂ , ∈ ( , ).
( )( , ) ≡ →∞
R , ≥ = × ( ,+∞)
( ) ( , )−∇ · ( ( )∇ ( , )) = , ( , ) ∈ ,
( ) ≥ , ( ) ≥( , ) = ( , ) + ε
( ) ( , ) − ∇ · ( ( )∇ ( , )) < ( , )= >
( , ) = ( , ) + ( , ), −∞ < < , > ,
, >
R , ≥ = × ( ,+∞)
( ) ( , )−∇ · ( ( )∇ ( , )) = ( ) ( ), ( , ) ∈ ,
( , ) = ( ), ∈ , ( , ) = , ∈ ∂ , ∈ ×( ,+∞).
( ) ≥ , ( ) ≥
( , ) = ( , ) − ( , )
( , ) =
( , ) = ( , )
( , )
( ) = ( − )
( , ) =∑∞ [ − (− ) ] exp[−( / ) ] cos .
( , )= ( , ) ==
( , ) =( , ) =
( , )
( , ) = ( , ), < < , > .
= ( ( , ) = )=
( , ) = , > ,
, >( , ) ( ( , ) = )
( , ) = ( − )− [( ) − ( ) + ] +∑∞
=exp[− / ] sin .
4
( , ) = ( , ) + , < < , > ,
( , ) = , ( , ) = + , > ,
( , ) = , ( , ) = , < < .
( , ) = ( , )− , < < , > ,
( , ) = / , ( , ) = − cos , > ,
( , ) = − , ( , ) = , < < .
( , ) = ( − ) − cos +∑∞
=(− ) − [ −cos − (cos −cos )
− ] sin( ).
( , ) = ( , ), < < , > ,
( , ) = , ( , ) = cos , > ,
( , ) = , ( , ) = , < < .
( , ) = ( , ), < < , > ,
( , ) = sin , ( , ) = , > ,
( , ) = , ( , ) = − , < < .
( , ) = ( − ) sin +∑∞
=[sin( )− sin ]
( − )sin( ).
( , ) = ( , ), < < , > ,
( , ) = , ( , ) = , > ,
( , ) = sin , ( , ) = + , < < .
( , ) = − ( , )− ( , ), < < , > ,
≥
=, = , , , ... .
( , ) =∞∑=
[sin
(+
√+
)
+ sin(
−√
+)]
,
< < , > .
=
( , ) + ( , ) = ( , ), < < , > ,
= (√
+ ) / ,
= (√
+ ) / ,
= /
/
>> = =√= . × , =
( , )
( , )
( , )
( , ) = ( ) ( ) ( , ),
( )( )
( ), ( ) ( , )( ( , ) ≡ ( ))
( ) ∂ ( , )
∂+
∂ ( , )
∂=
( , ).
( , ) = ,∂ ( , )
∂= ,
, , ) = ,∂ ( , )
∂= ,
∂ ( , )
∂,
∂ ( , )
∂= .
( , ) = ( ), ( , ) = ( ), < < .
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Απλή Υποστήριξη
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Πακτωμένο Οριζοντίως
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Ελεύθερο Σύνορο
( , )
( ) ≡
( , ) = − ( , ), < < , > ,
( , ) = , ( , ) = , ( , ) = , ( , ) = , > ,
( , ) = ( ), ( , ) = ( ), < < .
( , ) = ( ) ( ) = ( ) ( sin + cos ) ,
< < , > , >
( )( )− ( ) = , < < ,
( ) = cos√
+ sin√
+ cos√
+ sin√
,
< < , .....,
= = sin√
= , sin√
= .
= sin√
= .
= ( ) , = , , , ... ,
5
∇ ( , ) = − ( , ),
∇ ( , ) = − ( , ),
∇ ( ) = ,
∇ ( ) = − ( ),
∈� , = , >
= ( , ) × ( , )= , = , = , =
( = )( , )
, = , , , ...
( , , ) = , < < , < <, ( , , ) = , − < < , < < ( , ) =
( )
∂
∂
(∂
∂
)+
sin
∂
∂
(sin
∂
∂
)= , < < , < < ,
( , ) = ( ), < < .
( , , )
( , , ) = ( , )( , , )
( , ) = ( ) ( )
′′( ) + ′( )− ( ) = , < < ,
sin
(sin ′( )
)+ ( ) = , < < .
= cos
[( − )
( )]+ ( ) = , − < < ,
= ( + ), = , , , , ... ,
( ) = ( ) = (cos ), = , , , , ... .
= ( + )
′′( ) + ′( )− ( + ) ( ) = , < < ,
( ) = + −( + ), < < , = , , , , ... .
( ) =
( ) = , < < , = , , , ... .
( , ) =
∞∑=
(cos ), < < , < < ,
( , ) = ( ) =
∞∑=
(cos ), < < .
( )
=
(+
)∫( ) (cos ) sin , = , , , , ... .
=: + ( ),
= − �∇ = �
( )
( , ) = ( , , )=∫ ∫ ∫| ( , )| .
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞| ( , )| = .
( , )
� ( , ) = ( , ) =−� � ( , ) + ( ) ( , ), ∈ R , > .
( )( , )
( , ) = ( ) ( ), ∈ R , > .
( )+
�( ) = , > ,
−� � ( ) = ( ), ∈ R ,
( ) = exp
(−�
), > ,
( )
( , ) =∑
exp
(−�
)( ), > ,
( ) = ( ), =√+ + , ( , , ) ∈ R
∇ ( ) + [ − ( )] ( ) = , > ,
=�
∇
( , , ), ≤ < +∞, < < , − < <
[∂
∂
(∂
∂
)+
sin
∂
∂
(sin
∂
∂
)+
sin
∂
∂
]= −
�[ − ( )] .
