# Electromagnetic Field Variation of ELF Near-Region Excited by HED in a Homogeneous Half-Space Model
## 3 Analytical Approximate Expression of the Field Component
### Bessel Integral Solution  
The integral expressions of each field component from Equation (5)-(10) can be reduced to infinite integrals containing Bessel functions. The expressions of direct and reflected waves can be obtained by strictly solving the corresponding integral. According to the general solution of the integral formula given by [PERSON], the analytical expressions of the integrals \(F_{1}\) (\(k_{1}\), \(d_{0}\))-\(F_{5}\) (\(k_{1}\), \(d_{0}\)), \(G_{1}\) (\(k_{1}\), \(d_{1}\))-\(G_{5}\) (\(k_{1}\), \(d_{1}\)) can be obtained [8], and we have given the detailed solution in the Appendix A. Therefore, we only need to find the solution of the integral formulas \(H_{1}\) (\(d_{1}\))-\(H_{5}\) (\(d_{1}\)) to obtain the final expression of the electromagnetic field. As the integral formulas \(H_{1}\) (\(d_{1}\))-\(H_{5}\) (\(d_{1}\)) of the side wave component contain a slow decay oscillation function, which cannot be directly obtained analytically, we use the approximate method for asymptotic estimation. \(H_{1}\) (\(d_{1}\))-\(H_{5}\) (\(d_{1}\)) are expressed as follows [13]:  
\[H_{1}(d_{1})=\int_{0}^{\infty}\bigg{\{}\frac{\gamma_{2}-\gamma_{1}}{\gamma_{2}+ \gamma_{1}}[J_{0}(\lambda\rho)+J_{2}(\lambda\rho)]\bigg{\}}\gamma_{1}^{-1}e^{j \gamma_{1}d_{1}}\lambda d\lambda \tag{13}\]  
\[H_{2}(d_{1})=\int_{0}^{\infty}\bigg{\{}\frac{\gamma_{2}-\gamma_{1}}{\gamma_{2}+ \gamma_{1}}[J_{0}(\lambda\rho)-J_{2}(\lambda\rho)]\bigg{\}}\gamma_{1}^{-1}e^{ j\gamma_{1}d_{1}}\lambda d\lambda \tag{14}\]\[H_{3}(d_{1})=\int_{0}^{\infty}\biggl{\{}\frac{\gamma_{2}-\gamma_{1}}{\gamma_{2}+ \gamma_{1}}[J_{0}(\lambda\rho)-J_{2}(\lambda\rho)]\biggr{\}}e^{j\gamma_{1}(z+d_ {1})}\lambda d\lambda \tag{15}\]  
\[H_{4}(d_{1})=\int_{0}^{\infty}\biggl{\{}\frac{\gamma_{2}-\gamma_{1}}{\gamma_{2}+ \gamma_{1}}[J_{0}(\lambda\rho)+J_{2}(\lambda\rho)]\biggr{\}}e^{j\gamma_{1}(z+d _{1})}\lambda d\lambda \tag{16}\]  
\[H_{5}(d_{1})=\int_{0}^{\infty}\frac{\gamma_{2}-\gamma_{1}}{\gamma_{2}+\gamma_{ 1}}I_{1}(\lambda\rho)e^{j\gamma_{1}d_{1}}\gamma_{1}^{-1}\lambda^{2}d\lambda \tag{17}\]  
where \(J_{n}\) (\(\lambda\rho\)), \(n=0,1,2\), is the expression of the \(n\)-order Bessel function. In Equations (13)-(17), the integrand function contains the multivalued function term A(\(\lambda\)) = \((\gamma_{2}-\gamma_{1})/(\gamma_{2}+\gamma_{1})\). As it is known from the form text that the intermediate variable \(\gamma_{1}\) can be expressed by \(\gamma_{1}\approx j\lambda\), the absolute values for \(\gamma_{1}\), \(\gamma_{2}\), and \(k_{1}\) are approximately defined by a triangular relationship, as shown in Figure 3d. According to the quasistatic approximation (2),  
\[0<|\gamma_{2}-\gamma_{1}|\approx\biggl{|}-j\lambda-\sqrt{k_{1}^{2}-\lambda^{2 }}\biggr{|}<\sqrt{\lambda^{2}+k_{1}^{2}-\lambda^{2}}=|k_{1}| \tag{18}\]  
Based on Equation (18), the absolute value of vector \(\gamma_{2}-\gamma_{1}\) is proportional to the absolute value of vector \(k_{1}\), which can be deduced as follows:  
\[|\gamma_{2}-\gamma_{1}-k_{1}|d_{1}\approx\biggl{|}j\lambda-\sqrt{k_{1}^{2}- \lambda^{2}-k_{1}}\biggr{|}d_{1}\approx\xi|k_{1}d_{1}|<<1 \tag{19}\]  
where \(\mid\xi\mid\leq\)1.