Green function

For $L_k=k{\partial }_z+k^{-1}{\partial }_{\bar{z}}$, $k\ne 0$, $|k|\ne 1$, this kernel satisfies $L_k{G}_k={\delta }_0$. A real-linear change of complex coordinate reduces its derivation to the Cauchy-Green operator; the determinant changes sign at the unit circle. For $|k|=1$ the operator becomes a real directional transport equation, whose characteristic singularity requires limiting prescriptions. The kernel has $G_k\sim k/(\pi z)$ at zero and $G_k\sim 1/(\pi k\bar{z})$ at infinity.

A Dirichlet Green function for a differential operator satisfies homogeneous Dirichlet boundary conditions in one variable. It converts source and boundary data into an integral representation of the solution.

The density of expected Brownian occupation time before exit from a planar domain. For planar Brownian motion with infinitesimal generator $\frac{1}{2}\Delta$, it obeys $-\frac{1}{2}{\Delta }_w{G}_D(z,w)={\delta }_z(w)$ and has singularity $-{\pi }^{-1}\log |w-z|$. A conformal bijection preserves this Dirichlet Green function: the squared derivative in the conformal Brownian clock cancels the area Jacobian determinant.

For the positive Laplacian convention $\Delta G=\delta$, the method of images in the first quadrant reflects an interior pole $(a,b)$ across each axis with negative sign and across both with positive sign:The Dirichlet Green function vanishes on both axes. Its outward normal derivative on the horizontal axis isand the vertical-axis kernel follows by exchange of coordinates. Green second identity converts these derivatives into the quadrant Poisson integral for boundary data.

For the canonical Dirichlet Green function with logarithmic singularity, its harmonic correction satisfies $G_z(z)=-\log {\mathrm{crad}}_D(z)$. If $\varphi$ maps the unit disc to $D$ with $\varphi (0)=z$, thenThe quotient extends at zero to ${\varphi }^{\prime }(0)$. On unbounded domains the canonical Green function, rather than boundary values alone, fixes the harmonic correction.

If $G(x;\xi )$ satisfies the boundary conditions and $L_{x}G(x;\xi )=\delta (x-\xi )$, then a solution of $Ly=f$ isup to any homogeneous contribution required by nonhomogeneous boundary data.

For $\mathrm{Re}m>0$, the decaying solution ofisIt is continuous at zero and has derivative jump $G^{\prime }(0+)-G^{\prime }(0-)=-1$.

For $x,\xi >0$, the Green function satisfyingas $x\to \infty$ is

On $\mathbb{R}\times (0,a)$, separation into sine modes gives the Dirichlet Green function for the Laplacian

Let $y_1,y_2$ solve $y^{\prime \prime }+\alpha y^{\prime }+\beta y=0$ with $y_1^{\prime }(0)=0$ and $y_2^{\prime }(1)=0$. For Neumann boundary conditions, the Green function iswhere $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$. The derivative jump is one. If $\alpha =0$, the Abel identity makes $W$ constant and $G$ symmetric.

A causal Green function vanishes before the source time and has the derivative jump required by a delta source.

For $\mathcal{L}=D^3+a_2{D}^2+a_{1}D+a_0$ with smooth coefficients and leading coefficient one, a Green function solving $\mathcal{L}G={\delta }_{\xi }$ is continuous together with its first derivative at an interior source, and its second derivative jumps by one. Distributional differentiation shows why: a jump in $G$ would create a delta second derivative, a jump in $G^{\prime }$ a delta first derivative, and a jump in $G^{\prime \prime }$ the required delta. For a causal initial-value Green function, take the left branch zero and choose the right branch with initial values $0,0,1$ at the source.

For positive $k$, $e^{ik|x-x^{\prime }|}$ is outgoing on both sides of $x^{\prime }$ and obeys