Suppose we want to find Green's function in one of the $2$-dimensional quadrants, say the first one $D = \{(x,y) \in \mathbb{R}^n : x > 0, y > 0\}$. Let $x = (x_1, x_2)$ and $y = (y_1, y_2)$.
Using Evan's method of finding reflections and defining corrector functions: Let $$ x_0 = (x_1, x_2), \quad x_0^x = (x_1, -x_2), \quad x_0^y = (-x_1, x_2), \quad x_0^{xy} = (-x_1, -x_2),$$ with correctors $$\phi(y - x_0^x), \quad \phi(y - x_0^x), \quad \phi(y - x_0^y), \quad \phi(y - x_0^{xy})$$ where the fundamental solution $\phi$ in the two dimensional case is defined as, $\phi(z) := -\frac{1}{2\pi}|z|$, with $z \in \mathbb{R}^2$. Then Green's function is $$G(x_0, y) := \phi(y - x_0) + \phi(y - x_0^{xy}) - \phi(y - x_0^x) -\phi(y - x_0^y). \tag{$\dagger$}$$
However, I just noticed that in a few different notes that they take the opposite signs for $\phi$, i.e.
$$G(x_0, y) := -\phi(y - x_0) - \phi(y - x_0^{xy}) + \phi(y - x_0^x) +\phi(y - x_0^y). \tag{$\star$}$$
What's the reasoning for the difference between $(\dagger)$ and $(\star)$?