30 ΑΣΚΗΣΕΙΣ ΜΙΓΑΔΙΚΩΝ (+ 10 ΣΥΝΔΥΑΣΤΙΚΕΣ ΜΕ ΑΝΑΛΥΣΗ)

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: parmenides51 1 ( ) z : ( )( )4Re 2Re , 1 z zz| |+ = |\ . . z . . ( ) Re 0 z , : i. 4w zz= + 4 4 w . ii. 3 4 c z i = + + . . , c . . 1 2, z z 3z (1) ,

1 2 2 3 3 1 1 2 32 z z zz z z z z z + + = + + 2 ( ) 3( ) 4( ), zz z z i z z z C + = + (1) : . z . z 1z . , R , 1z 2104 z z + + =. 0z (1), 20122012 04 35 , 55z iw iw i| | += |\ ., w (0, 5) 21 = . 30 + 10 :10/01/2012 : parmenides51 3 ( ) z C , 1 z = 1 z a + = , a R . : . 0 2 a . 22( )2aRez=.2 21 3 z z a + = .2133 14a z z a + 4( pito) , zw 2 2| | 1,| | 3 z zw w zw + = + = . . | | 2 z w + = . . zw , . . zw. . , zw . 5 ( dennys) 22(cos) (5 4sin) 0, [0, ] z t z t t + = . : . 1 2, z z . 1 2| | z z . 1 2| | z z + 6( dennys) ( 1) , [0,1] z t t i t = + . . z. | | z 2 2( 2) ( 1) , w k k ik = + + , :. w . | | w. | | z w . | | w| | z w [0, 4] k : parmenides51 7 ( ) . 21 0 w w + + =. 1 2, z z 2 21 1 2 2 0 z z z z + + =i. :1 2z z =ii. :1 2 1 2z z z z + = =iii. *N 1 20 z z + , 1 21 2z zuz z =+ . : .- . ( ) 8 ( ) 1 2 3, , z z z , , A B , :1 2 32 3 z z z + = 1 3 21, 2 z z z = = =. 1 2Re( ) 0 zz =. i. 2 2 21 2 1 2z z z z = +ii. OAB . . 2 3Re( ) z z 1 3Re( ) zz. i. , , A B . ii. A B : . ( ) 9 ( ) 1 2, z a bi z c di = + = + , , , a b cd 1 21 z z = = . 21 22 2 0 x z z x + = 1 2, x x . : . 1 2, x x . . 1 22 x x = =. 2 21 2 1 24 8 x x z z + =. 1 22 1x xwx x= + . : parmenides51 10 ( ) ( ) ( ) 6 8 , 0 z t t i t ( = + + + . . 6 8 z i . . z . . z . 0 t , z 1 3 . 0 t = R , 11w zz= + ++ . 11 ( ) 2( )(1 ) x y izx yi+ +=+ *x, y R . . ( )2 22 222x y xyRe zx y+ += + ( )2 22 22x yImzx y=+ *x, y . . z . . z . . z . 12 ( pito) z 2| 1| | 9 20 | z z z = + | 4 | , 0 z = > . . : i. 2 2| | 164zz z + + =ii.2 2 2 2(1 ) | | (5 1)( ) 25 1 z z z + + = iii.| | 2 z . z . . z . 13 ( ) , zw C 2 22 2(3 4) 5 0, (4 3 ) 5 0 z i iz w i iw + + = + + = . . 22(4 3 ) 5 ( ) 0 z w i i z w + + + + = . . q 11 qz= + . : parmenides51 14 ( ) , zw C 1 zw = . , , , a b cdR 2 2 2 20 a b c d + + >( ) ( ) ( 1) ( 1) 0 az z ibz z czz dzz + + + + + = , : . z . z , w . . z , w . 15 ( pito) , * a C 1 2, z z 2 20 z az + + = . : . | | | | 1 a = = 1| | 2 z 2| | 2 z . . 1 2| | | | z z = , a . . 2aR , 12zz , 1 2| | | | z z = 16 ( ) *22( 2 ), z z Cz= (1) . 1z 2z (1) . v , 1 20v vz z + =. xy ,

