lesson 9 laurent series

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Lesson 9 Laurent series 1 Thursday, 5 September 13

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Lesson 9Laurent series

1Thursday, 5 September 13

2

� ;I�LEZI�WIIR�XLEX 8E]PSV�WIVMIW��

k=0

fkzk

GSVVIWTSRH�XS�*SYVMIV�WIVMIW�[MXL�SRP]�RSRRIKEXMZI�XIVQW

8LIWI�PMZI�MRWMHI�XLI�YRMX�HMWO

� 2S[� [I�MRZIWXMKEXI�WIVMIW�[MXL�SRP]�RSRTSWMXMZI�XIVQW�

�1�

k=��fkzk

8LIWI�PMZI�SYXWMHI�XLI�YRMX�HMWO

� ;I�EPWS�MRZIWXMKEXI 0EYVIRX�WIVMIW� [LMGL�LEZI�FSXL�TSWMXMZI�ERH�RIKEXMZI�XIVQW���

k=0

fkzk

8LIWI�PMZI�MR�ER�ERRYPYW

Thursday, 5 September 13

2

� ;I�LEZI�WIIR�XLEX 8E]PSV�WIVMIW��

k=0

fkzk

GSVVIWTSRH�XS�*SYVMIV�WIVMIW�[MXL�SRP]�RSRRIKEXMZI�XIVQW

8LIWI�PMZI�MRWMHI�XLI�YRMX�HMWO

� 2S[� [I�MRZIWXMKEXI�WIVMIW�[MXL�SRP]�RSRTSWMXMZI�XIVQW�

�1�

k=��fkzk

8LIWI�PMZI�SYXWMHI�XLI�YRMX�HMWO

� ;I�EPWS�MRZIWXMKEXI 0EYVIRX�WIVMIW� [LMGL�LEZI�FSXL�TSWMXMZI�ERH�RIKEXMZI�XIVQW���

k=0

fkzk

8LIWI�PMZI�MR�ER�ERRYPYW

Thursday, 5 September 13

2

� ;I�LEZI�WIIR�XLEX 8E]PSV�WIVMIW��

k=0

fkzk

GSVVIWTSRH�XS�*SYVMIV�WIVMIW�[MXL�SRP]�RSRRIKEXMZI�XIVQW

8LIWI�PMZI�MRWMHI�XLI�YRMX�HMWO

� 2S[� [I�MRZIWXMKEXI�WIVMIW�[MXL�SRP]�RSRTSWMXMZI�XIVQW�

�1�

k=��fkzk

8LIWI�PMZI�SYXWMHI�XLI�YRMX�HMWO

� ;I�EPWS�MRZIWXMKEXI 0EYVIRX�WIVMIW� [LMGL�LEZI�FSXL�TSWMXMZI�ERH�RIKEXMZI�XIVQW���

k=0

fkzk

8LIWI�PMZI�MR�ER�ERRYPYWThursday, 5 September 13

Analyticity at infinity

3Thursday, 5 September 13

4

� ;I�LEZI�HI½RIH�EREP]XMGMX]�IZIV][LIVI�I\GITX�EX �

� ;I�GEPP f(z) EREP]XMG�EX�MR½RMX] MJ

f

�1

z

MW�EREP]XMG�EX�^IVS

)\EQTPIW� 1, 1z �

1z2 � z+1

z�1 �1/z

� ;I�XLYW�WE] f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK � MJ�MX�MW�EREP]XMG�EX�IZIV]�½RMXITSMRX�SJ D ERH�EREP]XMG�EX �

� %R�MQTSVXERX�I\EQTPI�MW EREP]XMGMX]�SYXWMHI�XLI�YRMX�GMVGPI

Thursday, 5 September 13

5

� ;I�LEZI�HI½RIH�EREP]XMGMX]�IZIV][LIVI�I\GITX�EX �

� ;I�GEPP f(z) EREP]XMG�EX�MR½RMX] MJ

f

�1

z

MW�EREP]XMG�EX�^IVS

)\EQTPIW� 1, 1z �

1z2 � z+1

z�1 �1/z

� ;I�XLYW�WE] f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK � MJ�MX�MW�EREP]XMG�EX�IZIV]�½RMXITSMRX�SJ D ERH�EREP]XMG�EX �

