Chapter2 Analytic Functions

8/6/2019 Chapter2 Analytic Functions

1/23

70?C4A =0;HC82 D=2C8>=B

G4A28B4B

%064B >A 4027 >5 C74 5D=2C8>=B 14;>F 34B2A814 C74 3>5 34J=8C8>= C70C

8B D=34ABC>>30 "2

!

1 "2 +#!

2 "2 !!!

3 "2 !

=B 0 2 $ 1 2 (D??>B4 C70C "2 2 2 8= C74 5>A< "2 - 0 1 $. 0 1

F74A4 I G8H *B4 C74 4G?A4BB8>=B 0 !!

0=3 1 !!

C> FA8C4 5I 8=C4A5 I 0=3 B8


2/23


3/23

8B A= C> C74 B42C>A

(>;DC8>= 2 !4C / "2 2 )74= 852 +! 0=3 / ! F4 70E4! +! 0=3 + + )74 B42C>A + 8B A= C> C74 B42C>A

(7>F C70C C74 ;8=4B 0H G 0A4 =C> C74 B?8A0;B ! D=34A C74 CA0=B5>A= F 4G? I F74A4 / !$

(>;DC8>= / ! 2 ! 0 !$1 >A 0H G F4 70E4 !$ !1!$1 )7DB F4 E4A 0=H A40; =DD;3 14 2>AA42C F8C7 A4?;0243 1H & 5>A 0=H & >F4E4A C78B 3>4B =>C 270=64 C74 B70?4 >5 C74 8AA4B?>=38=6 ?>AC8>=B >5 C74 1>D=30A84B

(>;DC8>= / - $. !! !! )74 85 C74 ;8=4 G 8B C74 28A2;4>5 A038DB )74 85 C74 ;8=4 H F8C7 G 8B 8= C74 ?>B8C8E4 7>A8I>=C0;0G8B F8C7 - 0=3 C74 85 C74 ;8=4 1 8B C74 =460C8E4 A40; 0G8B F8C7- B> C74 8DCB834 C74D=8C 28A2;4

G4A28B4B %064B *B4 34J=8C8>= (42C8>= >5 ;8 ?A>E4 C70C


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0 !!

2 2

1 !!2 22

!!

!

(>;DC8>= 0 >A 0=H ;4C 5 2 2 C74= 2 2 2 2 2 2 B>

!! 2 2

(>;DC8>= 1 >A 0=H ;4C 5 2 2 C74=2 2 2 2 2 2 B>

!! 2 2

(>;DC8>= 2 >A 0=H ;4C 5 2 2 C74= !! !! 2 B> !!

!

!4C = 14 0 ?>B8C8E4 8=C464A 0=3 ;4C %I 0=3 &I 14 ?>;H=>A4< 8= (42C8>= 0=3 ;8=

C> J=30

!!

!2

1 !

!!

2 !!

!!

(>;DC8>= 0 (8=24 !!

2 C74 ;85 C78B 8B

!

(>;DC8>= 1 5 F4 A4?;024 I F8C7 8 C74= C74 4@D0C8>= 142>;DC8>= 2 !!

(7>F C70C C74 ;85 C74 5D=2C8>= "2

!!

0B I C4=3B C> 3>4B =>C

4G8BC > C78B 1H ;4CC8=6 =>=I4A> ?>8=CB I G 0=3 I GG 0??A>027 C74

>A868= -#>C4 C70C 8C 8B =>C BDN284=C C> B88=CB I G 0=3 I H 0B 8C F0B 8= G0


5/23

"2

$= C74 >C74A 70=3 0B 0 0 F4 70E4

"2

$

(8=24 C74B4 38O4A C74 ;8= (42C8>= >5 ;8 ?A>E4 C70C 85

!!"2 /

C74= !!

"2 / (D664BC8>= $1B4AE4 7>F 8=4@D0;8CH (42C8>= 4=01;4B >=4 C> FA8C4

"2 / "2 /

(>;DC8>= H 34J=8C8>= F4 70E4 C70C 5>A 0=H C74A4 8B 0 B> C70C 2 2 "2 / *B8=6 C74 CA80=6;4 8=4@D0;8CH F4 8= CDA=70E4 C70C"2 / "2 / B> 85 2 2 C74= "2 / "2 / )78B 4=3B C74 ?A>>5

(7>F C70C !!

"2#2 85 !!