( , , )( , , )
( , , )
| ( , , )| → + → +∞,
( ,− , ) = ( , , ), ( ,− , ) = ( , , ).
( , , ) = ( ) ( ) ( ), < < , < < , − < < .
( , , )
′′( ) + ( ) = , − < < , (− ) = ( ), ′(− ) = ′( ),
sin ′′( ) + sin cos ′( ) + ( − ) ( ) = , < < ,′′( ) + ′( ) + [ ( − ) − ] ( ) = , < < .
( ) =√
+ − √ , − < < .
, == , = ,± ,± ,± , ... .
( ) = , − < < ,
�
= cos
( − ) ′′( )− ′( ) +
(− −
)( ) = , − < < ,
=
( − ) ′′( )− ′( ) + ( ) = , − < < .
( ) = ( + ) ∈ N
=
( )
= ( + ), ∈ N
( ) = ( − ) / ( ), − < < .
( )> <
− ( ) = (− )( − )!
( + )!( ), − < < .
− ( ), ( )
( ) | | ≤
( + )= − , ...,− , , , ..., .
≥
( , ) =
[+ ( − )!
( + )!
] /
(− ) (cos ),
6
(−∞,+∞)
( = ) ( ,+∞) (−∞, )
( ) R
( )(− , ) ∈ R
+
( ) (− , )( ) R
( ) ∼∫ +∞ [∫ +∞
−∞( ) cos ( − )
].
cos ( − )
( ) ∼∫ +∞ [∫ +∞
−∞( ) cos
]cos
+
∫ +∞ [∫ +∞
−∞( ) sin
]sin .
( )R R
+
( ) ∼∫ +∞
( ) cos ,
( ) =∫ +∞
( ) cos( ) R
R+
( ) ∼∫ +∞
( ) sin ,
( ) =∫ +∞
( ) sin( )
( , ) = − ( , ), < <∞, > ,
( , ) = , ( , ), ( , ) →∞, > ,
( , ) = ( ), < <∞,
( )∫∞−∞ | ( )| < ∞
( , ) = − ( , ), < <∞, > ,
( , ) = , ( , ), ( , ) →∞, > ,
( , ) = sin , ( , ) = cos , < <∞,
( , ) = − ( , ), −∞ < <∞, > ,
( , ), ( , ) | | → ∞, > ,
( , ) = −| |, ( , ) = , −∞ < <∞,
( − )
≥≤ ≤ −∞ ≤ ≤ +∞
L{ ( , )| → } =( , ) >
L{ ( , )| → } = ( , )− ( , )− ( , ).
( , ), ( , ) =( , ), ( , )
=
( , )
L{ ( , )| → } =∫ ∞
− ( , ) = ( , ).
( , )
L{ ( , )| → } = ( , )− ( , ),
L{ ( , )| → } =
∫ ∞− ( , ) =
∂
∂
∫ ∞− ( , ) = ( , ),
L{ ( , )| → } = ( , ).
L{ ( , )| → } =∫ ∞
− ( , ) = ( , ).
( , ) = ( , ), < <∞, > ,
( , ) = ( ), ( , )→ , →∞, > ,
( , ) = , < <∞.
L{ ( , )| → } = ( , ), L{ ( )} = ( )
( , )− ( , ) = , < <∞,
( , ) = ( ), ( , )→ , →∞.
( , ) = ( ) −√
+ ( )√,
, , >
( , ) = − [ − ( /√
)]
( )R ( ) := F [ ( )]
( ) R
( ) = F [ ( )] := √∫ +∞
−∞( ) .
( ) := F− [ ( )]( ) R
( ) = F− [ ( )] := √∫ +∞
−∞( ) − .
( ) R+
R
( ) := F [ ( )]
( ) := F [ ( )] :=
√ ∫ +∞( ) sin( ) .
( ) := F− [ ( )]( ) R
+
( ) = F− [ ( )] :=
√ ∫ +∞( ) sin( ) .
( ) R+
R
( ) := F [ ( )]
( ) := F [ ( )] :=
√ ∫ +∞( ) cos( ) .
( ) := F− [ ( )]( ) R
+
( ) = F− [ ( )] :=
√ ∫ +∞( ) cos( ) .
( ) ( )>
= , −∞ < <∞, > ,
( , ) → , ( , )→ , | | → ∞, > ,
( , ) = ( ), ( , ) = ( ), −∞ < <∞.
∈ R
F{ ( , ); → } = ( , ), F{ ( ); → } = ( ), F{ ( ); →} = ( ), ∈ R
( , ) + ( , ) = , > , ∈ R,
( , ) = ( ), ( , ) = ( ), ∈ R.
( , ) = ( ) ( ) +( )
( ), ∈ R, > .
( , ) = √∫ +∞
−∞−
[( ) ( ) +
( )( )
],
−∞ < < ∞, > .
( , ) = + √∫ +∞
−∞
[− ( − ) + − ( + )
]( )
+ √∫ +∞
−∞
[− ( − ) − − ( + )
] ( ),
−∞ < < ∞, > .( ), ( )
( , ) = [ ( − ) + ( + )] +
∫ +
−( ) , −∞ < <∞, > .
R
( , ) + ( , ) = , < <∞, < < ,
( , ) = sinh , ( , ) = , < <∞,
( , ) = , ( , )→ , | | → +∞.
( , ) + ( , ) = , −∞ < <∞, < ,
( , ) , + → +∞,
( , ) =
{, | | < ,, | | > .