� %R�MQTSVXERX�I\EQTPI�MW EREP]XMGMX]�SYXWMHI�XLI�YRMX�GMVGPI

Thursday, 5 September 13

6

� -J f MW�EREP]XMG�IZIV][LIVI�SYXWMHI�XLI�YRMX�HMWO�MRGPYHMRK�EX�MR½RMX]� f(1/z) MW�EREP]XMGMRWMHI�XLI�YRMX�HMWO

� 8LYW�[I�GER�I\TERH�MX�MR�E�8E]PSV�WIVMIW

f(1/z) =��

k=0

ckzk

� -R�SXLIV�[SVHW�

f(z) =0�

k=��c�kzk

[LMGL�GSRZIVKIW�SYXWMHI�XLI�YRMX�GMVGPI

� 8LYW c�k = fk �

Thursday, 5 September 13

7

� -J f MW�EREP]XMG�IZIV][LIVI�SYXWMHI�XLI�YRMX�HMWO�MRGPYHMRK�EX�MR½RMX]� f(1/z) MW�EREP]XMGMRWMHI�XLI�YRMX�HMWO

� 8LYW�[I�GER�I\TERH�MX�MR�E�8E]PSV�WIVMIW

f(1/z) =��

k=0

ckzk

� -R�SXLIV�[SVHW�

f(z) =0�

k=��c�kzk

[LMGL�GSRZIVKIW�SYXWMHI�XLI�YRMX�GMVGPI

� 8LYW c�k = fk �

Thursday, 5 September 13

8

� 8LI�SRP]�JYRGXMSRW�XLEX�EVI�EREP]XMG�IZIV][LIVI�MR C� MRGPYHMRK�EX �� EVIGSRWXERX�

� &IGEYWI f(z) MW�EREP]XMG�MRWMHI�XLI�YRMX�GMVGPI� fk = 0 JSV k > 0

� &IGEYWI f(z) MW�EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI� fk = 0 JSV k < 0

� 8LYW f(z) = f0

Thursday, 5 September 13

9

� 7YTTSWI f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK�MR½RMX]�WS�XLEX 1/D MW�WMQTP]GSRRIGXIH� ERH � MW�E�GYVZI�MR D� -J f(z) = O

�z�2

�XLIR

�f(z) z = 0

� 2SXI�XLEX �

�f(z) z =

1/�

f(1/�)

�2�

� ;I�ORS[�XLEX f(1/�) = O��2

� 8LIVIJSVIf(1/�)

�2=

��

k=0

f�k�k�2

GSRZIVKIW

� -R�SXLIV�[SVHW� MX�MW�EREP]XMG�MR�XLI�YRMX�HMWO� ERH�

f(1/�)

�2� = 0

Thursday, 5 September 13

10

� =1

z

� 7YTTSWI f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK�MR½RMX]�WS�XLEX 1/D MW�WMQTP]GSRRIGXIH� ERH � MW�E�GYVZI�MR D� -J f(z) = O

�z�2

�XLIR

�f(z) z = 0

� 2SXI�XLEX �

�f(z) z =

1/�

f(1/�)

�2�

� ;I�ORS[�XLEX f(1/�) = O��2

� 8LIVIJSVIf(1/�)

�2=

��

k=0

f�k�k�2

GSRZIVKIW

� -R�SXLIV�[SVHW� MX�MW�EREP]XMG�MR�XLI�YRMX�HMWO� ERH�

f(1/�)

�2� = 0

Thursday, 5 September 13

11

� =1

z

� 7YTTSWI f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK�MR½RMX]�WS�XLEX 1/D MW�WMQTP]GSRRIGXIH� ERH � MW�E�GYVZI�MR D� -J f(z) = O

�z�2

�XLIR

�f(z) z = 0

� 2SXI�XLEX �

�f(z) z =

1/�

f(1/�)

�2�

� ;I�ORS[�XLEX f(1/�) = O��2

� 8LIVIJSVIf(1/�)

�2=

��

k=0

f�k�k�2

GSRZIVKIW

� -R�SXLIV�[SVHW� MX�MW�EREP]XMG�MR�XLI�YRMX�HMWO� ERH�

f(1/�)

�2� = 0

Thursday, 5 September 13

12

� =1

z

� 7YTTSWI f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK�MR½RMX]�WS�XLEX 1/D MW�WMQTP]GSRRIGXIH� ERH � MW�E�GYVZI�MR D� -J f(z) = O

�z�2

�XLIR

�f(z) z = 0

� 2SXI�XLEX �

�f(z) z =

1/�

f(1/�)

�2�

� ;I�ORS[�XLEX f(1/�) = O��2

� 8LIVIJSVIf(1/�)

�2=

��

k=0

f�k�k�2

GSRZIVKIW

� -R�SXLIV�[SVHW� MX�MW�EREP]XMG�MR�XLI�YRMX�HMWO� ERH�

f(1/�)

�2� = 0

Thursday, 5 September 13

13

� �

U

tk

t � zt = 0

MJ k � 0 ERH z MW�SYXWMHI�XLI�YRMX�GMVGPI�SV k < 0 ERH z MW�MRWMHI�XLI�YRMX�GMVGPI

� *SV k � 0 ERH z SYXWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�MRWMHI�XLI�YRMX�GMVGPI� LIRGI�XLI�MRXIKVEP�MW�^IVS

� *SV k < 0 ERH z MRWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI�ERH�FILEZIW�PMOI O�tk�1

�= O

�t�2

�� LIRGI�XLI

MRXIKVEP�MW�EPWS�^IVS

Thursday, 5 September 13

13

� �

U

tk

t � zt = 0

MJ k � 0 ERH z MW�SYXWMHI�XLI�YRMX�GMVGPI�SV k < 0 ERH z MW�MRWMHI�XLI�YRMX�GMVGPI

� *SV k � 0 ERH z SYXWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�MRWMHI�XLI�YRMX�GMVGPI� LIRGI�XLI�MRXIKVEP�MW�^IVS

� *SV k < 0 ERH z MRWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI�ERH�FILEZIW�PMOI O�tk�1

