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I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not: \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 $a, b, c \in [0, \infty)$.

I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not: \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 $a, b, c \in [0, \infty)$.

I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not: \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[\frac{1}{a} 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[\frac{1}{a} 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 $a, b, c \in [0, \infty)$.

I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not: \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 $a, b, c \in [0, \infty)$.

I have 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[\frac{1}{a} 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[\frac{1}{a} 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 $a, b, c \in [0, \infty)$.

I have another way to solve the problem by using the Taylor series of three variables and I am wondering whether it is correct or not: \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[\frac{1}{a} 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[\frac{1}{a} 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 $a, b, c \in [0, \infty)$.