( , ) ={arctan
(−
)+ arctan
(+
)}
(( )
)+
(−
)( ) = ,
> , ≥→ +
( ) = ( ), > , ≥ , ∈ R.
( )→ ∞
( ) ≈√
cos(− −
), > , ≥ , ∈ R,
( )/√
( )√
( )( ,∞) ( )
( +) + ( −)=
∫ ∞( ) ( ) , > , ≥ ,
( ) =
∫ ∞( ) ( ) , ≥ , ∈ R.
( ) ( )
H [ ( )] = ( ) =
∫ ∞( ) ( ) , ≥ , ∈ R.
( ) ( )
H− [ ( )] = ( ) =
∫ ∞( ) ( ) , > , ≥ .
√( )
( )
( ) =−
, = , ( ) = − , = ,
( ) =
{, < < ,, > ,
= .
H [ ( )] = ( ) =
∫ ∞ −( ) = √
+,
H [ ( )] = ( ) =
∫ ∞− ( ) =
( + ) /,
H [ ( )] = ( ) =
∫ ∞( ) ( ) =
∫+ ( )
( = ) =
∫ ( ) +( ) = +
∫+ ( )
= +
∫ [+
+ ( )]
= ++ ( ).
≥
H [ ( ) + ( )] = H [ ( )] + H [ ( )],
, ≥
∫ ∞( ) ( ) =
∫ ∞( ) ( ) ,
( ), ( ) ≥( ), ( )
( ) → → → ∞
H [ ′( )] = [( − ) + ( )− ( + ) − ( )] , ∈ N
( ) → , ( ) → →→∞
H[ (
( ))− ( )
]= − ( ), ∈ N.
∫ ∞( ) ( ) =
∫ ∞( )
∫ ∞( ) ( )
=
∫ ∞( )
∫ ∞( ) ( ) =
∫ ∞( ) ( ) .
H { (( )
)− ( )}
=
∫ ∞(
( )) ( ) −
∫ ∞( ) ( )
=( )
) ( )}∞ −∫ ∞
′ ( )( ) −
∫ ∞[ ( )] ( )
= −∫ ∞
[ ( )] ( ) = − ( ). ♦
7
( ) ′′( ) + ( ) ′( ) + ( ) ( ) = ( ), < < ,
, ,[ , ] ( ) �= , ∈ [ , ]
( )
( ) =∫ ( )
( ) ,
(( )
( ))+
( )
( )( ) ( ) =
( )( ) ( ), < < .
( ) =: ( ) ( )( ) ( ) =: ( )
( ) ( )
[ ( )] =:
(( )
( ))+ ( ) ( ) = ( ), < < ,
[ ( )]
=:
(( )
)+ ( ).
( ) �= ,
[ ( )] =: ′( ) + ( ) = ,
[ ( )] =: ′( ) + ( ) = ,
, ..., , ,
8
( − , − )( , )
( , ) ( , ) ( − , − )
( − , − )[ ( , )] =
∫ ∞
−∞
∫ ∞
−∞( − , − ) ( , ) = ( , ).
∫ ∞
−∞
∫ ∞
−∞( − ) ( − ) ( , ) =
∫ ∞
−∞( − ) ( , ) = ( , ),
( − ) ( − ) = ( − , − ).
∫ ∫( − , − ) ( , ) =
{( , ), ( , ) ∈ ,, ( , ) /∈ ,
⊂ R
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞( − , − , − ) ( , , ) = ( , , ),
( , , )
( − , − , − ) = ( − ) ( − ) ( − ).
∫ ∞
−∞...
∫ ∞
−∞( − , ..., − ) ( , ... ) ... = ( , ..., ),
( , ... )
( − , ..., − ) = ( − )... ( − ).
, ,
= ( , ), = ( , ),
,R = ( , )
= ( , ).
( − ) ( − ) = | |− ( − ) ( − ),
= [ , ]( , ) = ∂( ( , ), ( , ))∂( , )
[ , ]( , ) �= , ( , ) ∈ R
∫ ∫( − ) ( − ) ( , ) = ( , ).
∫ ∫( ( , )− ) ( ( , )− ) ( , )| | = ( , ).
( ( , )− ) ( ( , )− )| | = ( − ) ( − ).
[ , ]( , ) �= , ( , ) ∈ R
( − ) ( − ) = | |− ( − ) ( − )
♦( , )
[ , ]( , ) =
= cos , = sin == ( , ) =
= , =( , )
( , ) = ( )
( , ) =
∫ ∫( − ) ( − ) ( , )| |
=
∫ ∫( ) ( , )| | = ( ).
9
√−
+ + = +
√+
√ √
=
+ =
=√−
∞
sin, cos tan
∞ /∞
x
y
ab
c
da+b+c+d
θ
z=x-iy
z=x+iy
r=|z|
v=w-zw
, = + =(cos + sin ) | | = =
√+
arg= arg = sin− = cos− = tan−
× = ( + )×( + ) = (cos + sin )× (cos + sin ) =[cos( + )+ sin( + )]
| × | = | | × | |
x
y
θ2
θ1
θ1
+θ2
z1z2
z1z2
∈ N = (cos( ) + sin( )).
= (cos + sin )
= (cos + sin ) √= , = + , = + , ..., =
+( − )
= / == (cos + sin )
= {cos( )+ sin( )} = (cos + sin ).
= | | = =√
= + arg = = + , =,± ,± ,± , ... .
♦
√/ ∈ R
+ √= −√
= =
= ( − )− /
= ( − )− / =
(−
) /
=
(+
) /
=
√(+
)=
√+ .
