lesson 9 laurent series
TRANSCRIPT
2
� ;I�LEZI�WIIR�XLEX 8E]PSV�WIVMIW��
k=0
fkzk
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8LIWI�PMZI�MRWMHI�XLI�YRMX�HMWO
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k=0
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Thursday, 5 September 13
2
� ;I�LEZI�WIIR�XLEX 8E]PSV�WIVMIW��
k=0
fkzk
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k=��fkzk
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k=0
fkzk
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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
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�1�
k=��fkzk
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k=0
fkzk
8LIWI�PMZI�MR�ER�ERRYPYWThursday, 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
z
z� 1
2�i
�
Br(z)
f(t)
t � zdt = �f(z)
After switching orientations:
Zero due to analyticity
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
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
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
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
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
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
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
� ;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
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