פונקציות מרוכבות חוברת קורס 2010

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2010

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Microsoft Internet 6 Explorer, 0.7 Microsoft Word. Office</p> <p>.</p> <p> ? : \http://telem.openu.ac.il :</p> <p> ? . : , , , ", , . : , , , , . . , . . - . </p> <p> " " . , - . . . , . , : http://telem.openu.ac.il/personal_notes .</p> <p> ? . . , , , . . . ' , , "" .</p> <p> - . : . . ) (. , . , , . , . </p> <p> , . , , , , . , , . . . , . </p> <p> , . . !</p> <p> ? , . , . . , . ! . , . , 2222877-90, : infodesk@openu.ac.il " 1111877-90. ! . , .</p> <p> " ) (, . , . , . , ) (, , . . , , .</p> <p> 2222877-90, : infodesk@openu.ac.il : : 03:8 - 00:91 : 03:8 - 03:21 , " . : ) . " 1111877-90( ) ( ) URL( , . </p> <p> , 8 )"( - 4 )"(. " ", :</p> <p> 2 2 2 2 2 3 3 3 3 3 3 2 </p> <p> 1 2, 3 4 8 1, 2 5,6 6 7 1.8 ,2.8 8</p> <p> " 10 " 20 " 30 " 40 " 11 " 21 " 31 " 41 " 51 " 61 " 71 " 81</p> <p> 9 01,11 21 </p> <p> 02 .</p> <p> ! , , . : . . . . . ! 06 .</p> <p>, , . , . , - .</p> <p> . .</p> <p> )"( : , , , . " ". . . ) ( ) !( . . ) " (. ! </p> <p> . " " . " " ". : ! . " . " . " ". " , . , , . / . " ) (. . - . " " " ) (, ". , " , . .</p> <p> ! ) (. " 5 "(. ) </p> <p> " </p> <p> - "" " -" )" "(, . " . ". , , : . " . . ) ( . . " " .</p> <p> " . ". . , . , , , , " ". , . , . , : " ".</p> <p> " " ) (. " :</p> <p>www.openu.ac.il/sheilta " .</p> <p> " " ) . 1. www.openu.ac.il/sheilta .( 2. "". 3. , "" . 4. , " ". 5. " " " " ". 6. . ) (. 7. - "". 8. "" " .</p> <p> " " ", ) (. .</p> <p> )"( 1034202 : : 1 2 : : 51 1/0102 :</p> <p> : 9002.01.03</p> <p> " www.openu.ac.il/sheilta</p> <p> 1= 0102) (1 + i )2010 + (1 i 60012 1+ i 2</p> <p>.</p> <p>1+ i 2</p> <p>.</p> <p>. 1 i</p> <p>. i</p> <p>. 50012</p> <p>. 1</p> <p>. 0</p> <p> "" .</p> <p> 2 3 1 + i Re = 1 i. 3 2 + 24</p> <p>. 1</p> <p>. 2 </p> <p>+ 2</p> <p>3</p> <p>.</p> <p>. 4</p> <p>3 2</p> <p>.</p> <p>2</p> <p>.</p> <p> "" .</p> <p> 3= )3 Arg( 3 i. 0</p> <p>6</p> <p>.</p> <p>. 3</p> <p>4</p> <p>.</p> <p>3</p> <p>.</p> <p>7 4</p> <p>.</p> <p>11 6</p> <p>.</p> <p> "" .</p> <p>1</p> <p> 4= ) 0 &lt; , Arg(1 + cos i sin </p> <p>2</p> <p>. . . . .</p> <p>2</p> <p>2</p> <p>+ </p> <p>2</p> <p> "" .</p> <p> - - - - </p> <p> 5 51 :</p> <p> 5 1. 0 z C - 0 = ) 0102 . Re( z 2009 ) = Re( z 2. n - 0 = ) 1+ Im( z n ) = Im( z n z .