= + | | = = arg = tan−
=( ) / {
cos(
+)+ sin
(+
)}= , ,
= cos( )
+
sin( )
, = , , , ..., − =
= , ,
x
y
x
y
1
ω
x
y
1
ω
1
ω
ω2
ω3
ω2
ω4ω3
ω2
ω3
ω4
ω5ω6
ω7
= , ,
+
( − )(− + ) ( − ) ++ ( − )(− + )
−+ , ∈ R, ∈ C.
= + + + =
= = =
, ∈ C
= ( ) = || | − | || ≤ | − | | | = | || |
lim sup →∞ =, | − | <
+ > + | + − | < +
, , , ..., | − | <∈ N , , , ..., { , ∈ N}
lim inf →∞ . ♦
, , , ..., ∈ N
= + , , ∈ R
, , , ...,, , , ...,
, , , ...,, , , ...,
, , , ..., ♦
, , , ..., > ∈ N
| − | < , ≥
, , , ..., ∈ N
| − | < | | ≤ | | + ≥
| | ≤ ∈ N, = {| |, | |, ..., | − |, | | +♦
, , , ..., lim →∞ = > ∈ N
| − | < ≥ ≥ ≥| − | < | − | <
| − | = |( − )− ( − )| ≤ | − |+ | − | < .
, , , ...,
( )∞= .
( )∞= .lim→∞ =
( )∞=>
∈ N | − | <, ≥ ∈ N ≥
| − | <
| − | ≤ | − |+ | − | < + = ,
≥ lim →∞ = ♦
lim → [ /( + )]
lim → [ /( + )]
: C→ C = +
( ) =
{sin , = ,
, = ,
lim → [lim → ( )] lim → ( )lim → [lim → ( )]
: C→ C = +
( ) =( + )
+.
lim → [lim → ( )] = lim → [lim → ( )] =lim → ( )
: C→ C
( ) =+
+ +,
→ −: C→ C = +
( ) =− +
( − ) ++
( − ) +,
C =
C
∈ > ( , ) ⊂ ( , )( , ) = { ∈ C : | − | < }
C ˚
∈ >( , ) ⊂
∈ C
∈ C
∂
∈ R =: { ∈ C : > } ∈= − > − > −| − |
>| − | < ( , ) ⊂
C
=: { ∈ C : < } =: { ∈ C : < } =: { ∈ C : >}
( , ) C ˚
C
∈ ( , ) =: −| − | > > | − |∈ ( , ) | − | < | − |+ | − | < + | − | =∈ ( , ) ( , ) ⊂ ( , ) ( , )
C
˚ C
˚ ♦
¯( , ) =: { ∈ C : | − | > }C
♦
∅
A W
A ∈ W
∈ ¯ ∈ N} { ∈| − | < / ∈ N lim →∞ = ♦
¯
C ⊂¯ ⊂
∈ C \ ¯ >( , ) ∩ = ∅ ∈ ¯ ∈| − | < ≤ | − | ≤ | − | + | − | < | − | +
| − | > ( , ) ∩ = ∅ ¯¯
C ⊂ ∈ C \> ( , ) ∩ = ∅ ( , ) ∩ = ∅/∈ ¯ C\ ⊂ C\ ¯ ¯ ⊂♦
( ( ), ( )), ∈ [ , ],[ , ]
( ( ), ( )) = ( ( ), ( ))[ , ]
, ∈ [ , ], = ( ( ), ( )) = ( ( ), ( ))
C
( ) =:{ ∈ C : | − | < , > } ( ) =: { ∈ C : < | − | < , , > }
( ) =: { ∈ C : | − | ≤ , > }( ) =: { ∈ C : ≤ | − | ≤ , , > }
C
−
G
F
=: { ∈ C : < | − | < , , > }
) { ∈ C : | − + | = } ) { ∈ C : | − + | ≤ }) { ∈ C : ( + − | = } ) { ∈ C : | − |+ | + | = }) { ∈ C : | | = | + |} ) | | < ) < | − | < ) | + |+ | − | <
| | = | + |, ∈ C
⊂ ∂ ⊂⊂ ⊂˚
⊂ C
= { ∈ C : | | ≤ }∪{ ∈ C : | − | < } = [ , )∩{ + : ≤ }= C \ ( ∩ )
= [ ,∞) = { ∈ C : = cos , = , ≤ ≤ ∞}⊂ C
= { ∈ C : > } = { ∈ C : ≤ | − | ≤ }= { ∈ C : ≤ < } = { ∈ C : ≤ ≤ }= { ∈ C : ≥ | − |} = { ∈ C : ≥ | |}
+ ¯ ¯+ ¯
+ = ( + )(¯+ ¯ ) = ¯+ ¯ + ¯ + ¯ .
= ¯ + ¯ + ¯
= ¯+¯
+ ¯ .
= + , = + , ¯ = −
= − + ( + )−+
+ − .
= → = →→ =
′ ′ = →′ = → → = −
=
= ( ) =
+ = ( + ) = + + ( ) =+ ( )
=+ ( )
= + .
(= )
= ( ) = ′ = ′( ) =
C : �→C ( ) = ( + ) = ( , ) + ( , ) ( )
, , ,
( , ) = ( , ), ( , ) = − ( , ).
= +
′( ) = → , →+
+.
= → ′( ) = + = →′( ) = −
′( )
( , ), ( , )= +
( ) =( , ) + ( , )
′( ) = + = ( + ).