"2 0=3 85 C74A4 4G8BCB 0 ?>B8C8E4

=DF C70C 5>A 0=H

C74A4 8B BD27 C70C 2 2 "2 ,40;B> :=>F 1H 1>D=343=4BB C70C C74A4 8B 0 B> C70C # 2 5>A2 2 !4C *B8=6 C78B 8C 5>;;>FB C70C "2 # 2 "2 # 2 F74= 2 2 )7DB 1H34J=8C8>=

!!"2 # 2

*B4 C74 C74>A4< 8= (42C8>= C> B7>F C70C0

!!

!

1 !

!

2 !

!!

(>;DC8>= 0 )78B ;8= 1 )78B ;8;3B B8=24 !

2 B> C74 A428?A>20; 70B;8;DC8>= 2 ,4 70E4 !

!! !

!! !

!!

)74 34B8A43 ;820; 0=3 B> 8B ,8C7 C74 083 >5 C74 C74>A4< 8= (42C8>= B7>F C70C F74= 2 !

! 0

!2 85 2

1 !

2 0=3 !2 85


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(>;DC8>= 0 (8=24 2 5A>< F4 70E4 B> 1>C7 0 0=33 0A4 =>C 4@D0; C> #>F

!! (8=24 !

2 B> !

2

0=3 !

2 )7DB !

!

!!

(>;DC8>= >A 0 F4 =443 C> B7>F C70C !

! 85 2 =3443

!

! !

! !

!! >A C74 JABC ;8A C74 B42>=3 ;8 B7>F C70C !

!

=

3443 !

!

!!!

B8=24

1H 0BBD


7/23

*B8=6 A4BD;CB 8= (42C8>= B7>F C70C0 0 ?>;H=>5 346A44 = ( 8B 38O4A4=C801;4 4E4AHF74A4 F8C7 34A8E0C8E4 2 2 (2

1 C74 2>4N284=CB 8= C74 ?>;H=>;DC8>= 0 ,4 :=>F C70C

2

&2 5>A & 0=3 0;; 2 )7DB

2

2 &2 (> F4 70E4 2

2

&2

&2 !4C 9 : )74= 9 AD=B 5A>< C> = 0=3 :

9 )7DB 2

% 2

2 (2

(>;DC8>= 1 A>< 0 F4 70E4 2

2 F74A4 &

)7DB 2 2

2 F74A4 & & &

,4 2;08< C70C 5>A 0;; % 2

& , 2 ,4 DB4 8=3D2C8>=

C> ?A>E4 C78B 2;08


8/23

!

!!!

! (D??>B4 C70C "2 # 2 0=3 C70C " 2 0=3 # 2 4G8BC F74A4

# 2 4G8BC F74A4 # 2 *B4 34J=8C8>= (42C8>= >5 34A8E0C8E4 C>B7>F C70C

!!!!

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(>;DC8>= !!

!!

!!

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!!

!!

!!!!

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!

*B4 C74 3 8= G0


9/23

3 "2 !!(>;DC8>= 2 - 0 . 01 ,4 70E4 - 0=3 . 01 >A C74B4

C> 14 4@D0; F4 70E4 - 0=3 . 1 >AC74B4 C> 14 4@D0; F4 70E4 H 1DC C78B 2>=CA0382CB GH

(>;DC8>= 3 - ! 1 . ! 1 ,4 70E4 - . F7827 68E4B! 1 ! 1 )78B 68E4B 2>B H (8 B8= H 1DC F4 20==>C 70E4 2>B H B8= H

!4C C74 5D=2C8>= "0 1 $. 0 1 14 34J=43C7A>D67>DC B>A7>>3 >5 0 ?>8=C 2 0 $1 0=3 BD??>B4 C70CC74 JABC>A34A ?0AC80; 34A8E0C8E4B >5 C74 5D=2C8>=B D 0=3 E F8C7 A4B?42C C> G0=3 H 4G8BC 4E4AHF74A4 8= C70C =48671>A7>>3 5 C7>B4 ?0AC80; 34A8E0C8E4B0A4 2>=C8=D>DB 0C 0 1 0=3 B0C8B5H C74 0D27H'84=B- . -

. 0C 0 1 C74= " 2 4G8BCB

*B4 C74 C74>A4< C> B7>F C70C 5LI 0=3 8CB 34A8E0C8E4B 5MI 4G8BC 4E4AHF74A40=3 J=3 5MI F74=0 5I 8I1 "2 !!2 "2 23 "2 0 1 $ 0 1

