Non-Stationary Electromagnetics, 2nd Edition

as

${\displaystyle \int d{x}_1}\theta \left(t_2-t_1-\frac{\left|x_2-x_1\right|}{c}\right)\theta \left(t-t^{\prime }-\frac{\left|x_1-x^{\prime }\right|}{c}\right)=\theta \left(t_2-t^{\prime }-\frac{\left|x_2-x^{\prime }\right|}{c}\right){\displaystyle \underset{D_2}{\int }d{x}_1}$

(3.70)

As illustrated in Fig. 3.8a, the expression (3.69) can be rewritten in the form

$\langle x_3\left|{\widehat{\text{K}}}_2^n\right|x^{\prime }\rangle ={\left(\frac{{\omega }_{e2}^2}{2c}\right)}^2\theta \left(t-t^{\prime }-\frac{\left|x-x^{\prime }\right|}{c}\right){\displaystyle \underset{D}{\int }d{x}_{n-1}}{\displaystyle \underset{D_{n-1}}{\int }d{x}_{n-2}}\dots {\displaystyle \underset{D_2}{\int }d{x}_1},$

(3.71)

where the region D_k is defined by the intersection of the cones $t_{\kappa }-t_{\kappa -1}-\frac{|x_{\kappa }-x_{\kappa -1}|}{c}>0$ and $t_{\kappa -1}-t^{\prime }-\frac{1}{c}|x_{\kappa -1}-x^{\prime }|>0$ and represents a parallelogram. To calculate the integrals in Eq. (3.71), it is convenient to use the new variables $u_i=\frac{1}{\sqrt{2}}\left(c{t}_i+x_i\right)$ and $w_i=\frac{1}{\sqrt{2}}\left(c{t}_i-x_i\right)$, in which the regions of integration are transforming into rectangles D_i (Fig. 3.8b).

Figure 3.8 An example of regions for integration in (3.71).

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