Inverse Triple Laplace Transform of $\frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}}$ I want to find the inverse triple Laplace transform of $L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right]$. I did
\begin{align*} 
L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right] &= L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right] \right] \right]
\\
&= (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[\frac{1}{a} L^{-1}_{x_{1}} \left[ \frac{a}{s^2_{1} + a^2} \right] \right] \right], \ \  a^2 = s^2_{2} + s^2_{3} 
\\
&= (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[\frac{ \sin \left( x_{1} \sqrt{ \left( s^2_{2} + s^2_{3}\right)} \right) }{\sqrt{ \left( s^2_{2} + s^2_{3}\right)}}  \right] \right] 
\\
&= (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[ \frac{ \displaystyle\sum_{k = 0}^{\infty} \frac{(-1)^k \left(x_{1} \sqrt{s^2_{2} + s^2_{3}} \right)^{2k+1}}{(2k+1)!} }{\sqrt{ \left( s^2_{2} + s^2_{3}\right)}} \right] \right]
\\
&\approx (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[ \frac{ x_{1} \sqrt{s^2_{2} + s^2_{3}} - \frac{1}{6} \left(x_{1} \sqrt{s^2_{2} + s^2_{3}} \right)^3 }{\sqrt{ \left( s^2_{2} + s^2_{3}\right)}} \right] \right]
\\
&= (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[ x_{1} - \frac{1}{6} x_{1}^3 \left( s^2_{2} + s^2_{3} \right) \right] \right]
\\
&= (-1) L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[ \left( x_{1} - \frac{1}{6} x_{1}^3 s^2_{3} \right) - \frac{1}{6} x_{1}^3 s^2_{3} \right] \right]
\\
&= (-1) L^{-1}_{x_{3}} \left[ \left( x_{1} - \frac{1}{6} x_{1}^3 s^2_{3} \right) \delta(x_{2}) - \frac{1}{6} x_{1}^3 \delta^{"}(x_{2}) \right] 
\\
&= (-1) \left( \left( x_{1} \delta(x_{3}) - \frac{1}{6} x_{1}^3 \delta^{"}(x_{3}) \right) \delta(x_{2}) - \frac{1}{6} x_{1}^3 \delta^{"}(x_{2}) \delta(x_{3}) \right)
\end{align*}
I am wondering if this solution is correct or not? I would appreciate your help.
 A: I tried another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct :
\begin{align*} 
L^{-1}_{x_{3}} L^{-1}_{x_{2}} L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right] &= L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[L^{-1}_{x_{1}} \left[ \frac{-1}{s^2_{1} + s^2_{2} + s^2_{3}} \right] \right] \right]
\\
&= L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[L^{-1}_{x_{1}} \left[ \frac{-1}{a^2 + b^2 + c^2} + \frac{2}{a^2 + b^2 + c^2} \left[ a(s_{1} - a) + b(s_{2} - b) + c(s_{3} - c) \right] + \ldots \right] \right] \right]
\\
& \approx L^{-1}_{x_{3}} \left[ L^{-1}_{x_{2}} \left[L^{-1}_{x_{1}} \left[ \frac{-1}{a^2 + b^2 + c^2} + \frac{2}{a^2 + b^2 + c^2} \left[ a(s_{1} - a) + b(s_{2} - b) + c(s_{3} - c) \right] \right] \right] \right] \\
&\approx \frac{-1}{a^2 + b^2 + c^2} \delta(x_{1}) \delta(x_{2}) \delta(x_{3}) \\
& \ \ \ + \frac{2}{a^2 + b^2 + c^2} \Big[ a \left(\delta^{\prime}(x_{1}) \delta(x_{2}) \delta(x_{3}) - a \delta(x_{1}) \delta(x_{2}) \delta(x_{3}) \right) \\
& \ \ \ + b \left(\delta(x_{1}) \delta^{\prime}(x_{2}) \delta(x_{3}) - b \delta(x_{1}) \delta(x_{2}) \delta(x_{3})  \right) \\
& \ \ \ + c \left(\delta(x_{1}) \delta(x_{2}) \delta^{\prime}(x_{3}) - c \delta(x_{1}) \delta(x_{2}) \delta(x_{3})  \right) \Big], 
\end{align*}
for arbitrary $a, b, c \in [0, \infty)$.
