residues and applications in series summation€¦ · in series summation . created by t. madas...
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![Page 1: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/1.jpg)
Created by T. Madas
Created by T. Madas
RESIDUES and APPLICATIONS
in SERIES SUMMATION
![Page 2: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/2.jpg)
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 ± ±
![Page 3: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/3.jpg)
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
![Page 4: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/4.jpg)
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
![Page 5: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/5.jpg)
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
![Page 6: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/6.jpg)
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
![Page 7: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/7.jpg)
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
![Page 8: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/8.jpg)
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
![Page 9: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/9.jpg)
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
![Page 10: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/10.jpg)
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
![Page 11: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/11.jpg)
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
![Page 12: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/12.jpg)
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
![Page 13: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/13.jpg)
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
![Page 14: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/14.jpg)
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
![Page 15: RESIDUES and APPLICATIONS in SERIES SUMMATION€¦ · in SERIES SUMMATION . Created by T. Madas Created by T. Madas The Residue Theorem can often be used to sum various types of series](https://reader035.vdocuments.net/reader035/viewer/2022062603/5f3affc0239a9c625f2a59d8/html5/thumbnails/15.jpg)
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