(>;DC8>= 08E4= 5I 8I8G8H8GHH8G>8=CB 4A4

- . - . 0C 0;; ?>8=CB GH (> 1H C74>A4


10/23

!4C # 2 " 2 ! 1 $! 14A4 -

! 1 . ! 1

- ! 1 - ! 1 . ! 1 . ! 1)78B 8


11/23

!4C # 2 " 2 0 1 $ 0 1

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(>;DC8>= 1 (D??>B4 "0 $1 - 0 1 $. 0 1 5>A A40; 5D=2C8>=B D0=3 E A>< "2 2 2 F4 J=3 D 0=3 E H A>< - . 4E4AHF74A4 F4 B44 " 2 3>4B =>C 4G8BC 0C 0=H ?>8=C

(>;DC8>= 2 (D??>B4 "0 $1 - 0 1 $. 0 1 5>A A40; 5D=2C8>=B D0=3 E ,4 B44 D G 0=3 . 01 )74= - . . 1 0=3 . 015 - . C74= 1 0=3 B> H 5 C78B 8B CAD4 C74= . - )7DB - . 0=3 - . 3> =>C 1>C7 7>;3 0C 0=HF74A4 )7DB 5LI 3>4B=>C 4G8BC 0C 0=H ?>8=C

(>;DC8>= 3 (D??>B4 "0 $1 - 0 1 $. 0 1 5>A A40; 5D=2C8>=B D0=3 E ,4 J=3 - ! 1 ! 1 0=3 . ! 1 ! 1 )7DB - !

1 - ! 1 . ! 1 . ! 1 5- . 0=3 - . C74= 1 1 1420DB4 ! )78B 8B8BB81;4 1420DB4 1 1 5>A 0;; 1 )7DB 5LI 3>4B =>C4G8BC 0C 0=H ?>8=C


12/23

*B4 C74 C74>A4< 8= (42C8>= C> B7>F C70C 4027 >5 C74B4 5D=2C8>=B 8B38O4A4=C801;4 8= C74 8=3820C43 3>5 34J=8C8>= 0=3 C74= DB4 4G?A4BB8>=

8= C70C B42C8>= C> J=3 5LI0 5I I 2 1 5I

+! +

2 5I 4 + $! + + (>;DC8>= 1 !4C -

+ 0=3 .

+ )74= +-

+ . 0=3 B8= 8B 0;A403H B0C8BJ43C 8B CAD4 85 0 1 0 1 0 1 2 0 $1 $ $(> 5 8B 38O4A4=C801;4 >=;H F74= I 8" 2 - $. 0 $ 0

!4C D 0=3 E 34=>C4 C74 A40; 0=3 8=4=CB >5 C74 5D=2C8>= 5

34J=43 1H C74 4@D0C8>=B 5I

!

! F74= 2 F74= I

(>;DC8>= "2 !

!

5>A 2 )74= -

(8


13/23

)78B 8B - $ $. $ ! - $. 0 '420;; (42C8>= C70C 85 I G 8H C74= G !! 0=3 H

!!

H 0??;H8=6 C74 2708= AD;4 8= 20;2D;DB C> 0 5D=2C8>= GH >5 CF> A40;

E0A801;4B 34A8E4 C74 4G?A4BB8>= !

!

!

$

1 4J=4 C74 >?4A0C>A !

$

BD664BC43 1H ?0AC 0 C> B7>F

C70C 85 C74 JABC>A34A ?0AC80; 34A8E0C8E4B >5 C74 A40; 0=3 85 05D=2C8>= 5I DGH 8EGH B0C8B5H C74 0D27H'84=B C74=!

- - $ . - )7DB 34A8E4 C74 ! >5

C74

(>;DC8>=0 >A;;>FB 8


14/23

(>;DC8>= 2!4C "2 !

0

$!

0 - $.

- ! 0 . ! 0 - ! 0 - ! 0 . ! 0 . ! 0)7DB - - . . 0A4 0;; 2>=C8=D>DB 0C 0;; ?>8=CB GH 608= - . 0=3

- .D 0=3 E B0C8B5H 0D27H'84=B 0C 0;; ?>8=CB (> 5 LI 4G8BCB

4E4AHF74A4 5I 8B 0=0;HC82 4E4AHF74A4 5I 8B 4=C8A4(>;DC8>= 3!4C "2

2 ! 2 ! 2 !! ?>;H

=>

2 !! 8B 0=0;HC82 4E4AHF74A4

" # 0A4 0=0;HC82 8= 0 3>=0; 5D=2C8>= 2

2

2 2 $ 2 $ 2 $ $ ?>8=CB >5 5I 0A4 I 8 8 (8=24 8C 8B 0 A0C8>=0; 5D=2C8>= 8C 8B

0=0;HC82 4E4AHF74A4 4G24?C 0C C74B4 ?>8=CB


15/23

1 "2 !