,
= + + + ,
= + + + ,
→ , →
+ = ( + )( + ) + + ′ ,
= + , = + , →
+
+→ + ,
| || + | ≤ | |
| + | ≤ ′( )
( ) ♦
C
( ) ′( ) ⊂ C
( ) ∈ C
∈ C
( )
,
+ = − = , += − + = ,
( ) = ( + ) =( , ) + ( , )
( , ) ≡ ( , ) + ( , ) = , ( , ) ≡ ( , ) + ( , ) = ,
( , ) ( , )
= + = −
( , ) =
∫ ( , )
( , )( − )
∇ ( , ) = ×∇ ∇ ( , ) = ∇ × .
∇ = + , ∇ =+
+ = ( + )× = − ,
( )
10
( )= =
∫( )
∫( ) .
( ) = ( ) + ( ), ≤ ≤( ) = ( ) = ( ) = ( , )+ ( , )
( ( ), ( )), ( ( ), ( ))
( )
| − | = / −/ +∫
√ − ,√
− = exp
[( − ) + ( + )
]
( ) = ln[ + ( − ) / ]
( ), ∈ C ∫( ) =
( ) ( )′( ) = ( )
∫( ) = ( ) − ( ) = ∈
′( )
∫( ) =
∫( − ) +
∫( + ).
( , ), ( , )
=
∫( − ) =
∫ ∫ (−∂
∂− ∂
∂
),
=
=
∫( + ) =
∫ ∫ (−∂
∂− ∂
∂
)= .
∫( ) = + = .
R
C
D
′( )
′( )( )
∫( ) =
∫( ) +
∫( ) +
∫( ) +
∫( ) .
, = , , ,
∣∣∣∣∫
( )
∣∣∣∣ ≤∣∣∣∣∫
( )
∣∣∣∣ .
/ /
∣∣∣∣∫
( )
∣∣∣∣ ≤∣∣∣∣∫
( )
∣∣∣∣ < =
∫( ) = .
♦
C1
C
C2
C3C4
C5
C6C7
, ∗
z2
z1
z2
z1
C1
C2
C2
C1
( )
∫( )
∗∗ ( )
C1
C2
C1
C2*
~C
∫( ) +
∫∗( ) = ,
, ∗
∫( ) =
∫( ) ,
= C \ {( , )} −
− < <
∫= �=
∫ + ∫( − )
∫( + )
∫ +
∫ + ∫sin( )
∫− cos ( )
∫− cos ( )
∫− sinh ( )
∫ ( − ) ∫ / ∫∞ − , >
, ∈ Z∫− =
{, �= ,, = .
( )′( ) ∈
′( )
( )⊂ C ∈
∈
∫( )
− = ( ),
=
=
∮sin
( − )= .
= ( ) = sin
′′′( ) =!∮
sin
( − ).
∮sin
( − )= ′′′( ) = [− sin − cos ].
( )( ) = ∈ ( ) =
( ) =
∮( )
− .
( ) = − �= ( ) =
( ) =
∫( )
( ) ′( ) = ( )( )
( )
( )( )
′( ) = ( ) ( )
( )⊂ C
∫( ) =
( )
∈| − | <
( )( )
( ):= { ∈ C : | − | < , > =: { :
| − | < , < <
( ) =
∮( )
− .
( ) = + , ≤ ≤
( )
lim →∞( ) = ( )
( ) =( )− ( )
.
( ) C \ { }lim → ( ) = ′( ) ( ) = ′( )( ) C
( )
lim→∞ ( ) = lim→∞( )− ( )
= .
> | | >| ( )| < ( )
| | ≤( )
C
∈ C
( )− ( )= , ∈ C.
lim→∞( )− ( )
= .
=
( )− ( )= ( ) = ( ), ∈ C.
( ) = + + + ...+ , �= , ≥ , ∈ C.
∈ C
∈ C ( ) �= ( ) =/ ( ) ∈ C
( ) = / ( ) =: { ∈C : | | ≤ }
( ) =( )
< , �= , ≥ , ∈ C.
( ), ≥∈ R
+
( ) > , ≥ , ∈ C | | ≤ .
11
lim →∞ = > | | <∈ N ∑∞
=
∑∞=
> > > ... >
∞∑=
( − ) = + ( − ) + ( − ) + ...+ ( − ) + ... ,
∈C
∈ C
( ), ( ), ( )
| cos | ≤ + = | | == +
∣∣∣∣∣cos −∑=
(− )( )!
∣∣∣∣∣ = | ( )| ≤ +
−( )
=+
−( ) +
,
| | ≤ <
( ) =( ) = ( ) =
= +∞∑=
!, | | <∞.
( ) sin =∞∑=
(− ) +−
( − )!, | | <∞.
( ) sinh =∞∑=
−
( − )!, | | <∞.
( ) cosh = +∞∑=
( )!, | | <∞.
( )+
=∞∑=
(− ) , | | < .
( )+
=∞∑=
(− ) , | | < .
( ) − = + + + ...+ + ..., | | < .
( ) = arctan , =
( ) = cos( )− sin , = ( ) = cos( )− , =( ) = cosh( )− sinh( ), = ( ) =
+, = − +
, ,( > )
( ),
x
y
z0s
K
s
z
r1r2
C1 C2
( )
( ) =
∞∑( − ) +
∞∑( − )
,
12
( )= ∮
( ) = ,
( )=∮
( ) �= .
( )
( ) =
∞∑=
( − ) +
∞∑=
( − ),
< | − | < =( )
=
∮( ) .
∮( ) = ,
< | − | <=
( )( ) =
= = ( ).
