Heaviside formula
Let and be polynomials with the degree of the former less than the degree of the latter.
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If all complex zeroes (http://planetmath.org/Zero) of are simple, then
(1) -
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If the different zeroes of have the multiplicities , respectively, we denote ; then
(2)
A special case of the Heaviside formula (1) is
(3) |
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
(4) |
with constants . According to the linearity and the formula 1 of the parent entry (http://planetmath.org/LaplaceTransform), one gets
(5) |
For determining the constants , multiply (3) by . It yields
Setting to this identity gives the value
(6) |
But since , we see that ; thus the equation (5) may be written
(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 |
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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 |