</p> <p> 6 0 = 7 4 z 7 6 z 5 + 3 z 0 z . 1. z0 C </p> <p>2. z0 C 0 = z 2 + z 0 z .</p> <p> 7 . .ab 1 = 1 ab</p> <p> 1 = b = 1 , a 1 &lt; b &lt; 1 , a</p> <p>1. 2.</p> <p>ab 1 z = r1 2r z z 2 1 2r z + z 2 </p> <p>1. 2.</p> <p> 01 1 z3 , z2 , z - 1 Arg z3 &gt; Arg z2 &gt; Arg z- 0 3 , z1 = z2 = z . Arg . Arg3z2 z z 2 = Arg 1z3 z 1z</p> <p>1. 2.</p> <p>2z3 z 1 z 2 = Arg 1z3 z 2 1z</p> <p>): (</p> <p> 11 . z+w = z + w 1. , z = w, z + w = z + w</p> <p> 0 = z . Arg z = Arg w</p> <p>2.</p> <p> 21 . .1 Arg z</p> <p> n - 0 z n R</p> <p>1. 2.</p> <p>- }{ z n : n N</p> <p>1</p> <p> Arg z</p> <p>3</p> <p> 31 1. z1 , z2 , z3 , z4 C , - 3 z1 , z , 3. z4 = z1 z2 + z 2. z1 , z2 , z3 , z4 C 0 = 4 , z1 + z2 + z3 + z .</p> <p> 41 1. 1 z3 , z2 , z 0 = 3. z1 + z2 + z 2. 0 = 3 z1 + z2 + z- 0 3 z1 = z2 = z 1 z3 , z2 , z .</p> <p> 51 . a b , a , b C - } { z : z ia = z ib . - } { z : z + a = z b .</p> <p>:{z :{z</p> <p>za = zb za = zb</p> <p>} }</p> <p>1. 2. </p> <p>4</p> <p> )"( 1134202 : : 1,2 2 : : 8 1/0102 :</p> <p> : 9002.11.31</p> <p> : " "</p> <p> 1 )01 ( . a 1, a C = ) f ( z }1 = { z : z .za 1 za</p> <p> 2 )02 ( ABCD . H , G , F , E ) ( . EG- FH .</p> <p> 3 )01 ( - ln 1 + z z z - 1 .</p> <p> 4 )01 ( 1 + 2 p ( z ) = z 2009 + 243 z 24 212 z11 + 4 z 1 &lt; . z</p> <p>5</p> <p> 5 )01 ( . 0 &lt; Argz+i M z G - . z + z + z = M3</p> <p> 8 )51 ( C .</p> <p>6</p> <p> )"( 2034202 : : 2,3 2 : : 02 1/0102 :</p> <p> : 9002.21.11</p> <p> " www.openu.ac.il/sheilta</p> <p> : </p> <p> 1 4 := D</p> <p>&lt; 0 :{z</p> <p>}z &lt; </p> <p>} , B = { z : Im z &lt; Re z</p> <p>}, A = { z : 0 &lt; Im z Re z 1 B. A B. 1. 2.</p> <p> 2 B D. A R ), R (. 1. 2.</p> <p> 3 D. 1. 2. K - . A K</p> <p>7</p> <p> 4 . f ( z ) = exp z f-- - . D ) f 1 ( B. 1. 2.</p> <p> 5 - Int R . R A- B , : ) . Int( A B ) = (Int A) (Int B ) . Int( A B ) = (Int A) (Int B 1. 2.</p> <p> 6 lim.0 z</p> <p>Re z i Im z z</p> <p>1. 2.</p> <p>) lim ( z Arg z.0 z</p> <p> 7 Re z Im z 0, z 2z = ) f ( z . 0= 0 , z </p> <p>1.</p> <p>= ) f ( z .</p> <p>z Re z Im z</p> <p>2. </p> <p> 8 ) f ( z ) = z n + nei z ( n 2 , n N , R-- . 1. 2. ) f ( z-- D .</p> <p> 9 1. a S- f - a - S f - , S a . S 2. } { zk S z0 S f - S - zk - S ,2 ,1 = , k f - 0 z - . S</p> <p>8</p> <p> 01 1. ) sinh( x + iy ) = sin( x + iy . x = y sinh( x + iy ) = sinh 2 x + sin 2 y . x, y R2</p> <p>2.