Συνδυαστικές Ασκήσεις Μιγαδικών Με Ανάλυση - Νέα...
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z i
f (z)| z 2 | | z 1|
. z f (z) .
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. f (z) i , :
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iii. z , z
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z C {2i} 3z 8i
f (z)z 2i
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7
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f (z)z 2i
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. Im(f (1 i))
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f (z) R
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f (z 5i) f (z i) 10 (2)
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. z (2) ,
. 1z 2z (2) , 1 2z z 10
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f.
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z .
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.
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. z .
.
3
2x
z z 3 x xlim
z z 3 x x
. f x 'x x 0 x 1 z 2z ,
x
2
0f (t)dt 3x 6x 6 (0,1)
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, f 2z if
2w if 0 f f 0 . w z w z f f f .
. 0x , 0f x 0
. 1x , 1f x f
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. 1 2z ,z 1 2 1 2z z z z .
1 2w z z .
. f x
1z 1 i 2z 1 f x i
() f R f 0 0 f 0 0 .
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f
f
1 f e
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, f 1
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z f x f x i , f xz 2 1 e .
f0, 2 C ,
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) z .
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) 0x 0f x 0 , z 0
0 0x xz 0 2 1 e 0 e 1 .
f x 0 ,
f 0 2 0 , f x 0 x .
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x x
2 2 x
22 x
x
f x 0x
z 2 1 e f x f x i 2 1 e
f x f x 2 1 e
2f x 2 1 e
f x 1 e
f x 1 e , x
) z x yi ,
z f x f x i x yi f x f x i
x f xy x
y f x
, xx f x 1 e 1 , z
, y = x, x 1.
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f : R R 1
z C2
2 2f (x) x 2xf (x) x x 0
f (x) | z 2 |lli
x | 2zm
1|
.
. z .
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. 2x 0
f (x)lim
x x .
. g(x) f (x) x
( ,0) (0, )
. f .
. 3(| z 3 4i | 5)x x 10 1,2
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z 1 3x 3 3x i , x 0,2 .
. M z C
.
. x z .
z .
. 1 2x ,x x z
1 2M ,M z . f : R R
1 2M ,M . f
x 'x C .
13 ( 3 2002)
f, R. ,
2 2
22
x z x zf x
x z
z z = + i, ,R , 0 .
. x x
f x , f xlim lim
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. f, z 1 z 1 9
. f. 8
14 ( 4 2004)
f: IR IR f(1)=1. x IR,
3x
1
1g x z f t dt 3 z x 1 0
z
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z = + i C, , IR*, :
. g IR g.
. 1
z zz
. Re(z2) = 1
2
. f(2)= >0, f(3)= > , x0 (2, 3)
f(x0)=0. 5 + 8 + 6 + 6 =25
15 ( 3 2004)
f : , [, ] f(x) 0 x [, ]
z Re(z) 0, m(z) 0 Re z Im z .
1
z f ()z
2 22
1z f ()
z , :
. z 1 11
. f2() < f2() 5
. x3f() + f() = 0 (1, 1). 9
16
f : 1, 2f x x ln x .
i. f
ii. f (x) = 2012.
iii. 1 2z ,z 1 2z 1, z 1 1 1 1 2 2 2z z ln z z z ln z ln16 :
) 1 2z z 2
) 1 2
1 2
z z
4 z z
iv. z z ln z 1 , z
> 1.
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z i ,
f : :
2x 1
1
f (x) t z zdt 3x 2
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f 0x 1 , :
) 1
z2
) w 2z i
) w .
)
f , xx x = 0 x = 1.
18 ( : 36)
z i , R f x z xi , x R
(1, 2).
. 1 z 3 . z ;
. , f x 2 z x , (-1, 1).
19
f: , z = +i, z1 = +if(),
z2 = +if(). 2 2 1 23 z z 4izz 4iRe z z , Cf
xx.
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) z . z : 1 i z 1 i z 4 0 ,
z ().
) () () f
2
x
xf x x
e ,
, .
) f x
g x , x 0x
,
g.
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f : x x 2 f(x) x , x(-1,1).
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1. , :
x 0
1 2 x
2xlim
f 0
2. 0z i, , , (z), :
20zz Re zz 4Re z Im z z 51
22
() ,
:
2
1 1z 2 Re z Re z Im z Im z 0
1z 1z 1
() 0x , 0x
f x , : x x 3e 1 f x e x , x 1
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f [1,e] :
e e
1 1
f xdx 1 f x ln xdx
x
) f x e .
) z
: z 3i z 1 z i z 1 ;
) ,
z : 1z 2i f e Re z , 1z
().
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) xy
: z 1 2i z 7 2i
) , ,
2x 5x 10
f x2x
-, ()
x = -1.
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25 ( 3/ 2006)
z, (4z)10 = z10 f f(x)
= x2+x+, IR .
. z x=2.
. () f
x = 2 o = 3,
i. ().
ii.
f, (), xx x = 3
5 .
7 + 9 + 9 = 25
26
. 3 5
f z 2 i z i z2 2
, z x yi, x,y .
i. Re f z , Im f z
ii. M f z f z
.
iii. : f z x 2y 5
iv. z x yi, x,y
: f z 5
. z
2, 1
w zz
.
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. z,w
2 2 2z w z w 2Re zw
. f :
11 2z 3 f 2 i z f 2 3i
2 2 21 2 1z z 2Re z z .
. z,w
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2 2 2z w z w 2Re zw
. f :
11 2z 3 f 2 i z f 2 3i
2 2 21 2 1z z 2Re z z .
