Inverse Laplace Transform for Negative a^2?

[TheFerruccio]

There are lots of tables out there for finding the inverse laplace transform of:

$$\frac{1}{(s+b)^{2}+a^{2}}$$

or

$$\frac{s}{(s+b)^{2}+a^{2}}$$

but what if $$a^{2}$$ is negative?

I don't know what useful formula I should split it up into.

 

[matematikawan]

If a² is negative then you may further factored the denominator. Then you formed a partial fraction for the whole expression.

e.g. let a²=-c².

(s+b)²+a²=(s+b+c)(s+b-c)