</p> <p> 11 }0 &lt; R1 = { z : 2 &lt; Im z . exp } R2 = { z : 0 &lt; Re z &lt; 2 . exp 1. 2.</p> <p> 21 1. } R3 = { z : (Re z )3 &lt; Im z &lt; (Re z )3 + 2 . LR3 ( e ) = 1 + i = ) . LR4 (i2 } R4 = { z : 0 &lt; Im z &lt; i</p> <p>2.</p> <p> 31 1. z, z , . 2. exp z , R R .</p> <p> 412) ( z 0, z f ( z ) = z - 0 = . z 0= 0 , z </p> <p>1. 2.</p> <p> f D- ) 2 f ( z1 ) = f ( z z1 , z2 D 2Im z1 = Im z</p> <p> ) f ( z - . D</p> <p> 51 1. ) f ( x + iy ) = ( x 3)2 ( y + 5)2 sin( x + y . 2. f .</p> <p> 61 1. ) f ( x + iy ) = ( x3 3 xy 2 + x ) + i ( 3 x 2 y y 3 + y . = ) f ( x + iy .y + ix 2 x2 + y</p> <p>2.</p> <p>9</p> <p> 71 1. f R f - .R ) f ( z, z, 0 = )0( f (0) = 0 , f /</p> <p>2.</p> <p> , .</p> <p> 81 02 . ) f ( x, y 2 D R 2 - D ( x, y ) D : .2 f 2 f 0= 2 + 2 x y</p> <p> 81 1. ) f ( z D Im f , Re f - . D ) f - Im f , Re f , .( 2. v , u D ) f ( x + iy ) = u ( x, y ) + iv ( x, y - .D</p> <p> 91 v , u f = u + iv , D- g = v + iu - D u- v</p> <p>1.</p> <p> . 2. f = u + iv D g = v iu - . D</p> <p> 02 1. u D f D - . Re f = u 2. ) f ( x + iy ) = u ( x, y ) + iv ( x, y - D u v - . D</p> <p>01</p> <p> )"( 3034202 : : 4 2 : : 51 1/0102 :</p> <p> : 9002.21.52</p> <p> " www.openu.ac.il/sheilta</p> <p> - - - - </p> <p> 1 9 :</p> <p> 1 5 :)0 &gt; 1 (t ) = t + i R 2 t 2 , R t R , ( R</p> <p>)0 &gt; 2 (t ) = Reit , 0 t , ( R 1 3 ( t ) = 1 2t , 0 t 4 (t ) = e2 it 1 , 0 t 2 5 (t ) = i cos t , 0 t 2 6 (t ) = i cos t , 0 t </p> <p> - j . j</p> <p> 1 2 . 3 . 1. 2.</p> <p>11</p> <p> 2 4 -. 5 . 1. 2.</p> <p> 3 4 . 5 . 1. 2.</p> <p> 4 5. 6 = 1. 2. 1 = R 3 2 + -.</p> <p> 5 .5</p> <p> zdz = zdz6</p> <p>1. 2.</p> <p>1 2 . </p> <p> 6 .</p> <p> f ( z ) dz </p> <p> ) f ( z f ( z ) dz</p> <p>1. 2.</p> <p>.</p> <p>1= z</p> <p>8 = z 1 dz</p> <p> 7 1. , L . 1dz = L</p> <p>2. C 1 = z- ) z = 1 + i 1 (, . Im zdz Re zdz = 2 + iC C</p> <p>21</p> <p> 8 1. ) F ( z z C . F ( z ) = z&lt; 1 ) f ( z 1 f } 1 &lt; D = { z : z 2</p> <p>2.</p> <p> f-- - . D</p> <p> 9 . lim1 z 2 + 10 z 0 = dz )1 + R ( z 2)( z 3 + z z =R</p> <p>1.</p> <p>.</p> <p> Im z 1 z + z dz 1= z</p> <p>2.</p> <p> 01 C - z = i - , z = iC</p> <p>= z dz</p> <p>i i</p> <p>. . .</p> <p>02</p> <p>. "" .</p> <p> 11 C 0 , 1 , i , 1 + i ,</p> <p> C</p> <p>= z dz</p> <p>2</p> <p>i</p> <p>. . . .</p> <p>0 1 i 1+ i</p> <p> "" .