!! C 8B 0 A0C8>=0; 5D=2C8>= 2 2

2 2 2 ?>8=CB 0A4 I (8=24 8C 8B 0 A0C8>=0; 5D=2C8>= 8C 8B 0=0;HC82

4E4AHF74A4 4G24?C 0C C74B4 ?>8=CB

2 "2 !

!!! C 8B 0 A0C8>=0; 5D=2C8>=

2

2 2

2 >A 2 2 2 >A2

$ $ 2 $ $

(8=6D;0A ?>8=CB 0A4 I 8 8 (8=24 8C 8B 0 A0C8>=0; 5D=2C8>= 8C 8B0=0;HC82 4E4AHF74A4 4G24?C 0C C74B4 ?>8=CB

(>;DC8>= 1 =0;HC82 4G24?C 852 2 C70C 8B 4G24?C 0C I F7827 0A4 B8=6D;0A ?>8=CB

22>A38=6 C7 G4A28B4 1 (42C8>= C74 5D=2C8>=# 2

+! + 8B 0=0;HC82 8CB 3>F C70C C74 2>B8C4 5D=2C8>= I 6I8 8B

0=0;HC82 8= C74 70;5 ?;0=4 G F8C7 34A8E0C8E4 2 ! (D664BC8>= $1B4AE4 C70C '4I8 F74= G (>;DC8>= ,74= G 2 $ 0 B> 2 $ 8B 8= C74

3>5 34J=8C8>= >5 6I )7DB # 2 $ 8B 0=0;HC82 8= C78B 3>= C> E4A85H C70C C74 5D=2C8>= # 2 +

$8 + 8B 0=0;HC82 8= C74 8=3820C43 3>5 34J=8C8>= F8C7 34A8E0C8E4 6LI I)74= B7>F C70C C74 2>B8C4 5D=2C8>= 2 #

2

8B 0=0;HC82 8= C74

@D03A0=C G H F8C7 34A8E0C8E4 2 !! (D664BC8>= $1B4AE4 C70C ;DC8>= )74 5D=2C8>= "2 - + $. + 8B 0=0;HC82 8= 0 3>=C8=D>DB 5D=2C8>=B 8= C70C 3>


16/23

(>;DC8>=,4 70E4 - + 0=3 . )7DB +- + + .- +. (8=24 - + 8B 2>=C8=D>DB >= C74 68E4= A468>= 0=3 C74

>C74A JABC ?0AC80; 34A8E0C8E4B 0A4 CA8E80;;H 2>=C8=D>DB F4 B44 C70C 6 8B 0=0;HC828= C74 68E4= 3>C4 C70C

2

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0 1 $01 B0C8BJ4B C74 0D27H'84=B

- 0 . - 1 . 0=3 70B 2>=C8=D>DB JABC ?0AC80;B )7DB 8C 8B0=0;HC82 0=3 B8=24 C74 2>B8C8>= >5 CF> 0=0;HC82 5D=2C8>=B 8B 0=0;HC82 F4 0A4J=8B743

