Heaviside formula

Heaviside formula

Let

P

⁢

(

s

)

and

Q

⁢

(

s

)

be

polynomials

with the degree of the former less than the degree of the latter.

•

If all complex zeroes (

http://planetmath.org/Zero

)

a

1

,

a

2

,

…

,

a

n

of

Q

⁢

(

s

)

are simple, then

ℒ

-

1

⁢

{

P

⁢

(

s

)

Q

⁢

(

s

)

}

=

∑

j

=

1

n

P

⁢

(

a

j

)

Q

′

⁢

(

a

j

)

⁢

e

a

j

⁢

t

.

(1)

•

If the different zeroes

a

1

,

a

2

,

…

,

a

n

of

Q

⁢

(

s

)

have the

multiplicities

m

1

,

m

2

,

…

,

m

n

, respectively,
we denote

F

j

⁢

(

s

)

:=

(

s

-

a

j

)

m

j

⁢

P

⁢

(

s

)

/

Q

⁢

(

s

)

;  then

ℒ

-

1

⁢

{

P

⁢

(

s

)

Q

⁢

(

s

)

}

=

∑

j

=

1

n

e

a

j

⁢

t

⁢

∑

k

=

0

m

j

-

1

F

j

(

k

)

⁢

(

a

j

)

⁢

t

m

j

-

1

-

k

k

!

⁢

(

m

j

-

1

-

k

)

!

.

(2)

A special case of the

Heaviside formula

(1) is

ℒ

-

1

⁢

{

Q

′

⁢

(

s

)

Q

⁢

(

s

)

}

=

∑

j

=

1

n

e

a

j

⁢

t

.

(3)

Example.

Since the zeros of the

binomial

s

4

+

4

⁢

a

4

are

s

=

(

±

1

±

i

)

⁢

a

,  we
can calculate by (3) as follows:

ℒ

-

1

⁢

{

s

3

s

4

+

4

⁢

a

4

}

=

1

4

⁢

ℒ

-

1

⁢

{

4

⁢

s

3

s

4

+

4

⁢

a

4

}

=

1

4

⁢

∑

±

e

(

±

1

±

i

)

⁢

a

⁢

t

=

e

a

⁢

t

+

e

-

a

⁢

t

2

⋅

e

i

⁢

a

⁢

t

+

e

-

i

⁢

a

⁢

t

2

=

cosh

⁡

a

⁢

t

⁢

cos

⁡

a

⁢

t

Proof of (1).

Without hurting the generality, we can
suppose that

Q

⁢

(

s

)

is monic.  Therefore

Q

⁢

(

s

)

=

(

s

-

a

1

)

⁢

(

s

-

a

2

)

⁢

⋯

⁢

(

s

-

s

n

)

.

For

j

=

1

,

2

,

…

,

n

,  denoting

Q

⁢

(

s

)

:=

(

s

-

a

j

)

⁢

Q

j

⁢

(

s

)

,

one has

Q

j

⁢

(

a

j

)

≠

0

.  We have a partial fraction
expansion of the form

P

⁢

(

s

)

Q

⁢

(

s

)

=

C

1

s

-

a

1

+

C

2

s

-

a

2

+

…

+

C

n

s

-

a

n

(4)

with constants

C

j

.  According to the linearity and the
formula 1 of the parent entry (

http://planetmath.org/LaplaceTransform

),
one gets

ℒ

-

1

⁢

{

P

⁢

(

s

)

Q

⁢

(

s

)

}

=

∑

j

=

1

n

C

j

⁢

e

a

j

⁢

t

.

(5)

For determining the constants

C

j

, multiply (3) by

s

-

a

j

.  It yields

P

⁢

(

s

)

Q

j

⁢

(

s

)

=

C

j

+

(

s

-

a

j

)

⁢

∑

ν

≠

j

C

ν

s

-

a

ν

.

Setting to this

identity

s

:=

a

j

gives the value

C

j

=

P

⁢

(

a

j

)

Q

j

⁢

(

a

j

)

.

(6)

But since

Q

′

⁢

(

s

)

=

d

d

⁢

s

⁢

(

(

s

-

a

j

)

⁢

Q

j

⁢

(

s

)

)

=

Q

j

⁢

(

s

)

+

(

s

-

a

j

)

⁢

Q

j

′

⁢

(

s

)

,
we see that

Q

′

⁢

(

a

j

)

=

Q

j

⁢

(

a

j

)

;  thus the equation (5) may
be written

C

j

=

P

⁢

(

a

j

)

Q

′

⁢

(

a

j

)

.

(7)

The values (6) in (4) produce the formula (1).

References

1

K. Väisälä:

Laplace-muunnos

.  Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).

Title

Heaviside formula

Canonical name

HeavisideFormula

Date of creation

2014-03-19 9:14:46

Last modified on

2014-03-19 9:14:46

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

11

Author

pahio (2872)

Entry type

Topic

Classification

msc 44A10

Synonym

Heaviside expansion formula

Synonym

inverse Laplace transform of rational function

Related topic

HyperbolicFunctions

Related topic

ComplexSineAndCosine

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