Taylor's Series | Mathematics Optional Notes for UPSC PDF Download

 Page 1


Edurev123 
2. Taylor's Series 
2.1 If the function ?? (?? ) is analytic and one valued in |?? -?? |<?? , prove that for ?? <
?? <?? ,?? '
(?? ) =
?? ????
?
?? ?? ?? ??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
) . 
(2011: 15 marks) 
Solution: 
The function ?? (?? ) is given to be analytic in |?? -?? |<?? and ?? <?? , therefore ?? (?? ) is also 
analytic inside the circle ?? defined by |?? -?? |=?? . 
???? '
(?? )=
1
2????
??
?? ?
?? (?? )
(?? -?? )
2
???? (??) 
Expanding ?? (?? ) in a Taylor's series about ?? =?? , we have 
?? (?? )?=??
8
?? =0
??? ?? (?? -?? )
?? ?????????????????????????????????????????? (?? )?=?? (?? +?? ?? ????
)=??
8
?? =0
??? ?? ?? ?? ?? ?????? ?????????????????????????????????????????? (?? )
¯¯¯¯¯¯
?=??
8
?? =0
????
?? ?? ?? ?? -?????? ??????????????
1
2????
??
?? ?
?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? ?=
1
2????
? ?
2?? 0
?
S???
?? ?? ?? ?? -??????
?? 2
?? ??2?? ·???? ?? ????
????
 
=
1
2?? ? ???
?? ?? ?? -1
?
0
2?? ??? -?? (?? +1)????
???? =0 
From (i) and (ii), we have 
?? (?? )=
1
2????
??
?? ?
?? (?? )?? +?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? 
?=
1
????
? ?
2?? 0
?
 real part of ?? (???+?? ?? ????
)???? ?? ????
????
?? 2
?? ??2?? ?[??? =?? +?? ?? ????
]
?=
1
????
? ?
2?? 0
??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
)
 
2.2 Describe the maximal ideals in the ring of Gaussian integers ?? [?? ]={?? +???? |
?? ,?? ??? } . 
Page 2


Edurev123 
2. Taylor's Series 
2.1 If the function ?? (?? ) is analytic and one valued in |?? -?? |<?? , prove that for ?? <
?? <?? ,?? '
(?? ) =
?? ????
?
?? ?? ?? ??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
) . 
(2011: 15 marks) 
Solution: 
The function ?? (?? ) is given to be analytic in |?? -?? |<?? and ?? <?? , therefore ?? (?? ) is also 
analytic inside the circle ?? defined by |?? -?? |=?? . 
???? '
(?? )=
1
2????
??
?? ?
?? (?? )
(?? -?? )
2
???? (??) 
Expanding ?? (?? ) in a Taylor's series about ?? =?? , we have 
?? (?? )?=??
8
?? =0
??? ?? (?? -?? )
?? ?????????????????????????????????????????? (?? )?=?? (?? +?? ?? ????
)=??
8
?? =0
??? ?? ?? ?? ?? ?????? ?????????????????????????????????????????? (?? )
¯¯¯¯¯¯
?=??
8
?? =0
????
?? ?? ?? ?? -?????? ??????????????
1
2????
??
?? ?
?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? ?=
1
2????
? ?
2?? 0
?
S???
?? ?? ?? ?? -??????
?? 2
?? ??2?? ·???? ?? ????
????
 
=
1
2?? ? ???
?? ?? ?? -1
?
0
2?? ??? -?? (?? +1)????
???? =0 
From (i) and (ii), we have 
?? (?? )=
1
2????
??
?? ?
?? (?? )?? +?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? 
?=
1
????
? ?
2?? 0
?
 real part of ?? (???+?? ?? ????
)???? ?? ????
????
?? 2
?? ??2?? ?[??? =?? +?? ?? ????
]
?=
1
????
? ?
2?? 0
??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
)
 
2.2 Describe the maximal ideals in the ring of Gaussian integers ?? [?? ]={?? +???? |
?? ,?? ??? } . 
(2012: 20 Marks) 
Solution: 
Let us define 
?? (?? +???? )?=|?? +???? |
2
=?? 2
+?? 2
 for all ?? +???? ??? [??]
?? (?? )?=?? (?? +???? )=?? 2
+?? 2
>0 for all ?? ?0 in ?? [??]
 
