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