�= O

�t�2

�� LIRGI�XLI

MRXIKVEP�MW�EPWS�^IVS

Thursday, 5 September 13

13

� �

U

tk

t � zt = 0

MJ k � 0 ERH z MW�SYXWMHI�XLI�YRMX�GMVGPI�SV k < 0 ERH z MW�MRWMHI�XLI�YRMX�GMVGPI

� *SV k � 0 ERH z SYXWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�MRWMHI�XLI�YRMX�GMVGPI� LIRGI�XLI�MRXIKVEP�MW�^IVS

� *SV k < 0 ERH z MRWMHI�XLI�YRMX�GMVGPI�

tk

t � z

MW�EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI�ERH�FILEZIW�PMOI O�tk�1

�= O

�t�2

�� LIRGI�XLI

MRXIKVEP�MW�EPWS�^IVS

Thursday, 5 September 13

14

� 7YTTSWI f MW�EREP]XMG�MR�E�HSQEMR D GSRXEMRMRK�MR½RMX]�WS�XLEX 1/D MW�WMQTP]GSRRIGXIH� ERH � MW�E�GYVZI�MR D� -J f(z) = O

�z�2

�XLIR

f(z) =1

2�

f(t)

t � zt

[LIR � MW�SVMIRXIH�[MXL�XLI�MRXIVMSV�SJ�XLI�GSRXSYV�MRWMHI D

Thursday, 5 September 13

15

z

Thursday, 5 September 13

15

z

z

Thursday, 5 September 13

15

z

z

z

Thursday, 5 September 13

15

z

z

z� 1

2�i

Br(z)

f(t)

t � zdt = �f(z)

After switching orientations:

Thursday, 5 September 13

15

z

z

z� 1

2�i

Br(z)

f(t)

t � zdt = �f(z)

After switching orientations:

Zero due to analyticity

Thursday, 5 September 13

16

Laurent series

Thursday, 5 September 13

17

� 7YTTSWI f MW�WQSSXL�SR�XLI�YRMX�GMVGPI�FYX RIMXLIV EREP]XMG�IZIV][LIVI�MRWMHI�SVSYXWMHI�XLI�YRMX�HMWO

)\EQTPIW� 1/z+z ERH z+2z

� 8LIR�[I�ORS[�JSV�WYVI�XLEX�MX�GERRSX�FI�VITVIWIRXIH�F]�E�8E]PSV�WIVMIW

� ,S[IZIV� [I�GER�WXMPP�I\TERH�MR *SYVMIV�WIVMIW

f( �) =��

k=��fk

k�

� 8LMW�PIEHW�XS 0EYVIRX�WIVMIW SR�XLI�YRMX�GMVGPI

f(z) =��

k=��fkzk

� ;I�[MPP�WII�XLEX�XLMW�GER�FI�YWIH�XS�HIGSQTSWI f MRXS�E�JYRGXMSR �+ EREP]XMG�MRWMHIXLI�YRMX�GMVGPI�ERH �� EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI�WYGL�XLEX

�+(z) + ��(z) = f(z)

JSV z SR�XLI�YRMX�GMVGPI

Thursday, 5 September 13

18

� 7YTTSWI f MW�WQSSXL�SR�XLI�YRMX�GMVGPI�FYX RIMXLIV EREP]XMG�IZIV][LIVI�MRWMHI�SVSYXWMHI�XLI�YRMX�HMWO

)\EQTPIW� 1/z+z ERH z+2z

� 8LIR�[I�ORS[�JSV�WYVI�XLEX�MX�GERRSX�FI�VITVIWIRXIH�F]�E�8E]PSV�WIVMIW

� ,S[IZIV� [I�GER�WXMPP�I\TERH�MR *SYVMIV�WIVMIW

f( �) =��

k=��fk

k�

� 8LMW�PIEHW�XS 0EYVIRX�WIVMIW SR�XLI�YRMX�GMVGPI

f(z) =��

k=��fkzk

� ;I�[MPP�WII�XLEX�XLMW�GER�FI�YWIH�XS�HIGSQTSWI f MRXS�E�JYRGXMSR �+ EREP]XMG�MRWMHIXLI�YRMX�GMVGPI�ERH �� EREP]XMG�SYXWMHI�XLI�YRMX�GMVGPI�WYGL�XLEX

�+(z) + ��(z) = f(z)

JSV z SR�XLI�YRMX�GMVGPI

Thursday, 5 September 13

19

�+(z) ��(z)

�+(z) + ��(z) = f(z)

Thursday, 5 September 13

20

�+(z) + ��(z) = f(z)

�+(z) =��

k=0

fkzk ��(z) =�1�

k=��fkzk

Thursday, 5 September 13

21

�+(z) + ��(z) = f(z)

We now show

�+(z) =1

2�i

�f(t)

t � zdt ��(z) = � 1

2�i

�f(t)

t � zdt

Thursday, 5 September 13

22

� *SV f WYJ½GMIRXP]�WQSSXL�

�+(z) =1

2�

�f(t)

t � zt

ERH

��(z) = � 1

2�

�f(t)

t � zt

� *SV |z| < 0 [I�LEZI

1

2�

�f(t)

t � zt =

1

2�

��

k=��fk

�tk

t � zt

=1

2�

��

k=0

fk

�tk

t � zt =

1

2�

��+(z)

t � zt

= �+(z)

