Semi infinite body, steady 1-D Helmholtz.

X10 Semi-infinite body, 0 < x < $\infty$, with G = 0 (Dirichlet) at x = 0.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} e^{-mx^{\prime }}\sinh mx/m ... ...\\ e^{-mx}\sinh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} e^{-mx^{\prime }}\sinh mx/m & \text{for }x<x^{\prime } \\ e^{-mx}\sinh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} e^{-mx^{\prime }}\sinh mx/m &... ...\\ e^{-mx}\sinh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} }\right.$$

X20 Semi-infinite body, 0 < x < $\infty$, with $\partial$G/$\partial$x = 0 at x=0.

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} e^{-mx^{\prime }}\cosh mx/m ... ...\\ e^{-mx}\cosh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} e^{-mx^{\prime }}\cosh mx/m & \text{for }x<x^{\prime } \\ e^{-mx}\cosh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} e^{-mx^{\prime }}\cosh mx/m &... ...\\ e^{-mx}\cosh mx^{\prime }/m & \text{for }x>x^{\prime } \end{array} }\right.$$

X30 Semi-infinite body, 0 < x < $\infty$, with - k$\partial$G/$\partial$x + hG = 0 (convection) at x = 0. Note that B = h/(km).

$$\left\vert\vphantom{ \,x^{\prime }}\right. \prime \left.\vphantom{ \,x^{\prime }}\right. \left\{\vphantom{ \begin{array}{cc} \begin{array}{c} \left[ e^{-... ...-m(x+x^{\prime })}\right] /(2m) & \text{for }x>x^{\prime } \end{array} }\right. \begin{array}{cc} \begin{array}{c} \left[ e^{-m(x^{\prime }-x)}+\... ...}{1+B}e^{-m(x+x^{\prime })}\right] /(2m) & \text{for }x>x^{\prime } \end{array} \left.\vphantom{ \begin{array}{cc} \begin{array}{c} \left[ e^{-m... ...-m(x+x^{\prime })}\right] /(2m) & \text{for }x>x^{\prime } \end{array} }\right.$$