# About the symmetric nature of Green's function.

What is the significance of Green's function being symmetric ? How do I understand this intuitively? Thanks in advance .

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One example of the use of the symmetric property is in proving that Poisson's formula $u$ on the half space $\mathbb{R}^n$ belongs to the space \begin{equation*} C^{\infty}(\mathbb{R}^n_+)\cap L^{\infty}(\mathbb{R}^{n-1}) \end{equation*} (ie, smooth functions on the half plane extended to $L^{\infty}$ on the $n-1-$dimensional real numbers).
The intuition behind Green's function symmetry comes from odd/even functions. If $f:\mathbb{R}\to\mathbb{R}$ is a continuously differentiable function, then we can write two functions \begin{equation*} y(x)=f(x)-f(-x) \\ z(x)=f(x)+f(-x) \end{equation*} with initial conditions $y(0)=0$ and $z'(0)=0$. Observe that we can derive Dirichlet and Neumann type boundary conditions when we subtract mirror images of a function. This leads to the symmetry condition \begin{equation*} G(x,y)=G(y,x),~x,y\in\Omega,~x\neq y \end{equation*} where $\Omega$ is an open and bounded subset of $\mathbb{R}^n$, and $G$ is Green's function.