� 7MQMPEV�PSKMG�WLS[W�XLI�JSVQYPE�JSV ��(z)�Thursday, 5 September 13

23

� *SV f WYJ½GMIRXP]�WQSSXL�

�+(z) =1

2�

�f(t)

t � zt

ERH

��(z) = � 1

2�

�f(t)

t � zt

� *SV |z| < 0 [I�LEZI

1

2�

�f(t)

t � zt =

1

2�

��

k=��fk

�tk

t � zt

=1

2�

��

k=0

fk

�tk

t � zt =

1

2�

��+(z)

t � zt

= �+(z)

� 7MQMPEV�PSKMG�WLS[W�XLI�JSVQYPE�JSV ��(z)�Thursday, 5 September 13

24

� *SV f WYJ½GMIRXP]�WQSSXL�

�+(z) =1

2�

�f(t)

t � zt

ERH

��(z) = � 1

2�

�f(t)

t � zt

� *SV |z| < 0 [I�LEZI

1

2�

�f(t)

t � zt =

1

2�

��

k=��fk

�tk

t � zt

=1

2�

��

k=0

fk

�tk

t � zt =

1

2�

��+(z)

t � zt

= �+(z)

� 7MQMPEV�PSKMG�WLS[W�XLI�JSVQYPE�JSV ��(z)�Thursday, 5 September 13

25

Analyticity in an ellipse

Thursday, 5 September 13

26

0 1r R

• Suppose f is analytic in an annulus A

Thursday, 5 September 13

26

0 1r R

• Suppose f is analytic in an annulus A

• We will show for all z in A

f(z) = �+(z) + ��(z)

=��

k=��fkzk

Thursday, 5 September 13

27

z

f(z) =1

2�i

Br(z)

f(t)

t � zdt

Thursday, 5 September 13

27

z

f(z) =1

2�i

Br(z)

f(t)

t � zdt

z

Thursday, 5 September 13

27

z

f(z) =1

2�i

Br(z)

f(t)

t � zdt

z

=1

2�i

RU

f(t)

t � zdt � 1

2�i

rU

f(t)

t � zdt

z

Thursday, 5 September 13

28

����fk

��� = O�R�k

JSV k > 0 ERH ���fk

��� = O�r�k

JSV k < 0

� 0IX M HIRSXI�XLI�QE\MQYQ�SJ f � ;I�LEZI

���fk

��� =

����1

2�

Uf(z)z�k�1 z

���� =1

2�

�����

RUf(z)z�k�1 z

���� � MR�k

7MQMPEVP]� ���fk

��� =1

2�

�����

rUf(z)z�k�1 z

���� � Mr�k

Thursday, 5 September 13

29

����fk

��� = O�R�k

JSV k > 0 ERH ���fk

��� = O�r�k

JSV k < 0

� 0IX M HIRSXI�XLI�QE\MQYQ�SJ f � ;I�LEZI

���fk

��� =

����1

2�

Uf(z)z�k�1 z

���� =1

2�

�����

RUf(z)z�k�1 z

���� � MR�k

7MQMPEVP]� ���fk

��� =1

2�

�����

rUf(z)z�k�1 z

���� � Mr�k

Thursday, 5 September 13

30

����fk

��� = O�R�k

JSV k > 0 ERH ���fk

��� = O�r�k

JSV k < 0

� 0IX M HIRSXI�XLI�QE\MQYQ�SJ f � ;I�LEZI

���fk

��� =

����1

2�

Uf(z)z�k�1 z

���� =1

2�

�����

RUf(z)z�k�1 z

���� � MR�k

7MQMPEVP]� ���fk

��� =1

2�

�����

rUf(z)z�k�1 z

���� � Mr�k

Thursday, 5 September 13

31

Computed Laurent coefficients for f(z) =ez+1/z

(z + 1/2)(z � 3)

-40 -20 0 20 40

10-15

10-12

10-9

10-6

0.001

1

Thursday, 5 September 13

32

Analytic continuation

Thursday, 5 September 13

33

f is analytic in A g is analytic in B

Thursday, 5 September 13

33

f is analytic in A g is analytic in B

f = g in the intersection of A and B

Thursday, 5 September 13

33

f is analytic in A g is analytic in B

f = g in the intersection of A and B

Then g is the unique analytic

continuation of f

Thursday, 5 September 13

34

f and g are analytic in A

Thursday, 5 September 13

34

f and g are analytic in A

f = g on continuous curve

Thursday, 5 September 13

34

f and g are analytic in A

Then f = g on A

f = g on continuous curve

Thursday, 5 September 13

35

� ;I EREP]XMGEPP]�GSRXMRYI �+ F]�HI½RMRK�MX�EW

�+(z) =1

2�

RU

f(t)

t � zt

� &]�HIJSVQMRK�FEGO�XS�XLI�YRMX�GMVGPI� XLMW�IUYEPW �+(z) MRWMHI�XLI�YRMX�GMVGPI

� 7MQMPEVP]� [I�SFXEMR

��(z) = � 1

2�

rU

f(t)

t � zt

� 8LYW�JSV z � A�+(z) + ��(z) = f(z)