Again for any ?? ,?? ??? [??] 
?? (???? )=|?? |
2
|?? 2
=|?? |
2
=?? (?? ) 
Let ?? ,?? ??? [??],?? ?0 
Then,????????????????????????? =?? +???? ,?? =?? +???? ,?? ,?? ,?? ,?? ??? and (?? ,?? )?(0,0) 
???????????????????????????????
?? ?? =
?? +????
?? +????
=
(?? +???? )
(?? +???? )
×
?? +????
?? +????
 
=
(?? +???? )(?? +???? )
?? 2
+?? 2
=?? +???? (say)  
where ?? and ?? are rational numbers. Then we can find integers ?? and ?? such that |?? -
?? |=
1
2
 and |?? -?? |=
1
2
. 
? ??????????? =(?? +???? )?? =(?? +???? )?? +[(?? -?? )+??(?? -?? )]?? ? Now ???????? ?[(?? -?? )+??(?? -?? )]?? =[(?? +???? )-(?? +???? )]?? ???????? ?=(?? +???? )?? -(?? +???? )?? ???????? ?? -(?? +???? )?? ??? [??]
 
as ?? [??] is a ring and ?? ,?? ,?? +???? ??? [??] 
Let 
?? ?=[(?? -?? )+??(?? -?? )]?? ?? (?? )?=|?? 2
=[(?? -?? )
2
+(?? -?? )
2
]|?? 2
?=(
1
4
+
1
4
)|?? |
2
 
Taking ?? =?? +???? , we have, 
?? =???? +?? ?    (using (i) and (ii))  
where either ?? =0 or ?? (?? )<?? (?? ) 
Hence, ?? [??] is a Euclidean domain. 
As every Euclidean domain is a principal ideal domain (PID). 
Page 3


Edurev123 
2. Taylor's Series 
2.1 If the function ?? (?? ) is analytic and one valued in |?? -?? |<?? , prove that for ?? <
?? <?? ,?? '
(?? ) =
?? ????
?
?? ?? ?? ??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
) . 
(2011: 15 marks) 
Solution: 
The function ?? (?? ) is given to be analytic in |?? -?? |<?? and ?? <?? , therefore ?? (?? ) is also 
analytic inside the circle ?? defined by |?? -?? |=?? . 
???? '
(?? )=
1
2????
??
?? ?
?? (?? )
(?? -?? )
2
???? (??) 
Expanding ?? (?? ) in a Taylor's series about ?? =?? , we have 
?? (?? )?=??
8
?? =0
??? ?? (?? -?? )
?? ?????????????????????????????????????????? (?? )?=?? (?? +?? ?? ????
)=??
8
?? =0
??? ?? ?? ?? ?? ?????? ?????????????????????????????????????????? (?? )
¯¯¯¯¯¯
?=??
8
?? =0
????
?? ?? ?? ?? -?????? ??????????????
1
2????
??
?? ?
?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? ?=
1
2????
? ?
2?? 0
?
S???
?? ?? ?? ?? -??????
?? 2
?? ??2?? ·???? ?? ????
????
 
=
1
2?? ? ???
?? ?? ?? -1
?
0
2?? ??? -?? (?? +1)????
???? =0 
From (i) and (ii), we have 
?? (?? )=
1
2????
??
?? ?
?? (?? )?? +?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? 
?=
1
????
? ?
2?? 0
?
 real part of ?? (???+?? ?? ????
)???? ?? ????
????
?? 2
?? ??2?? ?[??? =?? +?? ?? ????
]
?=
1
????
? ?
2?? 0
??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
)
 
2.2 Describe the maximal ideals in the ring of Gaussian integers ?? [?? ]={?? +???? |
?? ,?? ??? } . 
(2012: 20 Marks) 
Solution: 
Let us define 
?? (?? +???? )?=|?? +???? |
2
=?? 2
+?? 2
 for all ?? +???? ??? [??]
?? (?? )?=?? (?? +???? )=?? 2
+?? 2
>0 for all ?? ?0 in ?? [??]
 
Again for any ?? ,?? ??? [??] 
?? (???? )=|?? |
2
|?? 2
=|?? |
2
=?? (?? ) 
Let ?? ,?? ??? [??],?? ?0 
Then,????????????????????????? =?? +???? ,?? =?? +???? ,?? ,?? ,?? ,?? ??? and (?? ,?? )?(0,0) 
???????????????????????????????
?? ?? =
?? +????
?? +????
=
(?? +???? )
(?? +???? )
×
?? +????
?? +????
 
