Post on 11-Jul-2020
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Created by T. Madas
Created by T. Madas
RESIDUES and APPLICATIONS
in SERIES SUMMATION
Created by T. Madas
Created by T. Madas
The Residue Theorem can often be used to sum various types of series.
The following results are valid under some restrictions on ( )f z , which more often
than not are satisfied when the series converges.
( )
r
f r
∞
=−∞
∑
use ( ) cot
n
f z z dz
Γ
π π∫� ,where nΓ is the square with vertices at ( )( )1 1 i2
n + ± ±
( ) ( )1r
r
f r
∞
=−∞
−∑
use ( ) cosec
n
f z z dz
Γ
π π∫� ,where nΓ is the square with vertices at ( )( )1 1 i2
n + ± ±
2 1
2r
rf
∞
=−∞
+ ∑
use ( ) tan
n
f z z dz
Γ
π π∫� ,where nΓ is the square with vertices at ( )1 in ± ±
( )2 1
12
r
r
rf
∞
=−∞
+ −
∑
use ( ) sec
n
f z z dz
Γ
π π∫� ,where nΓ is the square with vertices at ( )1 in ± ±
Created by T. Madas
Created by T. Madas
Question 1
( )( )
2
cot zf z
a z
π π=
+, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )( )2 2
2
1cosec
r
aa r
π π
∞
=−∞
=+∑ , a ∉� .
proof
Created by T. Madas
Created by T. Madas
Question 2
( )( )( )
cot
3 1 2 1
zf z
z z
π π=
+ +, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )( )
13
3 1 2 1r
r rπ
∞
=−∞
=+ +∑ .
proof
Created by T. Madas
Created by T. Madas
Question 3
( )2
cot
4 1
zf z
z
π π=
−, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
2
1
1 1
24 1r
r
∞
=
=−∑ .
proof
Created by T. Madas
Created by T. Madas
Question 4
( )2
cot zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
2
2
1
1
6r
r
π
∞
=
=∑ .
proof
Created by T. Madas
Created by T. Madas
Question 5
( )4
cot zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
4
4
1
1
90r
r
π
∞
=
=∑ .
proof
Created by T. Madas
Created by T. Madas
Question 6
( )( )
22
cot
1
zf z
z
π π=
+
, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )2 2
22
1
1 1 1 1cosech coth
4 4 21
r
r
π π π π
∞
=
= − −
+∑ .
proof
Created by T. Madas
Created by T. Madas
Question 7
( )( )
2
cosec zf z
a z
π π=
+, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )
( )( ) ( )2
2
1cosec cot
r
r
a aa r
π π π
∞
=−∞
−=
+∑ , a ∉� .
proof
Created by T. Madas
Created by T. Madas
Question 8
( )( )( )
cosec
2 1 3 1
zf z
z z
π π=
+ +, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )
( )( )( )
12 3 1
2 1 3 1 3
r
r
r r
π
∞
=−∞
−= −
+ +∑ .
proof
Created by T. Madas
Created by T. Madas
Question 9
( )2
cosec
4 1
zf z
z
π π=
−, z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )( )
2
1
1 12
44 1
r
r
rπ
∞
=
−= −
−∑ .
proof
Created by T. Madas
Created by T. Madas
Question 10
( )2
cosec zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( ) 2
2
1
1 1
12
r
r
rπ
∞
=
−= −∑ .
proof
Created by T. Madas
Created by T. Madas
Question 11
( )4
cosec zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )1 4
4
1
1 7
720
r
r
r
π
∞+
=
−=∑ .
proof
Created by T. Madas
Created by T. Madas
Question 12
( )4
tan zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )
4
4
0
1
962 1r
r
π
∞
=
=+∑ .
proof
Created by T. Madas
Created by T. Madas
Question 13
( )3
sec zf z
z
π π= , z ∈� .
By integrating ( )f z over a suitable contour Γ , show that
( )
( )
3
3
0
1
322 1
r
rr
π
∞
=
−=
+∑ .
proof