� *YVXLIVQSVI� XLI�VEHMYW�SJ�GSRZIVKIRGI�SJ �+ MW R �;L]# � LIRGI�JSV z � A

�+(z) =��

k=0

fkzk

� 7MQMPEVP]�

��(z) =�1�

k=��fkzk

Thursday, 5 September 13

36

� ;I EREP]XMGEPP]�GSRXMRYI �+ F]�HI½RMRK�MX�EW

�+(z) =1

2�

RU

f(t)

t � zt

� &]�HIJSVQMRK�FEGO�XS�XLI�YRMX�GMVGPI� XLMW�IUYEPW �+(z) MRWMHI�XLI�YRMX�GMVGPI

� 7MQMPEVP]� [I�SFXEMR

��(z) = � 1

2�

rU

f(t)

t � zt

� 8LYW�JSV z � A�+(z) + ��(z) = f(z)

� *YVXLIVQSVI� XLI�VEHMYW�SJ�GSRZIVKIRGI�SJ �+ MW R �;L]# � LIRGI�JSV z � A

�+(z) =��

k=0

fkzk

� 7MQMPEVP]�

��(z) =�1�

k=��fkzk

Thursday, 5 September 13

37

� ;I EREP]XMGEPP]�GSRXMRYI �+ F]�HI½RMRK�MX�EW

�+(z) =1

2�

RU

f(t)

t � zt

� &]�HIJSVQMRK�FEGO�XS�XLI�YRMX�GMVGPI� XLMW�IUYEPW �+(z) MRWMHI�XLI�YRMX�GMVGPI

� 7MQMPEVP]� [I�SFXEMR

��(z) = � 1

2�

rU

f(t)

t � zt

� 8LYW�JSV z � A�+(z) + ��(z) = f(z)

