Coefficients of Partial Fraction Expansion

Coefficients of Partial Fraction Expansion

Coefficients of Partial Fraction Expansion
Swapnil Sunil Jain
July 28 2006

Coefficients of Partial Fraction Expansion

Let us start with the

assumption

(or rather a Lemma) that any

rational

proper

function

F

⁢

(

s

)

of the form

F

⁢

(

s

)

=

P

⁢

(

s

)

(

s

-

q

)

r

⁢

(

s

-

p

1

)

⁢

(

s

-

p

2

)

⁢

…

⁢

(

s

-

p

i

)

⁢

…

⁢

(

s

-

p

n

)

(1)

has a partial fraction expansion given by

F

⁢

(

s

)

=

a

0

(

s

-

q

)

r

+

a

1

(

s

-

q

)

r

-

1

+

…

+

a

j

(

s

-

q

)

r

-

j

+

…

+

a

r

-

1

(

s

-

q

)

+

k

1

(

s

-

p

1

)

+

k

2

(

s

-

p

2

)

+

…

+

k

i

(

s

-

p

i

)

+

…

+

k

n

(

s

-

p

n

)

(2)

where

j

=

0

,

1

,

2

,

…

,

r

-

1

and

i

=

1

,

2

,

3

,

…

,

n

and

q

≠

p

1

≠

p

2

≠

…

≠

p

n

.

First, we determine the coefficient

k

i

. In order to do so, we multiply both sides of equation (2) by

(

s

-

p

i

)

which then gives us

(

s

-

p

i

)

⁢

F

⁢

(

s

)

=

a

0

(

s

-

q

)

r

⁢

(

s

-

p

i

)

+

a

1

(

s

-

q

)

r

-

1

⁢

(

s

-

p

i

)

+

…

+

a

j

(

s

-

q

)

r

-

j

⁢

(

s

-

p

i

)

+

…

+

a

r

-

1

(

s

-

q

)

⁢

(

s

-

p

i

)

+

k

1

(

s

-

p

1

)

⁢

(

s

-

p

i

)

+

k

2

(

s

-

p

2

)

⁢

(

s

-

p

i

)

+

…

+

k

i

(

s

-

p

i

)

⁢

(

s

-

p

i

)

+

…

+

k

n

(

s

-

p

n

)

⁢

(

s

-

p

i

)

(3)

If we then let

s

=

p

i

, all the terms on the R.H.S drop out except the one containing the coefficient

k

i

and we get

[

(

s

-

p

i

)

⁢

F

⁢

(

s

)

]

|

s

=

p

i

=

k

i

(4)

Now, in order to determine the coefficient

a

j

, we multiply both sides of (2) by

(

s

-

q

)

r

which yields

(

s

-

q

)

r

⁢

F

⁢

(

s

)

=

a

0

(

s

-

q

)

r

⁢

(

s

-

q

)

r

+

a

1

(

s

-

q

)

r

-

1

⁢

(

s

-

q

)

r

+

…

+

a

j

(

s

-

q

)

r

-

j

⁢

(

s

-

q

)

r

+

…

+

a

r

-

1

(

s

-

q

)

⁢

(

s

-

q

)

r

+

(

s

-

q

)

r

⁢

[

k

1

(

s

-

p

1

)

+

k

2

(

s

-

p

2

)

+

…

+

k

i

(

s

-

p

i

)

+

…

+

k

n

(

s

-

p

n

)

]

⇒

(

s

-

q

)

r

⁢

F

⁢

(

s

)

=

a

0

+

a

1

⁢

(

s

-

q

)

1

+

…

+

a

j

⁢

(

s

-

q

)

j

+

…

+

a

r

-

1

⁢

(

s

-

q

)

r

-

1

+

(

s

-

q

)

r

⁢

A

⁢

(

s

)

B

⁢

(

s

)

(5)

where we have defined

A

⁢

(

s

)

B

⁢

(

s

)

≡

k

1

(

s

-

p

1

)

+

k

2

(

s

-

p

2

)

+

…

+

k

i

(

s

-

p

i

)

+

…

+

k

n

(

s

-

p

n

)

Then if we take the

derivative

of the above equation with respect to

s

and we obtain

d

d

⁢

s

⁢

[

(

s

-

q

)

r

⁢

F

⁢

(

s

)

]

=

a

1

+

a

2

⁢

(

2

)

⁢

(

s

-

q

)

+

…

+

a

j

⁢

(

j

)

⁢

(

s

-

q

)

j

-

1

+

…

+

a

r

-

1

⁢

(

r

-

1

)

⁢

(

s

-

q

)

r

-

2

+

d

d

⁢

s

⁢

[

(

s

-

q

)

r

⁢

A

⁢

(

s

)

B

⁢

(

s

)

]

(6)

If we again take the derivative of both sides of the above equation with respect to

s

we get

d

2

d

⁢

s

2

⁢

[

(

s

-

q

)

r

⁢

F

⁢

(

s

)

]

=

2

⁢

a

2

+

…

+

a

j

⁢

(

j

)

⁢

(

j

-

1

)

⁢

(

s

-

q

)

j

-

2

+

…

+

a

r

-

1

⁢

(

r

-

1

)

⁢

(

r

-

2

)

⁢

(

s

-

q

)

r

-

3

+

d

2

d

⁢

s

2

⁢

[

(

s

-

q

)

r

⁢

A

⁢

(

s

)

B

⁢

(

s

)

]

(7)

If we keep taking derivatives this way until we have taken the derivative

j

times, we arrive at

d

j

d

⁢

s

j

⁢

[

(

s

-

q

)

r

⁢

F

⁢

(

s

)

]

=

a

j

⁢

(

j

)

⁢

(

j

-

1

)

⁢

(

j

-

2

)

⁢

…

⁢

(

2

)

⁢

(

1

)

⁢

(

s

-

q

)

j

-

j

+

…

+

a

r

-

1

⁢

(

r

-

1

)

⁢

(

r

-

2

)

⁢

…

⁢

(

r

-

j

)

⁢

(

s

-

q

)

r

-

j

-

1

+

d

j

d

⁢

s

j

⁢

[

(

s

-

q

)

r

⁢

A

⁢

(

s

)

B

⁢

(

s

)

]

(8)

If we then let

s

=

q

, all the terms on the R.H.S drop out except the one containing the coefficient

a

j

which yields

(

d

j

d

⁢

s

j

⁢

[

(

s

-

q

)

⁢

F

⁢

(

s

)

]

)

|

s

=

q

=

a

j

⁢

j

!

(9)

or

a

j

=

1

j

!

⁢

(

d

j

d

⁢

s

j

⁢

[

(

s

-

q

)

⁢

F

⁢

(

s

)

]

)

|

s

=

q

(10)

Title

Coefficients of Partial Fraction Expansion

Canonical

name

CoefficientsOfPartialFractionExpansion1

Date of creation

2013-03-11 19:26:06

Last modified on

2013-03-11 19:26:06

Owner

swapnizzle (13346)

Last modified by

(0)

Numerical id

1

Author

swapnizzle (0)

Entry type

Definition

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