!4C 0 5D=2C8>= 5I 14 0=0;HC82 8= 0 3>E4 C70C 5I =BC0=C C7A>D67>DC 85

0 5I 8B A40;E0;D43 5>A 0;; I 8= 1 "2 8B 2>=BC0=C C7A>D67>DC (D664BC8>= *B4 C74 0D27H'84=B 0=3 C74 C74>A4< 8= (42C8>=

C> ?A>E4 ?0AC 0 )> ?A>E4 ?0AC 1 >1B4AE4 C70C "2

! 85 "2 C74= DB4 C74


17/23

)74 B42>=3 68E4B 01 " 0 01 B> "0 )74 2>=9D60C4 8B . 1

01 1

(>;DC8>= 0 ,4 70E4 . - 0 )7DB . 0 1 $= C74>C74A 70=3 . - 1 )7DB 1 1 F7827 8


18/23

)> ?A>E4 C74 2>=E4AB4 F4 0BBD=9D60C4 >5 E )70C8B 1H )74>A4< 6 E 8D * 8+ 8B 0=0;HC82 )7DB F4 70E4 2>=C8=D8CH >5

JABC 34A8E0C8E4B 0=3 . - . -,4 C7DB B44 C70C 5 D 8E B0C8BJ4B C74 0D27H'84=B 0=3

C7DB 5 8B 0=0;HC82 ">A4>E4A 1H )74>A4< C78B 8


19/23

0 1 C A4?A4B4=CB 0 28A2;4 F8C7 24=C4A

0=3 A038DB >A 0

1 1 0 1 1

0 1

C A4?A4B4=CB 0 28A2;4 F8C7 24=C4A

0=3 A038DB

#>C4 7>F 8C 5>;;>FB 5A>< C74 4@D0C8>=B - 0 1 0=3. 0 1 C70C

0=3

(>;DC8>=,4 :=>F 5A>< 8=C >5 C74 3>A 85 =>C 8B ?4A?4=382D;0A C> C74 ;4E4; 2DAE4 C70C ?0BB4B C7A>D67C70C ?>8=C (8=24 8C F8;; BDN24 C> B7>F C70C C74 =>AAB 0A4 >AC7>6>=0;F4 =443 >=;H B7>F C70C - 0 1 . 0 1 -. -. (8=24 5 8B0=0;HC82 F4 :=>F C70C DE B0C8B5H C74 0D27H'84=B . - . - F7827 B7>FB C70C -. -. -- -- 0B 34B8A43

(>;DC8>=

!4C I G 8H ,4 70E4 "2 2 0 $1

$

>8=C GH A4B?42C8E4;H ,4 70E4 =>F

)74A45>A4

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4=24 ''

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20/23

. 0 1 0=3 85 " 2 C74= C74 ;8=4B C0=64=C C> C7>B4 2DAE4B 0C

0 1 0A4 ?4A?4=382D;0A #>C4 7>F 8C 5>;;>FB 5A>< C74 4@D0C8>=B

- 0 1 0=3 . 0 1 C70C 0=3 (7>F C70C F74= "2 2 C74 ;4E4; 2DAE4B - 0 1 0=3 . 0 1 >5 C74 2>=4=C 5D=2C8>=B 0A4 C74 7H?4A1>;0B 8=3820C438= C74 J6DA4 #>C4 C74 >AC7>6>=0;8CH >5 C74 CF> 50A868= 7>F4E4A C74H0A4 =>C >AC7>6>=0; C> 4027 >C74A ,7H 8B C78B 502C 8= 06A44E4BC0C4;DC8>="2 - $. 8B 0=0;HC82 8= 0 3>E8343 - 0=3 . 84 . 0=3

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C >A868= I G8H 88=C 85 "

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21/23

!4C C74 5D=2C8>= "2 - 0 1 $. 0 1 14 0=0;HC82 8= 03>=B834A C74 505 - 0 1 0=3

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2 C74= C74 ;8=4B C0=64=C C> C7>B4 2DAE4B 0C!4CC74 5D=2C8>= "2 - 0 1 $. 0 1 14 0=0;HC82 8= 0

3>=B834A C74 505 - 0 1 0=3. 0 1 F74A4 0=3 0A4 0A18CA0AH A40; 2>=BC0=CB 52 0 18B 0 ?>8=C 8= F7827 8B 2> CF> ?0AC82D;0A 2DAE4B - 0 1 0=3. 0 1 0=3 85 "

2 C74= C74 ;8=4B C0=64=C C> C7>B4 2DAE4B 0C0 1 0A4 ?4A?4=382D;0A #>C4 7>F 8C 5>;;>FB 5A>< C74 4@D0C8>=B- 0 1 0=3 . 0 1 C70C

0=3

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0=3

(:4C27 C74 505 ;4E4; 2DAE4B >5 C74 2>=4=C 5D=2C8>=B D 0=3 EF74= "2 2 0=3 =>C4 C74 >AC7>6>=0;8CH 34B2A8143 8= C74 5>;;>F8=6

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2 C74=C74 ;8=4B C0=64=C C> C7>B4 2DAE4B 0C 0 1 0A4 ?4A?4=382D;0A #>C4 7>F 8C5>;;>FB 5A>< C74 4@D0C8>=B - 0 1 0=3 . 0 1 C70C

0=3

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F74= 5I !! 0=3 =>C4 7>F C74 A4BD;C 8= G4A28B4 8B 8;;DBCA0C43 74A4

(>;DC8>= !4C C74 5D=2C8>= "2 - 0 1 $. 0 1 14 0=0;HC82 8= 0 3>=B834A C74 505 - 0 1 0=3

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. 0 1 0=3 85 " 2 C74= C74 ;8=4B C0=64=C C> C7>B4 2DAE4B 0C0 1 0A4 ?4A?4=382D;0A #>C4 7>F 8C 5>;;>FB 5A>< C74 4@D0C8>=B

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0=3

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22/23

= C78B ?A>1;4< C74 5D=2C8>="2

+! + 8B 38O4A4=C801;4 8= C74 8=3820C43

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Chapter2 Analytic Functions