</p> <p>31</p> <p> 21 ,0 t2 = ) , (t</p> <p>2</p> <p> ) 3 , f ( x + iy ) = ( x3 3 xy 2 ) + i (3 x 2 y y t + i sin t</p> <p>= f ( z ) dz i</p> <p>.</p> <p> 2 </p> <p>3</p> <p>.</p> <p>1</p> <p>.</p> <p>0</p> <p>.</p> <p> "" .</p> <p> 31</p> <p>= =1 e z dz || ze</p> <p>.</p> <p>.</p> <p>1</p> <p>.</p> <p>0</p> <p>.</p> <p> "" .</p> <p> 41 C 1 2 y = x )1 ,0( )0,1( , Log zdz -C</p> <p>2</p> <p>1 i</p> <p>. .</p> <p>2i2 ln</p> <p>. . ) (. "" .</p> <p> 51 lim - 0 rza =r</p> <p>f ( z ) dz ) f ( z , z = a za</p> <p>0</p> <p>. . . .</p> <p>)f (a) 2 if ( a</p> <p> "" .</p> <p>41</p> <p> )"( 2134202 : : 5,6 3 : : 7 1/0102 :</p> <p> : 0102.1.22</p> <p> : " "</p> <p> 1 )01 ( , C 1 ,1,0 .C</p> <p>e z dz )1 2 z ( z</p> <p> 2 )02 ( :2</p> <p>. .a r ,</p> <p>e 0</p> <p>) ( ei i</p> <p>d</p> <p>1. 2.</p> <p> z z a dz z =r</p> <p>3</p> <p> 3 )51 ( . v = Im f 1 &lt; . 0 &lt; r2 0</p> <p> ) f ( z , , u = Re f=2</p> <p>2 ( u ( rei ) ) d</p> <p>) ) ( v( rei 0</p> <p>2</p> <p>d</p> <p> )0( u (0) = v</p> <p> 4 )51 ( . a rdz az z =r</p> <p>51</p> <p> 5 )01 ( f f ( z ) e z . z C</p> <p> 6 )51 ( g , f M - ) ) Re ( f ( z ) ) M Re ( g ( z</p> <p> . z C . b , a - f ( z ) = ag ( z ) + b . z C . M1 M - ) ) Re ( f ( z ) ) M 1 Re ( g ( z . z C : g , f .</p> <p> 7 )51 ( ) f ( z 1 &lt; . z1 ) f ( z 1 z</p> <p>= ) f ( z an &lt; e 1 . n</p> <p> an z n 0= n</p> <p>61</p> <p> )"( 3134202 : : 6 3 : : 7 1/0102 :</p> <p> : 0102.2.91</p> <p> : " "</p> <p> 1 )02 ( z = : 0 = . 0 = . 1 = . 1 = . = ) , f (0) = 1 , f ( z = ), f ( z) 3 sin( z 3z</p> <p>. . . .</p> <p>z 21 z 2z2z 2 )1 + ( z</p> <p>= ), f ( z</p> <p>) 2 , f ( z ) = sin(2 z z</p> <p> 2 )01 ( :</p> <p> an z n 0=n</p> <p>1 = 0 3an + 4an 1 an 2 = 0 , a1 = 1 , a 2 . n . : , .</p> <p> 3 )01 ( 1 = z: = ). f ( z1 011 + z 2 + z 4 + z 6 + z 8 + z</p> <p>: .</p> <p>71</p> <p> 4 )02 ( z 0, z 1 . f ( z) = ez 0=1 , z </p> <p>. ) f ( z 0 = . z ? . )0( ) ) Bn = f ( n Bn Bernoulli , (. 2 n- 0 = 1+ B2 n n. n 1 n</p> <p> Bk k =0 k </p> <p> 0 =</p> <p> = ) g ( z 0 = z .</p> <p>0 z cot z , z 0=1 ,z</p> <p>.</p> <p>: ) g ( z- ) . f ( z</p> <p>. tan z 0 = . z ? 5 )51 ( ) f ( z D 0, ei . 2</p> <p>)1,0[ - )</p> <p> f - . D 6 )01 ( g , f 0 , z 0 zn z - f f = g g } . { zn c - ) f ( z ) = cg ( z z . 7 )51 ( : . f D , f - . D . 0 = z:1 1 , 0 = f + f n. n n 0 = )0( f (0) = f</p> <p> f 0 = z: 3 = , f 3 = f n. n 1+ n 1+ n n 1 n 1 n</p> <p>.