� *YVXLIVQSVI� XLI�VEHMYW�SJ�GSRZIVKIRGI�SJ �+ MW R �;L]# � LIRGI�JSV z � A

�+(z) =��

k=0

fkzk

� 7MQMPEVP]�

��(z) =�1�

k=��fkzk

Thursday, 5 September 13

38

Warning: theoretical equality doesn’t imply numerical

equality

Thursday, 5 September 13

39

-3 -2 -1 1 2 3

0.005

0.010

0.015

0.020

n = 10

-3 -2 -1 1 2 3

0.005

0.010

0.015

0.020

n = 10

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U

Thursday, 5 September 13

39

-3 -2 -1 1 2 3

1.¥10-7

2.¥10-7

3.¥10-7

4.¥10-7

5.¥10-7

6.¥10-7n = 20

-3 -2 -1 1 2 3

1.¥10-7

2.¥10-7

3.¥10-7

4.¥10-7

5.¥10-7

6.¥10-7n = 20

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U

Thursday, 5 September 13

39

-3 -2 -1 1 2 3

5.¥10-16

1.¥10-15

1.5¥10-15

2.¥10-15

2.5¥10-15

3.¥10-15

3.5¥10-15

n = 60

-3 -2 -1 1 2 3

5.¥10-16

1.¥10-15

1.5¥10-15

2.¥10-15

2.5¥10-15

3.¥10-15

3.5¥10-15

n = 60

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U

Thursday, 5 September 13

39

-3 -2 -1 1 2 3

1.¥10-15

2.¥10-15

3.¥10-15

4.¥10-15

n = 80

-3 -2 -1 1 2 3

1.¥10-15

2.¥10-15

3.¥10-15

4.¥10-15

n = 80

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U

Thursday, 5 September 13

40

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

n = 10

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

n = 10

f(z) = ez+1/zError in approximating by approximate Fourier series

z � 2U

Thursday, 5 September 13

40

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

n = 20

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

n = 20

f(z) = ez+1/zError in approximating by approximate Fourier series

z � 2U

Thursday, 5 September 13

40

-3 -2 -1 1 2 3

1.¥10-8

2.¥10-8

3.¥10-8

4.¥10-8

5.¥10-8

6.¥10-8

n = 60

-3 -2 -1 1 2 3

1.¥10-8

2.¥10-8

3.¥10-8

4.¥10-8

5.¥10-8

6.¥10-8

n = 60

f(z) = ez+1/zError in approximating by approximate Fourier series

z � 2U

Thursday, 5 September 13

40

-3 -2 -1 1 2 3

5.¥10-6

0.00001

0.000015

0.00002

0.000025

0.00003

n = 80

-3 -2 -1 1 2 3

5.¥10-6

0.00001

0.000015

0.00002

0.000025

0.00003

n = 80

f(z) = ez+1/zError in approximating by approximate Fourier series

z � 2U

Thursday, 5 September 13

40

-3 -2 -1 1 2 3

0.01

0.02

0.03

0.04

0.05

0.06

n = 100

-3 -2 -1 1 2 3

0.01

0.02

0.03

0.04

0.05

0.06

n = 100

f(z) = ez+1/zError in approximating by approximate Fourier series

z � 2U

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

0.05

0.10

0.15

0.20

0.25

0.30

0.35

n = 10

-3 -2 -1 1 2 3

0.05

0.10

0.15

0.20

0.25

0.30

0.35

n = 10

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035n = 20

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035n = 20

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

2.¥10-11

4.¥10-11

6.¥10-11

8.¥10-11

1.¥10-10

1.2¥10-10

n = 40

-3 -2 -1 1 2 3

2.¥10-11

4.¥10-11

6.¥10-11

8.¥10-11

1.¥10-10

1.2¥10-10

n = 40

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

2.¥10-8

4.¥10-8

6.¥10-8

8.¥10-8

1.¥10-7

1.2¥10-7

n = 60

-3 -2 -1 1 2 3

2.¥10-8

4.¥10-8

6.¥10-8

8.¥10-8

1.¥10-7

1.2¥10-7

n = 60

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

n = 80

-3 -2 -1 1 2 3

0.00005

0.00010

0.00015

0.00020

0.00025

n = 80

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

41

-3 -2 -1 1 2 3

0.05

0.10

0.15

0.20

n = 100

-3 -2 -1 1 2 3

0.05

0.10

0.15

0.20

n = 100

f(z) = ez+1/zError in approximating by approximate Fourier series

z � U2

Thursday, 5 September 13

42

Unit interval

Thursday, 5 September 13

� ;I�FIKER�XLI�GSYVWI�[MXL TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLI�TIVMSHMG�MRXIVZEPT F] *SYVMIV�WIVMIW

� -R�XLI�PEWX�PIGXYVI� [I�YWIH�XLI�QET z = � XS�HIZIPST�JYRGXMSR�ETTVS\MQEXMSRSR�XLI�YRMX�GMVGPI U F] 0EYVIRX�WIVMIW

� 2S[�[I�[MPP�VIPEXI�0EYVIRX�WIVMIW�XS RSR�TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLIYRMX�MRXIVZEP I = [�1, 1]

Thursday, 5 September 13

Periodic intervalT = [�⇡,⇡)

� ;I�FIKER�XLI�GSYVWI�[MXL TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLI�TIVMSHMG�MRXIVZEPT F] *SYVMIV�WIVMIW

� -R�XLI�PEWX�PIGXYVI� [I�YWIH�XLI�QET z = � XS�HIZIPST�JYRGXMSR�ETTVS\MQEXMSRSR�XLI�YRMX�GMVGPI U F] 0EYVIRX�WIVMIW

� 2S[�[I�[MPP�VIPEXI�0EYVIRX�WIVMIW�XS RSR�TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLIYRMX�MRXIVZEP I = [�1, 1]

Thursday, 5 September 13

Periodic intervalT = [�⇡,⇡)

Unit circleU = eiT

ei✓

� ;I�FIKER�XLI�GSYVWI�[MXL TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLI�TIVMSHMG�MRXIVZEPT F] *SYVMIV�WIVMIW

� -R�XLI�PEWX�PIGXYVI� [I�YWIH�XLI�QET z = � XS�HIZIPST�JYRGXMSR�ETTVS\MQEXMSRSR�XLI�YRMX�GMVGPI U F] 0EYVIRX�WIVMIW

� 2S[�[I�[MPP�VIPEXI�0EYVIRX�WIVMIW�XS RSR�TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLIYRMX�MRXIVZEP I = [�1, 1]

Thursday, 5 September 13

Periodic intervalT = [�⇡,⇡)

Unit circleU = eiT

ei✓

Unit intervalI = [�1, 1]

1

2

✓z +

1

z

� ;I�FIKER�XLI�GSYVWI�[MXL TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLI�TIVMSHMG�MRXIVZEPT F] *SYVMIV�WIVMIW

� -R�XLI�PEWX�PIGXYVI� [I�YWIH�XLI�QET z = � XS�HIZIPST�JYRGXMSR�ETTVS\MQEXMSRSR�XLI�YRMX�GMVGPI U F] 0EYVIRX�WIVMIW

� 2S[�[I�[MPP�VIPEXI�0EYVIRX�WIVMIW�XS RSR�TIVMSHMG JYRGXMSR�ETTVS\MQEXMSR�SR�XLIYRMX�MRXIVZEP I = [�1, 1]