( ) = cos ( ) =( + )
( ) = + +( + ) ( − )
( ) = sec( )
( ) = tan ( ) =+
= = − ( ) =+ +
( ) = cosh −cos = ( ) = −sin =
| | <( ) = − ( ) = −
− +( ) = −
+( ) = − − +
− +
( )
, , , ...,
∮( ) =
∑=
= ( ),
( )
∮( ) +
∮( ) +
∮( ) + ...+
∮( ) = ,
∮( ) =
∮( ) +
∮( ) + ...+
∮( ) ,
C1C2
Cn
a1a2
an
C
∮( ) =
∑=
= ( ). ♦
== == = =
= , =
=
∮ −− .
( ) = −−
= =
= ( ) =
[ −−]
=
= − , = ( ) =
[ −−]
=
= .
∮( ) = (− + ) = −∮( ) = (− + ) = −
∮( )
=: { ∈ C : | | = } =: { ∈ C : | − | =/ }
( ) = − ( ) = +( + )
( ) =( − )
( ) =( + + )
( ) =+ +
( ) =( + + ) ∮
( )=: { ∈ C : | | = } =: { ∈ C : | − | =
/ }( ) = tan ( ) = sin ( ) =
sin( ) =
−
( ) = / ( ) = cos
∫ +∞
−∞( ) ,
( ) R
lim→∞
∫−
( ) + lim→∞
∫( ) ,
. .
∫ +∞
−∞( ) = lim
→∞
∫−
( ) ,
( ) =
. .
∫ +∞
−∞= lim
→∞
∫−
= .
∫∞−∞ ( )
( )
( )
∫ +∞
−∞( ) cos( ) ,
∫ +∞
−∞( ) sin( ) ,
( )
| cos( )| | sin( )|sinh → ∞
∫ +∞
−∞
( )
( ),
−
( )
( )→ ∞
≤ ≤∫ +∞
−∞cos( ) ( ) = −
∑( )
,∫ +∞
−∞sin( ) ( ) = +
∑( )
,
♦
( ) =
∫ +∞
−∞cos
( + )= , ( ) =
∫ +∞
−∞sin
( + )= .
∫ +∞
−∞ ( + ).
( ) =( + )
= = − =− ≤ ≤ , =
| | = , >| | = , > ( )
∫ +
− ( + )= −
∫( + )
.
= ( )
= = ( ) =!lim→
( − ) ( ) =!lim→ ( + )
= − .
→ ∞∈ | + | ≥ ( − ) .
| | = | − | ≤ ≥∣∣∣∣∫
( + )
∣∣∣∣ ≤ ( − )
lim→∞
∫ +
− ( + )= .
lim→∞
∫ +
−cos
( + )= .
0
R
r
y
x
x
y
R-R
π
C1
C2
C3
C4 iπ/2
�= ∞C∪{∞}
( ) =: , ∈ ( , ) = , ≥ , ∈ (− , ]( )
= ≤
( − ) = − ( ) =
( − )
( )= �= /∈ Z
= / / ( − / ) = −/ ( / ) =
= ≤
( ), ∈ ( , )
∈ (− + , + ]= ( + ) = −
∈ (−∞,+∞)
= + = = ( + ), ∈ (− , ], ∈ Z
( )
= , ∈ Z
∈ Z
= ( + ) = cos + sin , ∈ (− , ], ∈ Z
C
∈ (− , ]>
< −
− −−
∫(cos , sin ) .
= , ≤ ≤
cos =+ −
=+ −
, sin =− −
=− −
, = .
( ) = (cos , sin )cos , sin
≤ ≤ ( ) ( )∫(cos , sin )
( )/
∫ ( − −,
+ − ).
| | =( )/
♦
=
∫+ sin
= √ − , − < < .
= �=
∫/
+ ( / ) − ,
| | =
=
(− +
√ −)
, =
(− −√ −
).
( ) =/
( − )( − ).
− < <
| | = +√ −| | > .
| | = | | <
= = ( ) = lim→ ( − ) ( ) =/
− = √ − .
=
∫/
+ ( / ) − = = √ − ,
=
∫− cos +
, < < .
= ∈ =: {| | = }
=
∫/
− ( / )( + ( / )) +=
∫( − )( − )
,
∫+cos
∫ sin( )cos +sin
∫cos +sin
∫sin+cos
=
∫cos cos( − ) =
!
∫(cos +sin ) =
∫cos sin = .
∫ [ +cos( )]− cos +
= − +− , < <
∫cos + sin
= , , ∈ R \ { }∫
[ cos + sin ]= ( + ) , , ∈ R \ { }.
( ), > , >
( )
[ ,+∞)
( )
( )∈ C
∈ C
( ) =( + ) = L{ ( )} ( ), ∈ R
+
∈ C
( )∈ [ , ], > ( ) = L{ ( )}
( ) = + ∈ C
>
L{ ( )} = ( ) = ( + ) = ( , ) + ( , )
=
∫ ∞( ) − =
∫ ∞( ) −( + ) .
( , ) =
∫ ∞( ) − cos( ) ( , ) =
∫ ∞( ) − sin( ) .
( + )
>
| ( ) − cos( )| ≤ | ( )| − | ( ) − sin( )| ≤ | ( )| −
( ) © ( ) ,> , | ( )| < > ≥ >
| ( ) − cos( )| < − ≤ ( − ) | ( ) − sin( )| < − ≤ ( − ) ,
( ) © ( )> >
| ( )| < ∈ ( ,+∞)
+∞∑−∞ ( )
+∞∑−∞
(− )
( ), .
( )
, , , ..., ( ) ≥
( )
+∞∑=−∞ ( )
= −∑=
=
(cot( )
( )
),
( )+∞∑=−∞
(− )
( )= −
∑=
=
(csc( )
( )
).
