<s>[INST] Inverse Laplace Transform via Circuit Analysis [HELP] Inverse Laplace Transform $\frac{1}{s^2 + \sqrt{2}s + 1}$
so what I did it changed the denominator to complete the square format which is $\left(s+\frac{\sqrt{2}}{2}\right)^2 + \frac{1}{2}$, then I can solve for $s$, it will make it as
$$
\left(\left(s+ \frac{\sqrt{2}}{2}\right) + \frac{\sqrt{2}}{2}i\right)
\left(\left(s+ \frac{\sqrt{2}}{2}\right) - \frac{\sqrt{2}}{2}i\right)
$$
So now, to the sheet of paper is to do Partial Fraction Decomposition of this which is absurd to me because of complex roots it has:
$$
\frac{1}{s^2 + s\sqrt{2} + 1}
= \frac{1}{\left(s+\frac{\sqrt{2}}{2}\right)^2 + \frac{1}{2}}
$$
Partial Fraction of Complex root will be
$$
\frac{K}{\left(s+ \frac{\sqrt{2}}{2}\right) + \frac{\sqrt{2}}{2}i}
 + \frac{K^*}{\left(s+ \frac{\sqrt{2}}{2}\right) - \frac{\sqrt{2}}{2}i}
$$
to follow the formula sheet.
which I got my K = -$i\frac{\sqrt{2}}{2}$ and $K^*$ = $i\frac{\sqrt{2}}{2}$
the problem I get is magnitude and $\theta$ is undefined it makes no sense at all.

 [/INST] Let us denote by $a$ the value $1/\sqrt 2$ to have an easy typing. The the partial fraction decomposition over complex numbers of the given fraction is
indeed
$$
\begin{aligned}
\frac{1}{s^2 + \sqrt{2}s + 1}
&=
\frac{1}{(s^2 + 2as + a^2)+a^2}
=
\frac1{2ia}\left(\frac 1{s+a-ia}-\frac 1{s+a+ia}\right) 
\\
&=\frac K{s+a-ia}-\frac {K^*}{s+a+ia}
\ .
\\[3mm]
\text{Here:} &
\\
K
&=\frac 1{2ia}
=-\frac 1{2a}i
=\frac 1{2a}\left(\cos\left(-\frac\pi2\right) +i\sin\left(-\frac\pi2\right)\right)
\\
&=a\left(\cos\left(-\frac\pi2\right) +i\sin\left(-\frac\pi2\right)\right)\ ,
\\
|K| &= a\ ,\\
\theta &=-\frac \pi 2\ .
\end{aligned}
$$
The inverse Laplace transform is thus using the table:
$$
2ae^{-at}\cos(at+\theta)
=
2ae^{-at}\sin(at)
=
\sqrt 2\cdot e^{-t/\sqrt 2}\sin(t/\sqrt 2)
\ .
$$
</s>