Thursday, 5 September 13

44

Joukowsky Map

Thursday, 5 September 13

45

� (I½RI�XLI .SYOS[WO]�QET EW

J(z) =1

2

�z +

1

z

� ;I�[MPP�½VWX�WLS[�XLEX�MX�QETW�XLI YRMX�GMVGPI XS�XLI YRMX�MRXIVZEP

z = x + iy

Thursday, 5 September 13

45

� (I½RI�XLI .SYOS[WO]�QET EW

J(z) =1

2

�z +

1

z

� ;I�[MPP�½VWX�WLS[�XLEX�MX�QETW�XLI YRMX�GMVGPI XS�XLI YRMX�MRXIVZEP

z�1 = x � iy

z = x + iy

Thursday, 5 September 13

45

� (I½RI�XLI .SYOS[WO]�QET EW

J(z) =1

2

�z +

1

z

� ;I�[MPP�½VWX�WLS[�XLEX�MX�QETW�XLI YRMX�GMVGPI XS�XLI YRMX�MRXIVZEP

z�1 = x � iy

z = x + iy

Thursday, 5 September 13

45

� (I½RI�XLI .SYOS[WO]�QET EW

J(z) =1

2

�z +

1

z

� ;I�[MPP�½VWX�WLS[�XLEX�MX�QETW�XLI YRMX�GMVGPI XS�XLI YRMX�MRXIVZEP

z�1 = x � iy

z = x + iy

J(z) = x

z = x + iy

Thursday, 5 September 13

45

� (I½RI�XLI .SYOS[WO]�QET EW

J(z) =1

2

�z +

1

z

� ;I�[MPP�½VWX�WLS[�XLEX�MX�QETW�XLI YRMX�GMVGPI XS�XLI YRMX�MRXIVZEP

z�1 = x � iy

z = x + iy

J(z) = x

z = x + iy

J(z) = x

z = x + iy

Thursday, 5 September 13

46

� 7YTTSWI�[I�[ERX�XS�ETTVS\MQEXI�E�JYRGXMSR f(x) HI½RIH�JSV x � I

� ;I�GER�I\TERH�XLI�JYRGXMSR

g(z) = f(J(z)) = f

�1

2

�z +

1

z

��=

��

k=��gkzk

HI½RIH�SR U MRXS 0EYVIRX�WIVMIW

� &IGEYWI J(z) MW�WQSSXL�SR�XLI�YRMX�GMVGPI� XLI�HMJJIVIRXMEFMPMX]�SJ g MW�MRLIVMXIH�JVSQf

� -R�TEVXMGYPEV� MJ f � �[�1, 1]� XLIR g � �[U] ERH�XLI�0EYVIRX�WIVMIW�GSRZIVKIWI\TSRIRXMEPP]�

Thursday, 5 September 13

47

� 2S[�[I�RIIH�XS�VIGSZIV f(x) JVSQ�ORS[PIHKI�SJ g(z) = f(J(z))

� 8LYW�[I�RIIH�XS�MRZIVX�XLI�.SYOS[WO]�QET

� 7SPZI

x = J(z) =1

2

�z +

1

z

� 6I[VMXI�MX�EWz2 � 2zx + 1 = 0

� -R�SXLIV�[SVHW�

z =2x ±

�4x2 � 4

2= x ±

�x2 � 1 = x ±

�1 � x2

� ;I�LEZI�X[S�MRZIVWIW� EW z ERH z�1 EVI�QETTIH�XS�XLI�WEQI�TSMRX

� ;I�[ERX�XS�GLSSWI�XLI�MRZIVWI�XLEX�LEW�TSWMXMZI�RIKEXMZI�VIEP�MQEKMREV]�TEVX�

J�1� (x) = x +

�1 � x2 ERH J�1

� (x) = x ��

1 � x2

Thursday, 5 September 13

47

� 2S[�[I�RIIH�XS�VIGSZIV f(x) JVSQ�ORS[PIHKI�SJ g(z) = f(J(z))

� 8LYW�[I�RIIH�XS�MRZIVX�XLI�.SYOS[WO]�QET

� 7SPZI

x = J(z) =1

2

�z +

1

z

� 6I[VMXI�MX�EWz2 � 2zx + 1 = 0

� -R�SXLIV�[SVHW�

z =2x ±

�4x2 � 4

2= x ±

�x2 � 1 = x ±

�1 � x2

� ;I�LEZI�X[S�MRZIVWIW� EW z ERH z�1 EVI�QETTIH�XS�XLI�WEQI�TSMRX

� ;I�[ERX�XS�GLSSWI�XLI�MRZIVWI�XLEX�LEW�TSWMXMZI�RIKEXMZI�VIEP�MQEKMREV]�TEVX�

J�1� (x) = x +

�1 � x2 ERH J�1

� (x) = x ��

1 � x2

Thursday, 5 September 13

47

� 2S[�[I�RIIH�XS�VIGSZIV f(x) JVSQ�ORS[PIHKI�SJ g(z) = f(J(z))

� 8LYW�[I�RIIH�XS�MRZIVX�XLI�.SYOS[WO]�QET

� 7SPZI

x = J(z) =1

2

�z +

1

z

� 6I[VMXI�MX�EWz2 � 2zx + 1 = 0

� -R�SXLIV�[SVHW�

z =2x ±

�4x2 � 4

2= x ±

�x2 � 1 = x ±

�1 � x2

� ;I�LEZI�X[S�MRZIVWIW� EW z ERH z�1 EVI�QETTIH�XS�XLI�WEQI�TSMRX

� ;I�[ERX�XS�GLSSWI�XLI�MRZIVWI�XLEX�LEW�TSWMXMZI�RIKEXMZI�VIEP�MQEKMREV]�TEVX�

J�1� (x) = x +

�1 � x2 ERH J�1

� (x) = x ��

1 � x2

Thursday, 5 September 13

47

� 2S[�[I�RIIH�XS�VIGSZIV f(x) JVSQ�ORS[PIHKI�SJ g(z) = f(J(z))