∮ [cot( )
( )
],
, = , , ...,
∮ [cot( )
( )
]=
∑ cot( )
( ),
lim→∞
∮ [cot( )
( )
]= ,
csc( )/ ( ) cot( )/ ( )♦
∑+∞=−∞ + , >
∑+∞=−∞
(− )+ , >
∑+∞=−∞ ( + )
∑+∞=−∞
(− )( + )
∑+∞= ( + )
∑+∞= ( + )
∑+∞=
(− )−
∑+∞=−∞
(− )+
∑+∞=−∞ −
∑+∞=
(− )+
∑+∞=−∞
(− )+∑+∞
=−∞ +
∑+∞=−∞ +
( )⊂ C
∮ ′( )
( )= # ( ) # ( ) .
x
y
-i
i
0-3π -2π -π 3π2ππ
( ) =( )
=
( ) =−
( − )+
− +
( − ) ++ ...+
−− + + ( − ) + ( − ) + ...,
→ ∞ ∞
L−( ) = L{ ( )} ( )
( ) =( )
( ), ∈ C,
( ), ( )= +
( )
( ) = L− { ( )}( )
( ) =( )
( )
+ cos , sin
− , �=( − ) + cos , sin
, > , ≤ ≤ −( + ) , > cos , sin , ≤ ≤ −( − ) , > , �= , ≤ ≤ −[( − ) + ] , > , �= cos , sin , ≤ ≤ −
( ) = L− { ( )/ ( )} ( ) =
( ) =( )
u
v
ω
arg Q(iω)
1 ω=0
ω=-1
ω=1
ω=∞
ω=-∞
ω=-2
ω=2
1 2 3 4 5
3π/2
5π/2
= ( ) = + + + ( )
=: { ∈ C : | − | ≤ } =: { ∈ C : | | ≤ }( )
+ + = + + + =+ + + + = + + + + =
( ) = ∞
∞( ) ∞
=∞ ( ) = −∮
( ) ,
= { ∈ C : | | = }| | ≥ =∞
=∞
( ) = ∞ − −− /
( ) = ∞ − =∞ ( )( / )
/ ( ) = − ( / )
=∞ ( ) = − = ( )
= /∮| |= /
( / ) =
∮| |=
( ) .
= ( ) =∞ ( )
= ( )
=∞ ( ) =∞= ( ) = lim → ( ) ( )
=
=∞ ( ) = − lim→∞ ( ), (∞) = .
(∞) = = ≥
( ) = − +−
+−
+ ...,
lim →∞ ( ) = −
( ) | | ≤∞
13
= +=
= −
y
x
y
x
v
u
v
u
y=1/2
y=1
x=1/2
x=1
y=1/2
y=1
x=1/2
x=1
v=1/2
v=1
u=1/2
u=1u=1/2
u=1
v=1/2
v=1
z - επίπεδο w- επίπεδο
−− =
= , = , , ...,≤ arg ≤
−
= − , =, , , ...,
= −
= − , ∈ C
= − , ∈ C
x
y
u
v
x=1/2
y=-1/2
x=-1/2
y=1/2
x=0
y=0
z - επίπεδο
w- επίπεδο
1 2-1-2
1
2
-1
-2
1-2 -1 2
1
-1
= −
( + ) + + + = , , , ∈ R,
= �=, ,
++
+−
+ = ,
/
++
+−
+ = ,
, ,
x
zy
u
wv
b Im z=b
w=ez
b
= == =
u0
v
x
y
0
z - επίπεδο w- επίπεδο
1 2 3 -e e e3-e3
e
e3
w=ez
3π4
π4
φ=3π4
φ=π4
= < ≤ , / ≤ < /
= ( + ) − , ∈ C
= , , , = − , , , | − | ≤= , ∈ C
| | > | | ≤ | | < / / < | | < √
≤ ≤ / − / ≤ ≤ /
: ⊂ C �→ ⊂ C,
= ( ) = ( , ) + ( , ), = + ∈ ⊂ C.
, = ,
: = ( ), = ( ), = ( ) = ( ) + ( ), = , ,
∗, = ,
∗ : = ( ), = ( ), = ( ) = ( ) + ( ) = , .
C1
θ
C2
D
C1*θ
C2*
G
w0=f(z0)
z0
z - επίπεδο w- επίπεδο
= ( )
( )
: �→
−
=
∫ ∫ ∣∣∣∣(
,
,
)∣∣∣∣ , =
∫ ∫ ∣∣ ′( )∣∣
= ( ) | | <≥ | ′( )|
( )∑∞= <
∫| ( )| =
∞∑=
| | .]
: C �→ C
= ( ) =+
+, − �= , , , , ∈ C
′ = ′( ) =−
( + ).
− �== ( )
− =′ = ′( ) = = ( ) =
= ( ) = + , = , = , = , ∈ C
= ( ) = , = , = , = , ∈ C
= ( ) = , = , = , = , =
= ( ) = + , = , = , , ∈ C
∞ C
+ �=∈ C �= = − /
+ = ∈ C
C =∞= − / ( �= )
∞(∞)
=�= , �= = ∞
= ∞
= − ( ) =− +
− , − �= , , , , ∈ C.
| | <| | <
− =
, , , ∈ Z − =( + )/( + ) = /(| + |
= ( ) =
(+
), = + ∈ C,
= −=
+
=
′ = ′( ) =
(−
),
= ±
= ±
�=
+ = + − =−
.
�=
= .
(| | < )(| | > )
| | =
=
( , ) =
(+
)cos , ( , ) =
(−
)sin .
�=
=
(+
)+
( − )(
+) − ( − ) ,
= − = − .
