One-dimensional modified Helmholtz Green function

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 $\kappa >0$, the Green function of $D_x^2-{\kappa }^2$ on $(0,L)$ with zero Dirichlet boundary conditions is the displayed negative kernel. Here $x_{<}=\min (x,\xi )$ and $x_{>}=\max (x,\xi )$. The left and right homogeneous solutions are $\sinh (\kappa x)$ and $\sinh (\kappa (L-x))$, with Wronskian $-\kappa \sinh (\kappa L)$. The kernel is continuous and its first derivative jumps by $+1$, giving $(D_x^2-{\kappa }^2)G=\delta (x-\xi )$. Reversing the operator sign reverses the kernel sign.

On the positive half-line, the Green function for $-d^2/d{x}^2+1$ with a zero Dirichlet boundary condition at the origin and decay at infinity isIt is continuous, has derivative jump $G_x(\xi +,\xi )-G_x(\xi -,\xi )=-1$, and is nonnegative. It is the negative of the Dirichlet half-line Green function for d2 minus 1; the sign of the leading differential term matters.

For $\mu >0$, the decaying solution of $-y^{\prime \prime }+y=1_{[0,\mu ]}$ on the positive half-line with $y(0)=0$ follows by integrating the Dirichlet half-line Green function for minus d2 plus 1:Both $y$ and $y^{\prime }$ are continuous at $\mu$, whereas $y^{\prime \prime }$ has the jump required by the bounded step in the forcing. Its common value there is $(1-e^{-\mu }{)}^2/2$.

If $H$ is continuous, piecewise twice differentiable, and has derivative jump $[H^{\prime }]=H^{\prime }(y+)-H^{\prime }(y-)$ at $y$, then its distributional second derivative is $H_{\mathrm{reg}}^{\prime \prime }+[H^{\prime }]{\delta }_y$. If the regular part is integrable and the boundary terms vanish, two integrations by parts give the displayed asymptotic expansion by the Riemann-Lebesgue lemma. A jump $[H]$ in $H$ itself instead produces $[H]e^{-iqy}/(iq)$ at leading order. This distinguishes the different decay rates of the streamfunction and streamwise velocity in linear inviscid damping in unbounded Couette flow.

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