� 8LYW�[I�RIIH�XS�MRZIVX�XLI�.SYOS[WO]�QET

� 7SPZI

x = J(z) =1

2

�z +

1

z

� 6I[VMXI�MX�EWz2 � 2zx + 1 = 0

� -R�SXLIV�[SVHW�

z =2x ±

�4x2 � 4

2= x ±

�x2 � 1 = x ±

�1 � x2

� ;I�LEZI�X[S�MRZIVWIW� EW z ERH z�1 EVI�QETTIH�XS�XLI�WEQI�TSMRX

� ;I�[ERX�XS�GLSSWI�XLI�MRZIVWI�XLEX�LEW�TSWMXMZI�RIKEXMZI�VIEP�MQEKMREV]�TEVX�

J�1� (x) = x +

�1 � x2 ERH J�1

� (x) = x ��

1 � x2

Thursday, 5 September 13

48

� 2S[�[I�RSXI�XLEX g(z) = f(J(z)) = f(J(1/z)) = g(1/z)

� -R�SXLIV�[SVHW� g VITIEXW f X[MGI

� 8LYW�[I�LEZIg(J�1

� (x)) = f(J(J�1� (x)) = f(x)

ERHg(J�1

� (x)) = f(x)

� 9WMRK�IMXLIV�I\TVIWWMSR� [I�GER�VIGSZIV f JVSQ g

� 8LYW�[I�GER�ETTVS\MQEXI f F]�GEPGYPEXMRK�XLI�ETTVS\MQEXI�0EYVIRX�WIVMIW gm SJg� HI½RMRK

fn(x) = gm(J�1� (x))

Thursday, 5 September 13

48

� 2S[�[I�RSXI�XLEX g(z) = f(J(z)) = f(J(1/z)) = g(1/z)

� -R�SXLIV�[SVHW� g VITIEXW f X[MGI

� 8LYW�[I�LEZIg(J�1

� (x)) = f(J(J�1� (x)) = f(x)

ERHg(J�1

� (x)) = f(x)

� 9WMRK�IMXLIV�I\TVIWWMSR� [I�GER�VIGSZIV f JVSQ g

� 8LYW�[I�GER�ETTVS\MQEXI f F]�GEPGYPEXMRK�XLI�ETTVS\MQEXI�0EYVIRX�WIVMIW gm SJg� HI½RMRK

fn(x) = gm(J�1� (x))

Thursday, 5 September 13

48

� 2S[�[I�RSXI�XLEX g(z) = f(J(z)) = f(J(1/z)) = g(1/z)

� -R�SXLIV�[SVHW� g VITIEXW f X[MGI

� 8LYW�[I�LEZIg(J�1

� (x)) = f(J(J�1� (x)) = f(x)

ERHg(J�1

� (x)) = f(x)

� 9WMRK�IMXLIV�I\TVIWWMSR� [I�GER�VIGSZIV f JVSQ g

� 8LYW�[I�GER�ETTVS\MQEXI f F]�GEPGYPEXMRK�XLI�ETTVS\MQEXI�0EYVIRX�WIVMIW gm SJg� HI½RMRK

fn(x) = gm(J�1� (x))

Thursday, 5 September 13

49

4SMRX[MWI�IVVSV�MR�ETTVS\MQEXMRK f F] gm(J�1� (x))

-1.0 -0.5 0.5 1.0

0.02

0.04

0.06

0.08

0.10m = 5

-1.0 -0.5 0.5 1.0

0.02

0.04

0.06

0.08

0.10m = 5

Thursday, 5 September 13

49

4SMRX[MWI�IVVSV�MR�ETTVS\MQEXMRK f F] gm(J�1� (x))

-1.0 -0.5 0.5 1.0

0.0001

0.0002

0.0003

0.0004

0.0005

m = 10

-1.0 -0.5 0.5 1.0

0.0001

0.0002

0.0003

0.0004

0.0005

m = 10

Thursday, 5 September 13

49

4SMRX[MWI�IVVSV�MR�ETTVS\MQEXMRK f F] gm(J�1� (x))

-1.0 -0.5 0.5 1.0

1.¥10-7

2.¥10-7

3.¥10-7

4.¥10-7

m = 15

-1.0 -0.5 0.5 1.0

1.¥10-7

2.¥10-7

3.¥10-7

4.¥10-7

m = 15

Thursday, 5 September 13

49

4SMRX[MWI�IVVSV�MR�ETTVS\MQEXMRK f F] gm(J�1� (x))

-1.0 -0.5 0.5 1.0

1.¥10-10

2.¥10-10

3.¥10-10

4.¥10-10

5.¥10-10

m = 20

-1.0 -0.5 0.5 1.0

1.¥10-10

2.¥10-10

3.¥10-10

4.¥10-10

5.¥10-10

m = 20

Thursday, 5 September 13

49

4SMRX[MWI�IVVSV�MR�ETTVS\MQEXMRK f F] gm(J�1� (x))

-1.0 -0.5 0.5 1.0

2.¥10-16

4.¥10-16

6.¥10-16

8.¥10-16

m = 30

-1.0 -0.5 0.5 1.0

2.¥10-16

4.¥10-16

6.¥10-16

8.¥10-16

m = 30

Thursday, 5 September 13