Singularity | Mathematics Optional Notes for UPSC PDF Download

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Edurev123 
3. Singularity 
3.1 Determine all entire functions ?? (?? ) such that 0 is a removal singularity of ?? (
?? ?? ) . 
(2017 : 10 Marks) 
Solution: 
For any complex valued function, ?? (?? ) 
?? (?? )=??
8
?? =0
?? ?? ?? ?? +??
8
?? =1
?? ?? ?? -?? 
Since, ?? (?? ) is entire, so all ?? ?? are zero. 
??
?? (?? )?=??
8
?? =0
??? ?? ?? ?? ?? (
1
?? )?=??
8
?? =0
??? ?? (
1
?? )
?? ?=?? 0
+
?? 1
?? +
?? 2
?? 2
+?.
 
?? (
1
2
) is a removable singularity. 
3.2 Let ?? be an entire function whose Tayior series expansion with centre ?? =?? 
has infinitely many terms. Show that ?? =?? is an essential singularity of ?? (
?? ?? ) . 
[2021 : 15 marks] 
Solution: 
Let ?? (?? ) be an entire function whose Taylor series expansion with centre ?? =0 has 
infinitely many terms. Clearly, ?? (?? )=?? ?? is an entire function and its Taylor's series 
expansion with centre ?? =0 has infinitely many terms, i.e., 
We have 
?? (?? )=?? ?? =1+?? +
?? 2
2!
+
?? 3
3!
+?. 
?? h?????? ??????????????????????????????????????????? (
1
?? )=?? 1/?? 
Page 2


Edurev123 
3. Singularity 
3.1 Determine all entire functions ?? (?? ) such that 0 is a removal singularity of ?? (
?? ?? ) . 
(2017 : 10 Marks) 
Solution: 
For any complex valued function, ?? (?? ) 
?? (?? )=??
8
?? =0
?? ?? ?? ?? +??
8
?? =1
?? ?? ?? -?? 
Since, ?? (?? ) is entire, so all ?? ?? are zero. 
??
?? (?? )?=??
8
?? =0
??? ?? ?? ?? ?? (
1
?? )?=??
8
?? =0
??? ?? (
1
?? )
?? ?=?? 0
+
?? 1
?? +
?? 2
?? 2
+?.
 
?? (
1
2
) is a removable singularity. 
3.2 Let ?? be an entire function whose Tayior series expansion with centre ?? =?? 
has infinitely many terms. Show that ?? =?? is an essential singularity of ?? (
?? ?? ) . 
[2021 : 15 marks] 
Solution: 
Let ?? (?? ) be an entire function whose Taylor series expansion with centre ?? =0 has 
infinitely many terms. Clearly, ?? (?? )=?? ?? is an entire function and its Taylor's series 
expansion with centre ?? =0 has infinitely many terms, i.e., 
We have 
?? (?? )=?? ?? =1+?? +
?? 2
2!
+
?? 3
3!
+?. 
?? h?????? ??????????????????????????????????????????? (
1
?? )=?? 1/?? 
???????????????????????????????????????????????????? (
1
?? )=1+
1
?? +
1
2!
·
1
?? 2
+
1
3!
·
1
?? 3
+?. 
Clearly, the principal part of Laurent's series contains infinitely many terms. 
??? =0 is an essential singular point.