2011 162 21 21 1 1( ) i ix yi z z+ = + ++ . z , 2 41 2z z z z = . 0z . . 7 z i + . : parmenides51 17( dennys) ( ) ( ) z k t k ti = + t R 1 k >.. z . . w ( 1) y x k = , k 5 2| | 12minz w = . k () z | | z z . k () | 3 4| w i min +. u( 1 ) ( 1 ) u m t m ti = + + + , m u . . , km () () , | | z u . 18 ( ) z, w 2 24 | | 2 1,| | 2 3 z zw w zw = = .. | 2 | 2 z w = .. zw , . . | 6 | z w + .. zw. 19 ( pito) z( )| 2 | | 1|if zz z= . z ( ) f z . . | ( ) | 1 f z .( ) f z i = , : i.N | 1| ( ) 1 z Rez + = ii. ( ) Rez . iii. z , z .i. 20 ( ) . ( 3 )( 4 2 ) z i z i + + +. 2(1 3 ) 14 2 0 z i z i + =(1). 1 2, z z (1) 1Re( ) 0 z >, , A B 1 2, z z 33 z i = +. AB . M( ) z z 2 2 2( ) 2( ) 2( ) 30 MA MB M + = + : parmenides51 21 ( pito) 1 2, z z 12zzz = 12, ( ) 0 z Imzz+ = . . z . . * N 2 21 2z z = . . 95 94... 1 0 z z z + + + + =. 2z 2 1 y x = + , 1z , . . OAB , , , OA B 1 20, , z z . . OAB () . 22 ( ) {2 } z C i 38( )2z if zz i+= . 2 z i , 2( ) 2 4 f z z iz = + . (1 ) f i +. 2( ) 2 4 f z z iz = + . ( ) 2 6 f z iz z = + . 1 z = , ( ) f z 7 = 23( ) z () ( )200520081 1 z z = . . z. 2z z =. (1) . z( ) Im 0 z > , (1) zwz += R . i. w ii. , w : parmenides51 24( pito) 1 2, z z 1( ) 0 Imz > 1 2 2 1| | | | 40(1) z z z z + = 1 225(2) z z = . . 1z 2z . . 1w z i = , . . * N 1 2z z z = . 1 2, z z z . 25 ( ) 20, , z z R + + = 12zi= 2z. , R 2z. v R , 1 216v vz z i = . z 2 21 216 z z z z + = (1). z (1), 4 4 z i 26 ( ) , 0 zw 0 zw zw + =. 2010zw| | |\ . .. , z w . . z w z w = + . . 2z wiw z+ = i. zw ( ) 1, 0. ii. 2012zw| | |\ .. 27 ( ) z : 2 2 2 21 2 | | | 1| 2 | 1| z z z + = + + +. z . . 2 2( 1) ( 1) 0 z z zz + + + = : parmenides51 . 21 0 z z + + =. z . . 2012 20142013 A z z = + + 28 ( ) z2 z i 24( )2zf zz i+=(1). Im( (1 )) f i +. z , ( ) f z R . ( ) 2 f z z i = +. z , ( 5 ) ( ) 10 f z i f z i + + = (2). z (2) , . 1z 2z (2) , 1 28 10 z z 29 ( ) 1 2z , z 2z z+9=0 , R 1 2z , z R . . . . ( )17 171 2z z R + . . 1 2z , z . . 1 22 1z z2z z+ = . .=0 ( )1Im 0 z > z 1 2z z 4 z z = + . 30 ( ) . , , :3 23 4 2 0 z z z + =. , : 3 24 3 2 z z z + = + 1032 32 0 z z + = : parmenides51 31 ( ) f , | |, a b , 0 a > ( ) , a b . :( ) ( )1 2z a if a z b if b = + = +. 1 2 1 2z z z z + = , ( )1, x a b , ( )10 f x =. A B , A B , : 1 2 1 2100 Azz Bz z + = . : i. 1 2zz ii.( ) ( ) f a f ba b= iii. ( ) ,ox a b , ( )( )ooof xf xx = iv. fC 32 ( ) f :[ , ] R ( ) 0 f > > , ( )( )ifzif += . : . ( ) f x x = ( , ) . 0( , ) x 0( ) 1 f x z . . i. f z , ii. . . Bolzano f [ 3, 3] . ( )23 ( 2 1)3w f z i = + + , w : parmenides51 34 ( ) z x yi = + , 22012 ( ) 12012 2012 3 021Imziz+ + =: . ( , ) Mxyz . . z ; .z , ( ) 0 Imz < . 35 ( ) f R ,( ) 0 f x x R 1( ) 0zf xdx = f(1) = 1 . . ( ) 0 f x >. z . . ( )( )323lim3xz z x xz z x x+ + + . f ' xx 0 x = 1 x = 2 z z + , 20( ) 3 6 6xf t dt x x = + (0,1) 36 ( ) | |, f ( )2z if = + ( )2w if = 0 ( ) ( ) 0 f f . w z w z + < ( ) () ( ) f f f < < . . ( )0, x ( )00 f x =. ( )1, x ( ) ()1f x f = 37 ( ) . 1 2, z z 1 2 1 2z z z z + . 1 2 w z z = . . ( )11f xz i = + ( ) ( )21 z f x i = + + () f R( ) 0 0 f =( ) 0 0 f . 2 3 < < : parmenides51 . g ( )1 2( ) g x Imz z = Rolle | |, ()()()()11fffefe+=+, () 1 f 38( ) f:R R (0, 2) A . ( ) ( ) z f x f x i = + 2( ) ( ) w f x f x i = 2( 1)xz e = +. f . . z x R . . ( ) Re( ) gx z w = . 39( pito) : f R R 12z C ` ) 2 2( ) 2 ( ) f x x xf x + = x 0( ) | 2 |,|m2 1|lixf x zllx z= =. . :i. | 2 | | 2 1| z z = ii. z . . 20( )limxf xx x. . ( ) ( ) gx f x x = ( , 0) (0, ) +. f . . 3(| 3 4| 5) 10 z i x x + + = + | |1, 2 40 ( ) ( )( )1 3 3 3 z x xi = + + + , | ) 0, 2 x . . Mz ( ) C . . xz . z . : parmenides51 . 1 2, x x x z

1 2, M M z . : f R R 1 2, M M . f ' xx ( ) C . + :29/12/2011 10/01/2012 (*)http://www.mathematica.gr/forum/viewtopic.php?f=51&t=21713 : dennys pito (*) : alexandropoulos dennys pito

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