/
= −
(+
) ( − )[(
+) − ( − ) ] .
u
v
x
y
α1
1
-1
-1 1
z - επίπεδο w- επίπεδο
+ = + + = ( + )
− = − + = ( − ) .
−+
=
( −+
)
= +
> =
−+
=
( −+
)
= ( )
| | <
| | <
= ( )
| | <
| | =
εσωτερικό
εσωτερικήγωνία
πλευρά
κορυφή
εξωτερικήγωνία
εξωτερικό
, , , , ..., − , = ( )
( ) >≥ = ( )
, , , ..., − ,
x
y
z=g(w)=f -1(w)
O
z - επίπεδο w- επίπεδο
Cn-1π
Cnπ
C1π C2π
C3π
C4π
Cn-1 CnC1 C2 C3 C4z . . .
= ′( ) .
= ( ) = ( )
= ( ) ,
= ( )
arg ′( ) = arg
arg ( − )
u
v
u
v
(a)
QP
R
a
b-bα α
(b)
Q
Ra
α
w- επίπεδο
= + = ( ) = sin = sin( + ) = sin cosh + cos
= ( , ) = sin cosh = ( , ) = cos sinh
(− / ) ≤ ≤ ( / ) ( )′( ) = cos ±( / )
( )
u
w
x
zy
v
π2
π2
-
Re z = -π2
Re z = π2i
w=sin(z)
sin(i)=isinh(1)
= sin = > , − / ≤ = ≤/
x
y
u
v
π2
π2
-
sin
(-K
1)
sin
(-K
2)-coshc sin
(K1)
sin
(K2)
coshc
y=C3y=C2
y=C1
isinhc
-isinhc
y=-C3y=-C2y=-C1
C1
-1
w=sinz
C2
C3
-C1
-C2
-C3
0
K1 K2-K1-K2
0 1
= sin = > , (− / ) ≤ ≤ ( / )
(− / ) < < ( / ), − < <
=
=
− < < , < < <
= , (− / ) < < ( / )
x
y
u
v
1
A
BC
D
E F-1
π2
π2
-A* B*C* D*
E* F*1-1
= sin (− / ) < < ( / ), − < <
x
y
u
v
1
π-πA*=B*
D*=C*
1-1
A B
CD
= sin − < < , < < <
( = )
= + = ( ) = cos = sin( + ( / ))
sin /
= + = ( ) = cos� = ±
= + , −∞ < <∞= + = cos = cos( + ) = cos cosh − sin sinh .
= cos cosh = − sin sinh
cos−
sin= ,
+ −cos �= , sin �==
cosh > > cos > <cos <
x
y
Ou
v
O
w=cosz
c
Re z = c
cos(c)>0
1
z - επίπεδο w- επίπεδο
= cos = cos ( ) > , sin( ) �=
x
y
Ou
v
O
w=cosz
c
cos(c)<0
-1
z - επίπεδο w- επίπεδο
= cos = cos ( ) < , sin ( ) �=
cos = = ( + ) / , ∈ Z = =(− ) + sinh (− ) + sinh
R
= cos =
x
y
u
v
w=cosz
z - επίπεδο w- επίπεδο
2Re z =
(2n+1)πRe w = 0
= cos = ( + ) /=
sin = = , ∈ Z = (− ) cosh =cosh ≥ R = +
[ ,∞)(−∞,− ]
x
y
u
v
w=cosz
z - επίπεδο w- επίπεδο
1Re z = 2kπ
= cos = [ ,+∞)
= =cos( + ), −∞ < < ∞ = cos( + ) =cos( ) cosh( )− sin( ) sinh( ) = ( ) = cos( ) cosh( ), =( ) = sin( ) sinh( ) �=
x
y
u
v
z - επίπεδο w- επίπεδο
-1
Re z = (2n+1)π
= cos = ( + ) (−∞,− ]
cosh+
sinh= ,
± = ==
= ( ) = cos( ) = ( ) == ( ) = cos( ) − +
R
x
y
u
v
w=cosz
z - επίπεδο w- επίπεδο
-coshb coshb
-i|sinhb|
i|sinhb|Im z = b ≠ 0
= cos = �=
= + =( ) = tan
= tan
x
y
u
v
w=cosz
z - επίπεδο w- επίπεδο
Im z = 0
1-1
= cos = =[− ,+ ]
sin , cos
= + = ( ) = tan =sin
cos=
( − − )/
− − =( − )/
+.
= / = −
= ( ) = ( ( ( )))− , = ( ) =−+
, = ( ) = .
= tan( ) ( )
( ) = /
: − << = tan
= ( ) = = − +
| | = − == − / , = , = /= − / , = , = /
=| | = − < > | | = − > <
| | <| | >
= ( ) =−+
.
∈ R ∈ R
| | ==
| | = | ( )| =∣∣∣∣
−+
∣∣∣∣=
∣∣∣∣
( − )
+
∣∣∣∣= .
| | == ( ) =
| | ==
= ( ) =−+
=/ − − /
/ + − /=
sin( / )
cos( / ).
= /= ( ) = −
= /
= tan: − < <
= + = ( ) = sinh = − sin( )
== sin = −
= + = ( ) = cosh = cos( )
== cos
x
y
u
v
z - επίπεδο t - επίπεδο w - επίπεδο s - επίπεδο
: − / < < /= tan
:≤ , ≤ ≤= cosh
cosh = == = ∈ R
+ cosh
≥
x
y
u
v
πw=coshz
z - επίπεδο w- επίπεδο
1-1
B* A*A
B
0
: ≤ , ≤ ≤= cosh
= cosh = cos− ≤ ≤
= = cosh = cos = −
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