=
(?? +???? )(?? +???? )
?? 2
+?? 2
=?? +???? (say)  
where ?? and ?? are rational numbers. Then we can find integers ?? and ?? such that |?? -
?? |=
1
2
 and |?? -?? |=
1
2
. 
? ??????????? =(?? +???? )?? =(?? +???? )?? +[(?? -?? )+??(?? -?? )]?? ? Now ???????? ?[(?? -?? )+??(?? -?? )]?? =[(?? +???? )-(?? +???? )]?? ???????? ?=(?? +???? )?? -(?? +???? )?? ???????? ?? -(?? +???? )?? ??? [??]
 
as ?? [??] is a ring and ?? ,?? ,?? +???? ??? [??] 
Let 
?? ?=[(?? -?? )+??(?? -?? )]?? ?? (?? )?=|?? 2
=[(?? -?? )
2
+(?? -?? )
2
]|?? 2
?=(
1
4
+
1
4
)|?? |
2
 
Taking ?? =?? +???? , we have, 
?? =???? +?? ?    (using (i) and (ii))  
where either ?? =0 or ?? (?? )<?? (?? ) 
Hence, ?? [??] is a Euclidean domain. 
As every Euclidean domain is a principal ideal domain (PID). 
??? [??] is PID. 
But in a PID, every non-zero not unit element ?? is prime element iff <?? > is a maximal 
ideal. 
? Maximal ideals in ???? ] are the ideals generated by prime elements. 
2.3 For a function ?? .??Ø
'
 arid ?? =?? , let fn denote the ?? th 
 derivative of ?? and 
?? (?? )=?? . Let ?? be an entire function such that for some ?? =?? ,?? ?? (
?? ?? )=?? for all ?? =
?? ,?? ,?? ,…. Show that ?? is a polynomial. 
(2017 : 15 Marks) 
Solution: 
Since ?? (?? ) is entire, ??? (?? ) is analytic. Hence, ?? (?? ) can be expressed as Taylor's series 
around ?? =0 as 
?? (?? )=???
8
?? =1
??? ?? 1
?? ?? + ??
8
?? =0
??? ?? ?? ?? ?? ?? (?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? 1
?? ?? +?? +
???
8
?? =?? ?(?? )(?? -1)…..(?? -(?? -1))?? ?? ?? ?? -?? ?? ?? (
1
?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? ·?? ?? +?? +
???
8
?? =?? ??? (?? -1)….(?? -?? +1)?? ?? (
1
?? )
?? -?? 
Now, 
As ?? ?? (
1
?? )?0 for all ?? =1,2,3,… 
Let us take ?? ?8 
We get ?? -?? ?0 and ?? ?? =0 
Similarly, if we take ?? +??,?? ?? +1
=0 
????? ?? =?? ?? +1
=?? ?? +2
=?…=0
????? (?? )=?? 0
+?? 1
?? +?? 2
?? 2
+?.+?? ?? -1
?? ?? -1
 
which is a polynomial function. 
Page 4


Edurev123 
2. Taylor's Series 
2.1 If the function ?? (?? ) is analytic and one valued in |?? -?? |<?? , prove that for ?? <
?? <?? ,?? '
(?? ) =
?? ????
?
?? ?? ?? ??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
) . 
(2011: 15 marks) 
Solution: 
The function ?? (?? ) is given to be analytic in |?? -?? |<?? and ?? <?? , therefore ?? (?? ) is also 
analytic inside the circle ?? defined by |?? -?? |=?? . 
???? '
(?? )=
1
2????
??
?? ?
?? (?? )
(?? -?? )
2
???? (??) 
Expanding ?? (?? ) in a Taylor's series about ?? =?? , we have 
?? (?? )?=??
8
?? =0
??? ?? (?? -?? )
?? ?????????????????????????????????????????? (?? )?=?? (?? +?? ?? ????
)=??
8
?? =0
??? ?? ?? ?? ?? ?????? ?????????????????????????????????????????? (?? )
¯¯¯¯¯¯
?=??
8
?? =0
????
?? ?? ?? ?? -?????? ??????????????
1
2????
??
?? ?
?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? ?=
1
2????
? ?
2?? 0
?
S???
?? ?? ?? ?? -??????
?? 2
?? ??2?? ·???? ?? ????
????
 
