Heaviside formula
Let
and
be polynomials with the degree of the former less than the degree of the latter.
A special case of the Heaviside formula (1) is
Example. Since the zeros of the binomial
are
, we obtain
Proof of (1). Without hurting the generality, we can suppose that
is monic. Therefore
For
, denoting
one has
. We have a partial fraction expansion of the form
 |
(3) |
with constants
. According to the linearity and the formula 1 of the parent entry,
one gets
 |
(4) |
For determining the constants
, multiply (3) by
. It yields
Setting to this identity
gives the value
 |
(5) |
But since
, we see that
; thus the equation (5) may be written
 |
(6) |
The values (6) in (4) produce the formula
(1).
- 1
- K. V¨AISÄLÄ: Laplace-muunnos. Handout Nr. 163. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1968).
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