General recipe for inverse Laplace transform
We obtain 
as a rational function:
as a rational function:
where 
is a polynomial of degree m, 
is a polynomial of degree
Here we use the "complex version" of the partial fraction decomposition, since this will also work for multiple complex roots.
Step 1: Find the roots 
of 
There may be
- multiple roots
- complex (nonreal) roots
Assume 
are the different roots, where 
is a root of multiplicity 
.
are the different roots, where is a root of multiplicity .Then 
.
.Note: For a complex (nonreal) root the complex conjugate 
is also a root.
the complex conjugate is also a root.Step 2: Set up the m terms of the partial fraction decomposition
For a root of multiplicity we use the terms
of multiplicity we use the terms
with unknown coefficients 
The partial fraction decomposition of 
is a sum of m terms, with m unknown coefficients.
is a sum of m terms, with m unknown coefficients.Note: For complex (nonreal) root : the coefficients for are the complex conjugates of the coefficients for .
: the coefficients for are the complex conjugates of the coefficients for .Step 3: Find the m coefficients of the partial fraction decomposition
Multiply the partial fraction decomposition 
by 
, yielding equation (eq1).
by , yielding equation (eq1).For all roots :
: In (eq1) plug in 
. This gives an equation which only contains 
, yielding .
. This gives an equation which only contains , yielding .If there are roots of multiplicity 
: take the derivative of (eq1), yielding equation (eq2).
: take the derivative of (eq1), yielding equation (eq2).For all roots of multiplicity :
of multiplicity : In (eq2) plug in . This gives an equation which contains 
and , yielding .
. This gives an equation which contains and , yielding .If there are of multiplicity 
: take the derivative of (eq2), yielding equation (eq3).
: take the derivative of (eq2), yielding equation (eq3).For all roots of multiplicity :
of multiplicity : In (eq3) plug in . This gives an equation which contains 
, , , yielding .
. This gives an equation which contains , , , yielding .If there are roots of even higher multiplicity: continue in the same way.
Note: If there is just one coefficient left to determine, you can plug in any new value for s. This is usually easier than taking a derivative.
Note: For complex (nonreal) roots you can skip the conjugate roots . We already know the coefficients (they are the complex conjugates of the coefficients for ).
you can skip the conjugate roots . We already know the coefficients (they are the complex conjugates of the coefficients for ).Step 4: Find the inverse Laplace transform for each term, and add everything together
For real roots :
:
For complex roots 
note that 
is also a root, yielding
note that is also a root, yielding 