=
1
2?? ? ???
?? ?? ?? -1
?
0
2?? ??? -?? (?? +1)????
???? =0 
From (i) and (ii), we have 
?? (?? )=
1
2????
??
?? ?
?? (?? )?? +?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? 
?=
1
????
? ?
2?? 0
?
 real part of ?? (???+?? ?? ????
)???? ?? ????
????
?? 2
?? ??2?? ?[??? =?? +?? ?? ????
]
?=
1
????
? ?
2?? 0
??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
)
 
2.2 Describe the maximal ideals in the ring of Gaussian integers ?? [?? ]={?? +???? |
?? ,?? ??? } . 
(2012: 20 Marks) 
Solution: 
Let us define 
?? (?? +???? )?=|?? +???? |
2
=?? 2
+?? 2
 for all ?? +???? ??? [??]
?? (?? )?=?? (?? +???? )=?? 2
+?? 2
>0 for all ?? ?0 in ?? [??]
 
Again for any ?? ,?? ??? [??] 
?? (???? )=|?? |
2
|?? 2
=|?? |
2
=?? (?? ) 
Let ?? ,?? ??? [??],?? ?0 
Then,????????????????????????? =?? +???? ,?? =?? +???? ,?? ,?? ,?? ,?? ??? and (?? ,?? )?(0,0) 
???????????????????????????????
?? ?? =
?? +????
?? +????
=
(?? +???? )
(?? +???? )
×
?? +????
?? +????
 
=
(?? +???? )(?? +???? )
?? 2
+?? 2
=?? +???? (say)  
where ?? and ?? are rational numbers. Then we can find integers ?? and ?? such that |?? -
?? |=
1
2
 and |?? -?? |=
1
2
. 
? ??????????? =(?? +???? )?? =(?? +???? )?? +[(?? -?? )+??(?? -?? )]?? ? Now ???????? ?[(?? -?? )+??(?? -?? )]?? =[(?? +???? )-(?? +???? )]?? ???????? ?=(?? +???? )?? -(?? +???? )?? ???????? ?? -(?? +???? )?? ??? [??]
 
as ?? [??] is a ring and ?? ,?? ,?? +???? ??? [??] 
Let 
?? ?=[(?? -?? )+??(?? -?? )]?? ?? (?? )?=|?? 2
=[(?? -?? )
2
+(?? -?? )
2
]|?? 2
?=(
1
4
+
1
4
)|?? |
2
 
Taking ?? =?? +???? , we have, 
?? =???? +?? ?    (using (i) and (ii))  
where either ?? =0 or ?? (?? )<?? (?? ) 
Hence, ?? [??] is a Euclidean domain. 
As every Euclidean domain is a principal ideal domain (PID). 
??? [??] is PID. 
But in a PID, every non-zero not unit element ?? is prime element iff <?? > is a maximal 
ideal. 
? Maximal ideals in ???? ] are the ideals generated by prime elements. 
2.3 For a function ?? .??Ø
'
 arid ?? =?? , let fn denote the ?? th 
 derivative of ?? and 
?? (?? )=?? . Let ?? be an entire function such that for some ?? =?? ,?? ?? (
?? ?? )=?? for all ?? =
?? ,?? ,?? ,…. Show that ?? is a polynomial. 
(2017 : 15 Marks) 
Solution: 
Since ?? (?? ) is entire, ??? (?? ) is analytic. Hence, ?? (?? ) can be expressed as Taylor's series 
around ?? =0 as 
?? (?? )=???
8
?? =1
??? ?? 1
?? ?? + ??
8
?? =0
??? ?? ?? ?? ?? ?? (?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? 1
?? ?? +?? +
???
8
?? =?? ?(?? )(?? -1)…..(?? -(?? -1))?? ?? ?? ?? -?? ?? ?? (
1
?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? ·?? ?? +?? +
???
8
?? =?? ??? (?? -1)….(?? -?? +1)?? ?? (
1
?? )
?? -?? 
Now, 
As ?? ?? (
1
?? )?0 for all ?? =1,2,3,… 
Let us take ?? ?8 
We get ?? -?? ?0 and ?? ?? =0 
Similarly, if we take ?? +??,?? ?? +1
=0 
????? ?? =?? ?? +1
=?? ?? +2
=?…=0
????? (?? )=?? 0
+?? 1
?? +?? 2
?? 2
+?.+?? ?? -1
?? ?? -1
 
