Rectangle, steady GF, single- and double-sum form.

The GF for the 2D rectangle satisfies:

$${\frac{\partial ^{2}G}{\partial x^{2}}} {\frac{\partial ^{2}G}{\partial y^{2}}} \delta \prime \delta \prime$$ = (8) 0 < x < L;  0 < y < W

$${\frac{\partial G}{\partial n_{i}}}$$ = 0 for faces i = 1, 2,..., 4

Each of the four faces of the rectangle may have 3 possible boundary condition types, so the above expression represents a total of 3⁴ = 81 different GF for cases XIJYKL; for example umber X11Y11 represents a rectangle with all four faces of type 1. The 2D GF may be stated in any of three alternative forms.

Double-summation Form for the Rectangle
The GF may be found from separation of variables in both coordinate directions resulting in a double-summation form:

$$\left\vert\vphantom{ \,x^{\prime },y^{\prime }}\right. \prime \prime \left.\vphantom{ \,x^{\prime },y^{\prime }}\right. \sum_{m=0}^{\infty } \sum_{n=0}^{\infty} {\frac{X_m(x^{\prime}) X_m(x)}{N_x(\lambda_m)}}$$ =

$${\frac{Y_{n}(y^{\prime })Y_{n}(y)}{N_{y}(\gamma_{n})}} \left\{\vphantom{ \begin{array}{cc} 0; & m=n=0 \\ (\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise} \end{array}}\right. \begin{array}{cc} 0; & m=n=0 \\ (\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise} \end{array} \left.\vphantom{ \begin{array}{cc} 0; & m=n=0 \\ (\lambda_m^2 + \gamma_n^2)^{-1}; & \mbox{ otherwise} \end{array}}\right.$$ x (9)

Here N_x($\lambda_{m}^{}$) and N_y($\gamma_{n}^{}$) denote the norms of the m^th x-direction eigenfunction and the n^th y-direction eigenfunction, respectively. Eigenfunctions X_m(x) and eigenvalues $\lambda_{m}^{}$ are given in the Tables 2 and 3 above. Eigenfunctions Y_n(y) and eigenvalues $\gamma_{n}^{}$ may be found in the same tables by substituting y, W, and $\gamma_{n}^{}$ in place of x, L, and $\lambda_{m}^{}$, respectively. Generally the double-sum form of the 2D GF has poor convergence properties; the single-sum forms are recommended for calculating numerical values. The double-sum form can appear in temperature expressions involving the steady 3D GF or the transient 2D GF.

Single-Summation Form for the Rectangle
The GF for the rectangle has a single-summation form with previously described eigenfunction Y_n and norm N_y, and with kernel function P_n, as follows:

$$\left\vert\vphantom{ \,x^{\prime },y^{\prime }}\right. \prime \prime \left.\vphantom{ \,x^{\prime },y^{\prime }}\right. \sum_{n=0}^{\infty} {\frac{Y_{n}(y^{\prime })Y_{n}(y)}{N_{y}(\gamma_{n})}} \prime$$ (10)

The n = 0 term of the series is needed only when Y22 is part of the GF number (when zero is an eigenvalue). Special case X22Y22 is discussed later. Kernel function P₀ is identical to the polynomial form of the 1D GF for the slab and is given in Table 1. That is, P₀ = G_1D(x, x^$\prime$). Kernel functions P_n for n $\neq$ 0 are given by

$$\prime \gamma_{n}^{} \left\vert\vphantom{ x-x^{\prime} }\right. \prime \left.\vphantom{ x-x^{\prime} }\right\vert$$ = (11)

$$\gamma_{n}^{} \prime$$

$$\gamma_{n}^{} \left\vert\vphantom{ x-x^{\prime} }\right. \prime \left.\vphantom{ x-x^{\prime} }\right\vert$$

$$\gamma_{n}^{} \prime$$

$$\gamma_{n}^{} \gamma_{n}^{}$$ ÷

where coefficients S₁⁺, S₁^-, S₂⁺, and S₂^- are given in Table 4. Kernel functions P_n for n $\neq$ 0 have nine different forms, one for each of the boundary condition combinations XIJ for I,J = 1, 2, 3. (The kernel functions may also be stated with hyperbolic sines and cosines; however these can cause ``numerical overflow errors'' when evaluated on a computer).

Table 4. Coefficients for kernel functions.

geometry	S1+	S1-	S2+	S2-
X11	1	-1	1	-1
X12	1	-1	1	1
X13	1	-1	$\gamma_{n}^{}$L + B2	$\gamma_{n}^{}$L - B2
X21	1	1	1	-1
X22	1	1	1	1
X23	1	1	$\gamma_{n}^{}$L + B2	$\gamma_{n}^{}$L - B2
X31	$\gamma_{n}^{}$L + B1	$\gamma_{n}^{}$L - B1	1	-1
X32	$\gamma_{n}^{}$L + B1	$\gamma_{n}^{}$L - B1	1	1
X33	$\gamma_{n}^{}$L + B1	$\gamma_{n}^{}$L - B1	$\gamma_{n}^{}$L + B2	$\gamma_{n}^{}$L - B2

Alternative Single-sum GF for the Rectangle
In the above equation kernel functions P_n are placed along the x axis. An alternate GF may be constructed by placing the kernel functions along the y axis. The convergence behavior of the alternate single-sum forms of the rectangle GF are complementary. Where one converges slowly, the other generally converges rapidly, so that between them the GF may be computed anywhere in the rectangle.