</p> <p>81</p> <p> )"( 4134202 : : 7 1.8, 2.8 8 3 : : 7 1/0102 :</p> <p> : 0102.3.21</p> <p> : " "</p> <p> 1 )51 ( . .= )f ( z 1 cos z 2 z 3 + iz</p> <p>.</p> <p>f ( z) = e</p> <p>cot z</p> <p>.</p> <p>= )f ( z</p> <p>1 + ei z 2 )1 4 ( z</p> <p>.</p> <p> 2 )01 ( , , . 1= . z f ( z ) = sin z 1 z</p> <p>. . .</p> <p> 0 = . z 1 = . z</p> <p>f ( z ) = Log z f ( z ) = tan 1 1 z</p> <p> 3 )51 ( 0 ? z )f ( z 1 z</p> <p> f }0{ \ C</p> <p>91</p> <p> 4 )02 ( f 1 = z- 3 = z f1</p> <p> . f</p> <p> an ( z 2)n = n</p> <p>. , , f . . , f f- an .: - ) . g ( z ) = ( z + 1)( z 3) f ( z</p> <p> 5 )01 ( f , 0 &lt; z &lt; R 0 &gt; M 0 &lt; r &lt; R . r f ( rei ) d &lt; M0 2</p> <p> 0 = z . f</p> <p> 6 )51 ( f . : 1 = ) f ( z ), ( 0 , z = ei f .</p> <p>&gt; . z: z . z</p> <p>{</p> <p>1 2</p> <p> }</p> <p> 7 )51 (. ) f ( z</p> <p>)f ( z 1 ) f ( z 1 = z 3z</p> <p> f ( z ) z 1 . z3 ' 1 &gt; 0 z - 0 f ( z0 ) = z 3 f ( z ) = z</p> <p>3</p> <p>.</p> <p>&gt; . z</p> <p>1 2</p> <p>02</p> <p> )"( 4034202 : : 8 2 : : 02 1/0102 :</p> <p> : 0102.3.62</p> <p> " www.openu.ac.il/sheilta</p> <p> 1 3 D 1 i . f ( z ) = e z e 1 f- D:2e</p> <p>.</p> <p>+e</p> <p>1 e</p> <p>.</p> <p>e</p> <p>.</p> <p>e</p> <p>1 e</p> <p>.</p> <p>1</p> <p>.</p> <p> "" . 2 f- D: . . . . . . . . D "" . 3 f- : D . . D . . D . D . . . D "" .</p> <p>12</p> <p> 4 7 )4 g ( z ) = exp( z 2 + 2 z</p> <p> 4 :. 61e . 21e</p> <p>{z: z</p> <p> g } 2 </p> <p>. e</p> <p>. 1</p> <p>. 0</p> <p> "" .</p> <p> 5 :</p> <p>{z: z</p> <p> g } 2 </p> <p>. 2 = . z . 2 &lt; . z , 2 = z 2&lt; .z . 2 = . z "" . .</p> <p> 6 :</p> <p>{z: z</p> <p> g } 2 z = e z 0 &gt; . Re z</p> <p> 7 )01 ( } D = { z : z &lt; R f D D . D z D - . f ( z ) = z</p> <p>82</p> <p> )"( 6134202 : : 01,11 3 : : 8 1/0102 :</p> <p> : 0102.5.41</p> <p> : " "</p> <p> 1 )51 ( : . , f , D }) { pn ( z - f - . D .</p> <p> 0=n</p> <p>cos nz !n</p> <p>.</p> <p> 2 )51 ( g , f . 1 &lt; z &lt; r :1 2 i 1 z f ( w) g dw w w w =r</p> <p> - r ) h ( z , = ) . h( z</p> <p> anbn z n 0=n</p> <p>= ) ) g ( z 1 &lt; ( z</p> <p> bn z n 0=n</p> <p>= ), f ( z</p> <p> an z n 0=n</p> <p> 3 )01 ( = ) . h( z</p> <p> et 0</p> <p>) 2 cos(t dt + zt</p> <p> - R h }. { z : z &lt; R</p> <p>92</p> <p> 4 )01 ( . f ( x) = arccos x ). f (x .</p> <p>&lt; , D1 = z : &lt; arg z &lt; . D2 = z : 0 &lt; arg z</p> <p>{ {</p> <p>3 2</p> <p> } }2</p> <p> 5 )01 ( f1 ( z...</p>