which is a polynomial function. 
2.4 Show that an isolated singular point ?? ?? of a function ?? (?? ) is a pole of order ?? if 
and only if ?? (?? ) can be written in the form ?? (?? )=
???
(?? )
(?? -?? ?? )
?? where ?? (?? ) is analytic and 
non-zero at ?? ?? . 
Moreover 
?????? ?? =?? ?? ??? (?? )=
?? (?? -?? )
(?? ?? )
(?? -?? )!
 if ?? =?? 
(2019 : 15 Marks) 
Solution: 
Since ?? (?? ) has a pole of order, ?? , then by definition, for 0<|?? -?? 0
|<?? . 
?? (?? )=??
8
?? =0
??? ?? (?? -?? 0
)
?? +
?? 2
?? -?? 0
+
?? 2
(?? -?? 0
)
2
+
?? ?? (?? -?? 0
)
?? ,?? ?? +0
???? (?? )=
1
(?? -?? 0
)
?? [??
8
?? =0
??? ?? (?? -?? 0
)
?? +?? +?? 1
(?? -?? 0
)
?? -1
+?? 2
(?? -?? 0
)
?? -2
+?? 2
(?? -?? 0
)
?? -2
+?? ?? ]
????? (?? )=
???(?? )
(?? -?? 0
)
?? 
Clearly, ?? (?? 0
)=?? ?? ?0 and is analytic at ?? 0
, as it has Taylor series expansion about ?? 0
. 
Conversely, suppose ?? (?? ) can be written in the form: 
?? (?? )=
?? (?? )
(?? -?? 0
)
?? , then 
?? ?
(?? )=?? (?? 0
)+?? (?? 0
)(?? -?? 0
)+
?? ?? (?? 0
)
2!
(?? -?? 0
)
2
+?.+
?? (?? -1)
(?? 0
)
(?? -1)!
(?? -?? 0
)
?? -1
+?.
 
Since, 
?? (?? ) has a pole of order ?? . 
?? (?? 0
)?0 
With residue, 
?? 1
=
?? (?? -1)
(?? 0
)
(?? -1)!
 
In case of simple pole, i.e., ?? -1, 
Res??? =?? 0
,?? (?? 0
)=?? (?? 0
) . Hence, proved.  
Page 5


Edurev123 
2. Taylor's Series 
2.1 If the function ?? (?? ) is analytic and one valued in |?? -?? |<?? , prove that for ?? <
?? <?? ,?? '
(?? ) =
?? ????
?
?? ?? ?? ??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
) . 
(2011: 15 marks) 
Solution: 
The function ?? (?? ) is given to be analytic in |?? -?? |<?? and ?? <?? , therefore ?? (?? ) is also 
analytic inside the circle ?? defined by |?? -?? |=?? . 
???? '
(?? )=
1
2????
??
?? ?
?? (?? )
(?? -?? )
2
???? (??) 
Expanding ?? (?? ) in a Taylor's series about ?? =?? , we have 
?? (?? )?=??
8
?? =0
??? ?? (?? -?? )
?? ?????????????????????????????????????????? (?? )?=?? (?? +?? ?? ????
)=??
8
?? =0
??? ?? ?? ?? ?? ?????? ?????????????????????????????????????????? (?? )
¯¯¯¯¯¯
?=??
8
?? =0
????
?? ?? ?? ?? -?????? ??????????????
1
2????
??
?? ?
?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? ?=
1
2????
? ?
2?? 0
?
S???
?? ?? ?? ?? -??????
?? 2
?? ??2?? ·???? ?? ????
????
 