) 1 2z z
) f ,g g x f 2x f x x, x ,
) f , , 12x f x f x
2,3
) 1 2z z
) f ,g g x f 2x f x x, x ,
) f , , 12x f x f x
2,3
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2 2 21 2 1z z 2Re z z
():
) ,
f , 2
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(1) + (2) :
f : (3)
, (4)
(3) (4) :
g . .
) ,
H h [2, 3]
h(x)=0 (2, 3) Bolzano.
, h . . , f ,
. f,- . , 2. h(x)=0
(2, 3)
, h(x)=0 (2, 3).
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z, w : 2 2z w 10i 2w wz 6i .
z
w, , :
) : z 2w
) z, w.
) f [1, 2] f x 0 x 1,2
[1, 2]
x x
2011 2011
2011
1 2
f tg x z dt w f t dt , x 1,2
2 ,
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xx
ii. g .
) z
p 0 z pww
. = 2 = - 1/3 .
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) z 2w w x yi, x,y .
: z 2 2i w 1 i z 2 2i w 1 i
.
) i. Rolle g [1, 2].
ii. ,
x x
1 2
f t dt 0 f t dt 0
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1 2z z ,
2012 2012
1 2 1 2z z z z .
1 2f x xz z , x . :
) 1
2
zRe 0
z
) f
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0 0f x 3x 1
) f 2z
)
: 1 11 2 1 22 2
z zz z z z 0 0
z z
.
) , 2 2 22
1 2 1 2 1 2f x xz z xz z ... z x z
) Rolle 3g x f x x x [-1, 1]
) () x,
2 2 22
1 2 1 2 2 2f x xz z z x z 0 z z f 0
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1 2z ,z f
2x
1 2
0
f x z t z dt x x
:
) 21
z2
) f x 2020 0,
) x x
21
0
z xz t dt
2 4
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) Fermat 2x
1 2
0
g x z t z dt x 0 g 0
) , 1 2f x 2 2z x z 0 f x 0 x.
, x
f xlim
f x x , ,
) ,
tu2x x u t2 x
21 2 1 2 1
010 0dt du2
z xf x z t z dt x z 2u z 2du x z t dt
2 4
31(..)
f : f 0 1 z .
2012 2010 2007 2009f x i 2x i 2z 2 i z 1 i x , :
) 2f x x 1,x z 1 2z 2
) z .
) f
O(0, 0) , A( , 0) , ( , 2 +1) (0, 2 +1) , > 0.
f .
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K ,03
4
3 .
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32 ( 2011)
. xe x 1 , x . xe x 1 ;
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: x 1f t f x xt
0 0z e dt ix e dt
xt
0
zf t e dt f a 1
2 , >0 .
:
. i. z
Re z Im z 01 i
x 0 .
ii. f x xe f x e x 0 .
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. f .
. f .
. f 0, , , , 0,a
, a f ' 1 .
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t 0 : u x
t 1 u 0
,
1 0 x xf x xt f u f u f t
0 x 0 0x e dt e du e du e dt
x 0, : x 1 x xf t f x xt f t f t
0 0 0 0z e dt ix e dt e dt i e dt
x f t
0Re z Im z e dt (1)
,
x x xf t f t f t
0 0 0z e dt i e dt 1 i e dt
x f t
0
ze dt
1 i
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0e dt 0 , :
1
x f t
0
z ze dt 0 Re z Im z 0
1 i 1 i
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0
ze dt 0
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, :
x xf t f t
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ze dt e dt
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x f t f x0 f t e dt f x e
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(2) x : f x xe f x e
x 0
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1 2x x ,
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f x f xx x
1 2 1 2 1 2 1 2x x e e e f x e f x g f x g f x f x f x 0
[0, ] (0, )
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f f 0 1 0f f f 1
0 0
:
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33
xR x
t0
2g(x) dt
e
>0
z g(x) x.i z i z 1 .
. :
) g . ) z 1g .
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) Re(z) Im(z) xR.
) =1.
) 2 1
2 t t0 0
1 1 1 1dt dt
1 e1 e e e
.
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z , Im z 1 . f :
xf x x ln e z
. x
f xlim
. f .
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z zf x 1 f x
e z e z
. z : 1
2
0
x f x 2xf x dx ln 2
35
222f x x z x zz x Im z , x ,z .
f fC x 'x , :
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. z.
. z 4 z 3 3i 6 .
. f .
. z 2i
g x x 2 f x , x = 4.
36
f, f x z x i , x z
) . f .
. f 0x Im z .
. f f x Re z , x .
) Rolle f
[-1, 3], ,
. z .
. 1,3 f
xx.
. z,
23 Re z
0
f 1 x dx 126
.
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xz e x 1 i, x
) : 2
Re z Im z Re z Im z x
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) z, , .
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0
Re z Im z dx
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z, w, :
2
2x 1
z 3 4i x w 3 6i x 2 1lim
2x 1
) z.
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) w.
) |z-w|.
39
:
2
2
2x 3x 1z 1 , x 1
x 1
f (x)
1z i x , x 1
2
z .
) z.
) z (),
z w , 0 0 0 0 0 0w x iy , x , y x y 0, .
) 0z , (),
2012z (
x0, y0)
) f x0 = 1, z .
40
2z z 0 1 , (1)
.
) 1 2z , z (1) .
) : 3 31 1 2 2z 6z 9 z 6z 9
)
3 3
1 2f x x z x z , x