=
1
2?? ? ???
?? ?? ?? -1
?
0
2?? ??? -?? (?? +1)????
???? =0 
From (i) and (ii), we have 
?? (?? )=
1
2????
??
?? ?
?? (?? )?? +?? (?? )
¯¯¯¯¯¯
(?? -?? )
2
???? 
?=
1
????
? ?
2?? 0
?
 real part of ?? (???+?? ?? ????
)???? ?? ????
????
?? 2
?? ??2?? ?[??? =?? +?? ?? ????
]
?=
1
????
? ?
2?? 0
??? (?? )?? -????
???? , where ?? (?? ) is the real part of ?? (?? +?? ?? ????
)
 
2.2 Describe the maximal ideals in the ring of Gaussian integers ?? [?? ]={?? +???? |
?? ,?? ??? } . 
(2012: 20 Marks) 
Solution: 
Let us define 
?? (?? +???? )?=|?? +???? |
2
=?? 2
+?? 2
 for all ?? +???? ??? [??]
?? (?? )?=?? (?? +???? )=?? 2
+?? 2
>0 for all ?? ?0 in ?? [??]
 
Again for any ?? ,?? ??? [??] 
?? (???? )=|?? |
2
|?? 2
=|?? |
2
=?? (?? ) 
Let ?? ,?? ??? [??],?? ?0 
Then,????????????????????????? =?? +???? ,?? =?? +???? ,?? ,?? ,?? ,?? ??? and (?? ,?? )?(0,0) 
???????????????????????????????
?? ?? =
?? +????
?? +????
=
(?? +???? )
(?? +???? )
×
?? +????
?? +????
 
=
(?? +???? )(?? +???? )
?? 2
+?? 2
=?? +???? (say)  
where ?? and ?? are rational numbers. Then we can find integers ?? and ?? such that |?? -
?? |=
1
2
 and |?? -?? |=
1
2
. 
? ??????????? =(?? +???? )?? =(?? +???? )?? +[(?? -?? )+??(?? -?? )]?? ? Now ???????? ?[(?? -?? )+??(?? -?? )]?? =[(?? +???? )-(?? +???? )]?? ???????? ?=(?? +???? )?? -(?? +???? )?? ???????? ?? -(?? +???? )?? ??? [??]
 
as ?? [??] is a ring and ?? ,?? ,?? +???? ??? [??] 
Let 
?? ?=[(?? -?? )+??(?? -?? )]?? ?? (?? )?=|?? 2
=[(?? -?? )
2
+(?? -?? )
2
]|?? 2
?=(
1
4
+
1
4
)|?? |
2
 
Taking ?? =?? +???? , we have, 
?? =???? +?? ?    (using (i) and (ii))  
where either ?? =0 or ?? (?? )<?? (?? ) 
Hence, ?? [??] is a Euclidean domain. 
As every Euclidean domain is a principal ideal domain (PID). 
??? [??] is PID. 
But in a PID, every non-zero not unit element ?? is prime element iff <?? > is a maximal 
ideal. 
? Maximal ideals in ???? ] are the ideals generated by prime elements. 
2.3 For a function ?? .??Ø
'
 arid ?? =?? , let fn denote the ?? th 
 derivative of ?? and 
?? (?? )=?? . Let ?? be an entire function such that for some ?? =?? ,?? ?? (
?? ?? )=?? for all ?? =
?? ,?? ,?? ,…. Show that ?? is a polynomial. 
(2017 : 15 Marks) 
Solution: 
Since ?? (?? ) is entire, ??? (?? ) is analytic. Hence, ?? (?? ) can be expressed as Taylor's series 
around ?? =0 as 
?? (?? )=???
8
?? =1
??? ?? 1
?? ?? + ??
8
?? =0
??? ?? ?? ?? ?? ?? (?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? 1
?? ?? +?? +
???
8
?? =?? ?(?? )(?? -1)…..(?? -(?? -1))?? ?? ?? ?? -?? ?? ?? (
1
?? )=???
8
?? =1
?(-1)
?? (?? )(?? +1)…..(?? +(?? -1))?? -?? ·?? ?? +?? +
???
8
?? =?? ??? (?? -1)….(?? -?? +1)?? ?? (
1
?? )
?? -?? 
Now, 
As ?? ?? (
1
?? )?0 for all ?? =1,2,3,… 
Let us take ?? ?8 
We get ?? -?? ?0 and ?? ?? =0 
Similarly, if we take ?? +??,?? ?? +1
=0 
????? ?? =?? ?? +1
=?? ?? +2
=?…=0
????? (?? )=?? 0
+?? 1
?? +?? 2
?? 2
+?.+?? ?? -1
?? ?? -1
 
which is a polynomial function. 
2.4 Show that an isolated singular point ?? ?? of a function ?? (?? ) is a pole of order ?? if 
and only if ?? (?? ) can be written in the form ?? (?? )=
???
(?? )
(?? -?? ?? )
?? where ?? (?? ) is analytic and 
non-zero at ?? ?? . 
Moreover 
?????? ?? =?? ?? ??? (?? )=
?? (?? -?? )
(?? ?? )
(?? -?? )!
 if ?? =?? 
(2019 : 15 Marks) 
Solution: 
Since ?? (?? ) has a pole of order, ?? , then by definition, for 0<|?? -?? 0
|<?? . 
?? (?? )=??
8
?? =0
??? ?? (?? -?? 0
)
?? +
?? 2
?? -?? 0
+
?? 2
(?? -?? 0
)
2
+
?? ?? (?? -?? 0
)
?? ,?? ?? +0
???? (?? )=
1
(?? -?? 0
)
?? [??
8
?? =0
??? ?? (?? -?? 0
)
?? +?? +?? 1
(?? -?? 0
)
?? -1
+?? 2
(?? -?? 0
)
?? -2
+?? 2
(?? -?? 0
)
?? -2
+?? ?? ]
????? (?? )=
???(?? )
(?? -?? 0
)
?? 
Clearly, ?? (?? 0
)=?? ?? ?0 and is analytic at ?? 0
, as it has Taylor series expansion about ?? 0
. 
Conversely, suppose ?? (?? ) can be written in the form: 
?? (?? )=
?? (?? )
(?? -?? 0
)
?? , then 
?? ?
(?? )=?? (?? 0
)+?? (?? 0
)(?? -?? 0
)+
?? ?? (?? 0
)
2!
(?? -?? 0
)
2
+?.+
?? (?? -1)
(?? 0
)
(?? -1)!
(?? -?? 0
)
?? -1
+?.
 
Since, 
?? (?? ) has a pole of order ?? . 
?? (?? 0
)?0 
With residue, 
?? 1
=
?? (?? -1)
(?? 0
)
(?? -1)!
 
In case of simple pole, i.e., ?? -1, 
Res??? =?? 0
,?? (?? 0
)=?? (?? 0
) . Hence, proved.  
2.5 Find the Laurent series expansion of ?? (?? )=
?? ?? -?? +?? ?? (?? ?? -?? ?? +?? )
 in the powers of (?? +?? ) in 
the region |?? +?? |>?? . 
[2021 : 20 marks] 
Solution: 
?? (?? )=
?? 2
-?? +1
?? (?? 2
-3?? +2)
 
Let ?? +1=?? , then                                   ?? =?? -1 
  
?? (?? )=
(?? -1)
2
-(?? -1)+1
(?? -1)[(?? -1)
2
-3(?? -1)+2]
? ?? (?? )=
?? 2
+1-2?? -?? +1+1
(?? -1)[?? 2
+1-2?? -3?? +3+2]
 
???????????????????????????????????????? (?? )=
?? 2
-3?? +3
(?? -1)(?? 2
-5?? +6)
?????????????????????????????????????????????? (?? )=
?? 2
-3?? +3
(?? -1)(?? -2)(?? -3)
????????????????????????????????????????????????????=
?? ?? -1
+
?? ?? -2
+
?? ?? -3
?????????????????????????????????????????????????????(1)
?????????????????????? 2
-3?? +3=?? (?? -2)(?? -3)+?? (?? -1)(?? -3)+?? (?? -1)(?? -2)
 if ?? =2, then, ?4-6+3=?? (1)(-1)
????? =-1
 if ?? =1, then, ?1-3+3=?? (-1)(-2)
????? =
1
2
 if ?? =3, then, ?9-9+3=?? (2)(1)
????? =
3
2
?? (?? )=
1
2
?? -1
+
-1
?? -2
+
3
2
?? -3
 
Corresponding to 
1
?? -1
 : 
1
?? -1
=
1
?? (1-
1
?? )
-1
=
1
?? (1+
1
?? +
1
?? 2
+?.) is valid for |
1
?? |<1 and |?? |>0. 
1
?? -1
=
1
?? +
1
?? 2
+
1
?? 3
+?. is valid for |?? |>1 
Corresponding to 
1
?? -2
 :